State-Space Modeling of Electrochemical Processes · ETH - Ceramics State-Space Modeling of...

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State-Space Modeling of Electrochemical Processes

„Who uses up my battery power ?“

Michel Prestat

ETH-ZürichInstitute for Nonmetallic Materials

Head: Prof. L.J. Gauckler


Outline

Electrochemistry

Electrochemical Impedance Spectroscopy and State-Space Modeling

Oxygen reduction at solid oxide fuel cell cathodes

Comparison modeling – experiments

Summary


Electrochemistrycorrosion

electropolishing

coloration of titanium

batteries, fuel cells


Solid Oxide Fuel Cell (SOFC)

cathode

O2-

e-

e-

H2OH2

O2 O2

anode

electrolyte

O2 reduction

O2

O2-

½O2 + 2e- O2-

mechanism?rate-limiting steps?

fuel oxidation

O2-

H2 + O2- H2O + 2e-


H2OH2

single cell

Sulzer Hexis SOFCmicrostructure

½O2 + 2e- O2-

H2 + O2- H2O + 2e-

overall: H2 + ½O2 → H2O

800-1000°C


Electrochemistry and Materials

cathode

O2-

O2 O2

electrolyte

O2 reduction

O2

O2-

O2 + 4e- 2O2-


Overpotential η = E - Eeq

~ electrokinetic losses

Potential (V)

Current (A)

Eeq

La0.85Sr0.15MnO3 (LSM)

Au

ideal electrode:no kinetic losses

ηηLSM < ηAu

real electrode

η strongly dependant on the material

Aim:find the appropriate material to gethigh current and low overpotential(reduction of electrokinetic losses)

What limits the performanceof my system ?


Electrochemical Impedance Spectroscopy (EIS)


EIS principle

Potential (V)

Current (A)

E~

I~

steady-stateoperating point

Small amplitude (5-10 mV) input signal→ Linearization

I =Z(jω)

1E~~

Admittance Transfer Function

Z = impedance

complex ( j2 = -1)

frequency dependant ( ω = 2π f )


EIS spectra and equivalent circuits

Re (Z)

Im (Z)

R-6

-4

-2

00 2 4 6 8 10 12

Re (Z) / Ω

Im (Z

) /Ω

110

102103

104

f (Hz)

experimental EIS spectra

Re (Z)

Im (Z)

R

Cf = 12π RC

R

Re (Z)

Im (Z)

R2

C2

R1

How to interpretthe experimental equivalent circuit ??

Re (Z)

Im (Z)

R2

C2

R1 R3

C3

R1 R2+R3


Alternative approch: modeling the impedance

Re (Z)

Im (Z)

Re (Z)

Im (Z)

experimental impedance

reaction modelnew model

O2 Oads O2-O2 O2,ads O2-

state-space modeling

faradaic impedance

validation of the modelassessment of kinetics


State-Space Modeling (SSM)


State-Space Model

electrode potential

E (input)

faradaic current

IF (output)O2(g) Oads O2-

electrochemical system

Kb

KfKdes

Kads

K = model parameters (Kads, Kdes, Kf , Kb …)

θ = state variable (Oads concentration)

= f (θ , E, K) → state equation

IF = g (θ , E, K) → output equation

dθdtState-Space Model


State-Space Modeling

time domain frequency domain

I

E

θ = Aθ +B E

IF = Cθ +D E

.

linearization

θ = 0.

steady-state analysis

ZF(jω)

Laplace transform varying ω

Re(ZF)*

* * * ** **

**

*Im(Z

F)θ = Kads(1- θ)2 +...

IF = -Kf θ e-fE + …

.

state-space model*

Simulink®: easy implementation of the model.

