State-Space Modeling of Electrochemical Processes. Michel Prestat

Transcription

1 State-Space Modeling of Electrochemical Processes Who uses up my battery power? Michel Prestat ETH-Zürich Institute for Nonmetallic Materials Head: Prof. L.J. Gauckler

2 Outline Electrochemistry Electrochemical Impedance Spectroscopy and State-Space Modeling Oxygen reduction at solid oxide fuel cell cathodes Comparison modeling experiments Summary

3 corrosion Electrochemistry electropolishing coloration of titanium batteries, fuel cells

4 Solid Oxide Fuel Cell (SOFC) single cell microstructure reduction Sulzer Hexis SOFC C e - cathode ½ + 2e ½ + 2e- O2- mechanism? rate-limiting steps? electrolyte - fuel oxidation - e - anode H H 2 O + 2e - H 2 H 2 O H H 2 O + 2e - H 2 H 2 O overall: H 2 + ½ H 2 O mechanism? rate-limiting steps?

5 reduction Electrochemistry and Materials cathode + 4e - 2- electrolyte - - mechanism? rate-limiting steps? Overpotential η = E - E eq Current (A) ideal electrode: no kinetic losses La 0.85 Sr 0.15 MnO 3 (LSM) real electrode Au ~ electrokinetic losses η strongly dependant on the material Aim: find the appropriate material to get high current and low overpotential (reduction of electrokinetic losses) E eq η LSM η < η Au Potential (V) What limits the performance of my system?

6 Electrochemical Impedance Spectroscopy (EIS)

7 Current (A) EIS principle ~ I steady-state operating point Small amplitude (5-10 mv) input signal Linearization ~ 1 I = Z(jω) E ~ E ~ Potential (V) Admittance Transfer Function Z = impedance complex ( j 2 = -1) frequency dependant ( ω = 2π f )

8 Im (Z) EIS spectra and equivalent circuits Re (Z) R -6 experimental EIS spectra f (Hz) Im (Z) f = 1 2π RC C Im (Z) /Ω R Re (Z) R Re (Z) / Ω Im (Z) C 2 Re (Z) R 1 R 2 Im (Z) C 2 C 3 How to interpret the experimental equivalent circuit?? R 1 R 2 +R 3 Re (Z) R 1 R 2 R 3

9 Alternative approch: modeling the impedance new model reaction model -, ads - state-space modeling experimental impedance faradaic impedance Im (Z) Im (Z) Re (Z) Re (Z) validation of the model assessment of kinetics

10 State-Space Modeling (SSM)

11 State-Space Model electrochemical system E (input) electrode potential K ads K b (g) - K des K f K = model parameters (K ads, K des, K f, K b ) θ = state variable ( concentration) I F (output) faradaic current State-Space Model dθ dt = f (θ, E, K) state equation I F = g (θ, E, K) output equation

12 State-Space Modeling time domain frequency domain state-space model* linearization Laplace transform varying ω. θ = K ads (1- θ) I F = -K f θ e -fe +. θ = Aθ +B E I F = Cθ +D E Z F (jω) Im(Z F ) * * * * * * * * * Re(Z F ) I E. θ = 0 steady-state analysis Simulink : easy implementation of the model. Matlab : state-space calculations and computing

13 Oxygen Reduction at Solid Oxide Fuel Cell Cathodes

14 Dissociative adsorption: (g) + 2s 2 Reaction Models Surface diffusion: (tpb) Charge transfer at the tpb: + 2e - - electrode s = adsorption site Electrolyte = -.. x conductor (V o and O o ) Typically YSZ (Y 2 O 3 -Zr ) x - ( O o ) electrolyte x.. p O2, [O o ] and [V o ] are constant triple phase boundary (tpb) Model 1: surface diffusion negligible Model 2: with surface diffusion

15 Dissociative adsorption: Model 1 (without surf. diffusion) K ads Charge transfer: (g) + 2s 2 K des electrode K f (E) + 2e - - K b (E) - x ( O o ) θ ads electrolyte consecutive reaction steps state variable θ ads = scalar state-space model dθ ads dt = K ads p O2 (1-θ ads ) 2 K des θ ads2 K f (E) θ ads + K b (E) (1-θ ads ) I F = K i [ K f (E) θ ads + K b (E) (1-θ ads )]

