Weierstrass Institute for Applied Analysis and Stochastics Electrochemical processes and porous media: mathematical and numerical modeling Jürgen Fuhrmann Alfonso Caiazzo, Klaus Gärtner, Hartmut Langmach, Alexander Linke, Hong Zhao Mohrenstrasse 39 · 10117 Berlin · Germany · Tel. +49 30 20372 0 · www.wias-berlin.de RICAM Workshop · Linz · 2011-10-05
Electrochemistry � Corrosion � Electroplating � Biological processes (heart beat etc.) � Batteries � Conversion of chemical energy stored in compounds within the cell into electrical 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 · Page 2 (46)
Polymer electrolyte fuel cell - membrane electrode assembly (MEA) Void space Polymer electrolyte Catalyst particles Porous matrix Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 3 (46)
Polymer electrolyte fuel cell - membrane electrode assembly (MEA) Void space Anode channel Polymer electrolyte Porous layer Carbon fiber + teflon Reaction zone Carbon fiber + teflon + nafion + catalyst Nafion (proton conducting polymer) Membrane Carbon fiber + teflon + nafion + catalyst Reaction zone Porous layer Carbon fiber + teflon Catalyst particles Cathode channel Porous matrix Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 3 (46)
Polymer electrolyte fuel cell - membrane electrode assembly (MEA) Hydrogen fuel cell ( H 2 -PEMFC) Void space H 2 Anode channel Polymer electrolyte Porous layer 2 H 2 → 4 H + + 4 e − Reaction zone Membrane Reaction zone Porous layer Catalyst particles Cathode channel Porous matrix Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 3 (46)
Polymer electrolyte fuel cell - membrane electrode assembly (MEA) Hydrogen fuel cell ( H 2 -PEMFC) Void space H 2 Anode channel Polymer electrolyte Porous layer e − 2 H 2 → 4 H + + 4 e − Reaction zone – H + Membrane Load Reaction zone + e − Porous layer Catalyst particles Cathode channel Porous matrix Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 3 (46)
Polymer electrolyte fuel cell - membrane electrode assembly (MEA) Hydrogen fuel cell ( H 2 -PEMFC) Void space H 2 Anode channel Polymer electrolyte Porous layer e − 2 H 2 → 4 H + + 4 e − Reaction zone – H + Membrane Load Reaction zone + e − Porous layer Catalyst particles Cathode channel O 2 , N 2 Porous matrix Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 3 (46)
Polymer electrolyte fuel cell - membrane electrode assembly (MEA) Hydrogen fuel cell ( H 2 -PEMFC) Void space H 2 Anode channel Polymer electrolyte Porous layer e − 2 H 2 → 4 H + + 4 e − Reaction zone – H + Membrane Load 4 H + + 4 e − + O 2 → 2 H 2 O Reaction zone + e − Porous layer Catalyst particles Cathode channel O 2 , N 2 Porous matrix Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 3 (46)
Polymer electrolyte fuel cell - membrane electrode assembly (MEA) Hydrogen fuel cell ( H 2 -PEMFC) Void space H 2 Anode channel Polymer electrolyte Porous layer e − 2 H 2 → 4 H + + 4 e − Reaction zone – H + Membrane Load 4 H + + 4 e − + O 2 → 2 H 2 O Reaction zone + e − Porous layer Catalyst particles Cathode channel O 2 , N 2 H 2 O , N 2 Porous matrix Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 3 (46)
Polymer electrolyte fuel cell - membrane electrode assembly (MEA) Hydrogen fuel cell ( H 2 -PEMFC) Void space H 2 Anode channel Polymer electrolyte Porous layer e − 2 H 2 → 4 H + + 4 e − Reaction zone – H + Membrane Load 4 H + + 4 e − + O 2 → 2 H 2 O Reaction zone + e − Porous layer Catalyst particles Cathode channel O 2 , N 2 H 2 O , N 2 Porous matrix Overall reaction: 4 H 2 + O 2 → 2 H 2 O + energy Several MEA “sandwiches” combined into a stack. Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 3 (46)