Matlab®: state-space calculations and computing


Oxygen Reduction

at

Solid Oxide Fuel Cell Cathodes


Reaction Models

Oads Oads (tpb)Surface diffusion:

Oads + 2e- O2-Charge transfer at the tpb:

O2(g) + 2s 2OadsDissociative adsorption:

s = adsorption site

electrolytexO2- ( Oo )

electrode

Oads

O2

Oads

O2

pO2 , [Oo] and [Vo] are constantx ..

Electrolyte = O2- conductor (Vo and Oo).. x

Typically YSZ (Y2O3 - ZrO2)

triple phase boundary (tpb)

Model 1: surface diffusion negligible

Model 2: with surface diffusion


Model 1 (without surf. diffusion)Dissociative adsorption:

O2(g) + 2s 2Oads

Kads

Kdes

O2- ( Oo )x

Oads

O2

electrolyte

electrode

θads

Charge transfer:

Oads + 2e- O2-

Kf(E)

Kb(E)

→ consecutive reaction steps

→ state variable θads = scalar

Kads pO2(1-θads) 2 – Kdesθads2 – Kf(E) θads + Kb(E) (1-θads)

IF = Ki [– Kf(E) θads + Kb(E) (1-θads)]

dθads

dt =→ state-space model


Model Implementation in Simulink®

Kads pO2(1-θads) 2 – Kdesθads2 – Kf(E) θads + Kb(E) (1-θads)

IF = Ki [– Kf(E) θads + Kb(E) (1-θads)]

dθads

dt =

+-

1

-++

Kads pO2

u2Kdes

x

x

-+

integrator

Einput

Ki IFoutput

kf exp(-2 β f E)

kb exp (2 (1-β) f E)

function Kf (E)

function Kb (E)

1-θadsθads

θads

u2


Model 2 (with surf. diffusion)

O2(g) + 2s 2Oads

Kads

Kdes

tpb

Oads reservoirθeq

electrolyte

diffusionlayer

θ1

θ2

θ3

O2- ( Oo )x

OadsO2

OadsO2

OadsO2

OadsO2

Oads Oads

Kdif

Oads + 2e- O2-

Kf(E)

Kb(E)

2

2θ θt

∂ ∂=

∂ ∂zDiffusion processes

2nd Fick‘s law:

→ Finite difference approach toestimate time and space derivatives

→ state variable θ (θ1, θ2, θ3) = vectorθ1 ≤ θ2 ≤ θ3

→ Parallel reaction pathways


Model Implementation in Simulink®Subsystems: as many as compartments

i( ) ( ) 2

i-12 2 dif i 1iads O i des i 2i-2

Kθ 2K 1 θ K θ θ ( + chg transfer kinetics)

t 3 32θθ +∂

= − − + − +∂

p

Einput

IFoutput

θ1

θ1

θ3

θ2θ2

θ2 θ3

tpb (1)

(2)(3)

in-port out-port


Model Implementation in Simulink®

Compartment n°2

( ) ( )2

2 2 dif 32 1ads O 2 des 2 2

K θθ 2θK 1 θ K θ θt 4 3 3

∂= − − + − +

∂p

+-

1

u2

-++

Kads pO2

u2Kdes

1s

Integrator (θ2)

θ2out-port to

compartment (1) and (3)

Kdif/4

-

+

+

1/3

2/3θ1

θ3

in-ports


Modeling results

Oads

Oads

O2

O2

O2-

electrode

electrolyte

ads. – diff. - charge transfer(model 2)

-0.6

-0.4

-0.2

02 2.5 3 3.5

Re(ZF) / Ω

Im(Z

F ) /

Ω

Oads

Oads

O2

O2

O2-

electrode

electrolyte

diffusion - charge transfer(model 2)

-0.6

-0.4

-0.2

02 2.5 3 3.5

Re(ZF) / Ω

Im(Z

F ) /

Ω

45°

Oads

Oads

O2

O2

O2-

electrode

electrolyte

adsorption - charge transfer(model 1 = diffusion slow)