16 Model Implementation in Simulink dθ ads dt = K ads p O2 (1-θ ads ) 2 K des θ ads2 K f (E) θ ads + K b (E) (1-θ ads ) I F = K i [ K f (E) θ ads + K b (E) (1-θ ads )] K ads p O2 u 2 K des u θ ads θ ads θ ads integrator function K f (E) E input k f exp(-2 β f E) k b exp(2(1-β) f E) x x - + K i I F output function K b (E)

17 K ads Model 2 (with surf. diffusion) (g) + 2s 2 K des K dif reservoir θ eq K f (E) + 2e - - K b (E) θ 3 diffusion layer Diffusion processes 2 nd Fick s law: θ t = 2 z θ 2 θ 2 Finite difference approach to estimate time and space derivatives - x ( O o ) θ 1 tpb electrolyte state variable θ (θ 1, θ 2, θ 3 ) = vector θ 1 θ 2 θ 3 Parallel reaction pathways

18 Subsystems: as many as compartments Model Implementation in Simulink θi 2 2 Kdif 2θ i-1 θi+ 1 = KadspO ( 1 θ ) ( i ) 2 i Kdesθi + θ + ( + chg transfer kinetics) 2i-2 t in-port out-port θ 1 θ 2 θ 3 (3) θ 3 θ 2 (2) θ 2 θ 1 E input tpb (1) I F output

19 Compartment n 2 Model Implementation in Simulink θ2 2 2 Kdif 2θ1 θ3 = KadspO ( 1 θ ) ( ) 2 2 Kdesθ2 + θ2 + t K ads p O2 u 2 K des u 2 1 θ 1 θ 3 in-ports 2/3 1/ K dif / s Integrator (θ 2 ) θ out-port to compartment (1) and (3)

20 -0.6 Modeling results Im(ZF) / Ω R 1 electrode charge transfer (model 1 = diffusion slow) Re(ZF) / Ω - electrolyte -0.6 C 2 Im(ZF) / Ω R 1 R 2 electrode adsorption - charge transfer (model 1 = diffusion slow) Re(ZF) / Ω - electrolyte -0.6 Im(ZF) / Ω Re(ZF) / Ω - electrode electrolyte diffusion - charge transfer (model 2) -0.6 Im(ZF) / Ω electrode ads. diff. - charge transfer (model 2) Re(ZF) / Ω - electrolyte

21 Comparison Modeling - Experiments

22 Electrode / electrolyte interfaces porous ~10-30 µm industrial application dense ~100 nm -1 µm microstructured ~20 µm -100 µm control of the electrode dimensions top view

23 potentiostat (DC) frequency response analyzer (AC) reference electrode working electrode counter electrode E+E ~ I+I ~ Experimental C DL C F Im (Z TOT ) / Ω C DL high C DL moderate C DL =0 (Z F ) Z TOT = R Ω R 1 R Z F Re (Z TOT ) / Ω C DL maymaskz F Berthier s method in Corrosion 51 (1995) 105

24 -1.5 La 0.85 Sr 0.15 MnO 800 C η= -270mV, I =150 ma.cm -2 Comparison modeling - experiments Im(ZEXP) / Ω K dif << K ads, K f Re(Z EXP ) / Ω slow surface diffusion = Model 1 -(R Ω, C DL ) -1.5 La 0.85 Sr 0.15 MnO 800 C η= -270mV, 150 ma.cm -2 Im(Z F ) / Ω electrode Re(Z F ) / Ω - electrolyte mixed control adsorption - charge transfer

25 Ongoing SOFC projects Mixed ionic-electronic electrodes Modeling the whole SOFC - electrode ½ + 2e - - cathode electrolyte electrolyte anode H H 2 O + 2e - Additional reaction pathway through the electrode bulk. Oxygen reduction, fuel oxidation, transport in the electrolyte. Intermediate T SOFC ( C). Typical material: La x Sr 1-x Co y Fe 1-y O 3 (LSCF).

26 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.

27 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 cathodes Part 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 cathodes Part Ii: surface diffusion limited reactions A. Bieberle and L.J. Gauckler, Solid State Ionics, 146 (2002) 23 State-Space Modeling of the anodic SOFC system Ni, H 2 -H 2 O YSZ A. Mitterdorfer and L.J. Gauckler, Solid State Ionics, 117 (1999) 187 Identification of the reaction mechanism of the Pt, (g) Yttria-Stabilized Zirconia system, Part I: general framework, modelling and structural investigation A. Mitterdorfer and L.J. Gauckler, Solid State Ionics, 117 (1999) 203 Identification of the reaction mechanism of the Pt, (g) Yttria-Stabilized Zirconia system, Part II: model implementation, parameter estimation and model validation