Polymer electrolyte fuel cell - membrane electrode assembly (MEA) Direct methanol fuel cell (DMFC) Void space CH 3 OH , H 2 O CO 2 Anode channel Polymer electrolyte Porous layer e − 2 CH 3 OH + 2 H 2 O → 2 CO 2 + 12 H + + 12 e − Reaction zone – H + Membrane Load 12 H + + 12 e − + 3 O 2 → 6 H 2 O Reaction zone + e − Porous layer Catalyst particles Cathode channel O 2 , N 2 H 2 O , N 2 Porous matrix Overall reaction: 2 CH 3 OH + 3 O 2 → 4 H 2 O + 2 CO 2 + energy Several MEA “sandwiches” combined into a stack. Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 3 (46)
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 reactions General case: ⇒ numerical modeling J. Newman, K. Thomas-Aleya (2004):Electrochemical systems; A. Kulikovsky (2010): Analytical Fuel Cell Modeling Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 4 (46)
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 Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 5 (46)
Charge transport: Nernst Planck Poisson system � Transport of i -th dissolved species ( i = 1 ... n ) due to diffusion, electromigration, advection in Variables dilute solution ⇒ Nernst-Planck equation : n number of species electromigration diffusion advection φ electrostatic potential � �� � � �� � ���� � N i = − z i u i Fc i ∇ φ − D i ∇ c i + c i � v c i species concentration ∂ t c i + ∇ · � � N i = j i N i molar flux z i charge u i mobility � Distribution of charged species D i diffusion coefficient ⇒ self-consistent electric field ε electrostatic permeability ⇒ Poisson equation : F Faraday constant j i Reaction n ∑ − ∇ · ε ∇ φ = F z i c i � v Substrate velocity i = 1 Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 6 (46)
Assumptions behind Nernst-Planck Dilute solution theory: � Ignore interactions (collisions) between different dissolved species c i . 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 changes Otherwise: variable density flow Special case: semiconductor device equations. Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 7 (46)
Bulk electroneutrality � ε << F ⇒ electroneutrality in bulk (away from interfaces and boundaries) n ∑ z i c i = 0 i = 1 � Sum up equations multiplied by z i : � � � � n n n n n ∑ ∑ z 2 ∑ ∑ ∑ ∂ t − ∇ · i u i Fc i ∇ φ + z i D i ∇ c i + � = z i c i v z i c i z i j i i = 1 i = 1 i = 1 i = 1 i = 1 � Express e.g. c 1 : n ∑ z 1 D 1 c 1 = − D 1 z i c i i = 2 � No bulk reactions ⇒ � � n n Fz 2 ∑ ∑ ∇ · i u i c i ∇ φ + z i ( D i − D 1 ) ∇ c i = 0 i = 1 i = 2 � Equal diffusion coefficients or small concentration gradients ⇒ Ohm’s law : � � n ∑ z 2 ∇ · κ ∇ φ = 0 κ = F i u i c i : conductivity i = 1 Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 8 (46)
Nernst Planck Ohm system Variables � Transport of i -th dissolved species ( i = 2 ... n ) due to diffusion, electromigration, advection in n number of species dilute solution φ electrostatic potential ⇒ Nernst-Planck equation : c i species concentration electromigration diffusion advection � N i molar flux � �� � � �� � ���� � N i = − z i u i Fc i ∇ φ − D i ∇ c i + c i � v z i charge ∂ t c i + ∇ · � N i = j i u i mobility D i diffusion coefficient κ conductivity � Self-consistent electric field from Ohm’s Law : F Faraday constant j i Reaction ∇ · κ ∇ φ = 0 � v Fluid velocity Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 9 (46)
Special case: solid electrolyte/electrode c 1 : mobile charge carriers c 2 : immobile charges in solid lattice ( u 2 = D 2 = 0 ) � v = 0 ⇒ c 2 = const Electroneutrality , z 1 = − z 2 ⇒ c 1 = c 2 small ∇ c 2 ⇒ κ = z 2 1 u 1 Fc 1 � Electrodes (graphite, metal): c 1 ↔ free e − � Polymer electrolytes in fuel cells: c 1 ↔ free H + . Electrochemistry and porous media · RICAM Workshop · Linz · 2011-10-05 · Page 10 (46)
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