-0.6

-0.4

-0.2

02 2.5 3 3.5

Re(ZF) / Ω

Im(Z

F ) /

Ω

R2

C2

R1

Oads

Oads

O2

O2

O2-

electrode

electrolyte

charge transfer(model 1 = diffusion slow)

-0.6

-0.4

-0.2

02 2.5 3 3.5

Re(ZF) / Ω

Im(Z

F ) /

ΩR1


Comparison Modeling - Experiments


Electrode / electrolyte interfaces

porous

industrial application~10-30 µm

dense

~100 nm -1 µm

control of theelectrode dimensions

microstructured ~20 µm -100 µm

top view


Experimental

E+E~

I+I~

potentiostat(DC)

workingelectrode

frequencyresponseanalyzer

(AC)counter

electrode

referenceelectrode

ZF

ZTOT =R2

CF

R1RΩ

CDL

0 2 3 4 5Re (ZTOT) / Ω

1

-1

-2

-3

0Im

(ZTO

T) / Ω

CDL high CDL moderate CDL=0(ZF)

CDL may mask ZF → Berthier‘s method in Corrosion 51 (1995) 105


Comparison modeling - experiments

Kdif << Kads, Kf

slow surface diffusion = Model 1

-1.5

-1

-0.5

02 3 4 5

Re(ZEXP) / Ω

Im(Z

EXP)

/ Ω

La0.85Sr0.15MnO3 @ 800°Cη= -270mV, I =150 mA.cm-2

102

103

104

- (RΩ, CDL)

Oads

Oads

O2

O2

O2-

electrode

electrolyte

mixed controladsorption - charge transfer

-1.5

-1

-0.5

00 1 2 3

Re(ZF) / Ω

Im(Z

F) /

Ω

La0.85Sr0.15MnO3 @ 800°Cη= -270mV, 150 mA.cm-2


Ongoing SOFC projects

Mixed ionic-electronic electrodes Modeling the whole SOFC

cathode

O2-

anode

electrolyte

½O2 + 2e- O2-

H2 + O2- H2O + 2e-

electrolyteO2-

Oads

Oads

O2

electrodeO2-

O2-

O2-

Additional reaction pathway throughthe electrode bulk.

Oxygen reduction, fuel oxidation, transport in the electrolyte.

Intermediate T° SOFC (600-800°C).

Typical material: LaxSr1-xCoyFe1-yO3(LSCF).


Summary

• electrochemical reactions yield sophisticated impedance behavior⇒ necessity of a modeling approach (analytical or numerical).

• SSM (with modern computation tools) enables to simulate the faradaic impedance of such reactions.⇒ “fingerprint” of reaction models.

• SSM approach applicable to any other field of electrochemistry.


Many thanks to ...

SOFC group Prof. L.J. Gauckler

References

M. Prestat and L.J. Gauckler, Solid State Ionics, submitted (2003)Faradaic impedance of oxygen reduction at solid oxide fuel cell cathodesPart I: adsorption and charge transfer limited reactions

M. Prestat and L.J. Gauckler, Solid State Ionics, submitted (2003)Faradaic impedance of oxygen reduction at solid oxide fuel cell cathodesPart Ii: surface diffusion limited reactions

A. Bieberle and L.J. Gauckler, Solid State Ionics, 146 (2002) 23State-Space Modeling of the anodic SOFC system Ni, H2-H2O|YSZ

A. Mitterdorfer and L.J. Gauckler, Solid State Ionics, 117 (1999) 187Identification of the reaction mechanism of the Pt, O2 (g)|Yttria-Stabilized Zirconia system,Part I: general framework, modelling and structural investigation

A. Mitterdorfer and L.J. Gauckler, Solid State Ionics, 117 (1999) 203Identification of the reaction mechanism of the Pt, O2 (g)|Yttria-Stabilized Zirconia system,Part II: model implementation, parameter estimation and model validation

State-Space Modeling of Electrochemical Processes · ETH - Ceramics State-Space Modeling of...

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