Porous MediaElectrodes and electrolytes in batteries and fuel cells

Chapters 2, 8, 9 and 10

Microstructure of CL-PEMFC• Carbon: for conduction of electrons and support of the platinum nano-particles ;

• Ionomer: typically Nafion®, for proton transport;

• Platinum: for electrochemical reactions;

• Pore: for transport of reactant and product gases;

Illustration of Microstructure in SOFC

CFL= cathode functional layer

AFL= anode functional layer

YSZ= yttria stabilized zirconia

Ni= nickel

TPB= triple phase boundary

Microstructure of a battery

Transport Phenomena in general porous media

• Fluid flow• Heat transfer• Mass transfer• Phase change• Unsaturated and multi‐phase flow• Solid‐fluid interaction• Non‐equilibrium phenomena• Chemical and electro‐chemical reactions• Ion transport• Current transport

TerminologyVolume averaged velocity, temperature

Fluid pressure

Saturation

Mass fractions

Improved models: Phase velocity and temperature

Parameters arising from averaging

Porosity

Permeability

Tortuosity

Path lengths in a porous microstructure

𝜏 =𝑠

𝐿ε is the porosity of the porous mediumtortuosity τ

The Bruggeman correlation relates the tortuosity and porosity and reads,

𝜏𝐵𝑟𝑢𝑔𝑔𝑒𝑚𝑎𝑛2 = 𝜀1−𝛼

𝜀 =𝑣𝑜𝑖𝑑 𝑣𝑜𝑙𝑢𝑚𝑒

𝑡𝑜𝑡𝑎𝑙 𝑣𝑜𝑙𝑢𝑚𝑒

Parameter estimation

• Governing equations can be solved by FVM, FEM, or related numerical techniques.

• In the context of porous media, determining parameters is more important than solving the mass‐momentum‐energy equations.

• Porosity

• Permeability (absolute, relative)

• Capillary pressure

• Dispersion

• Inhomogeneities and anisotropy

Complexity

• Flow path tortuos

• Geometry is three dimensional and not clearly defined

• Original approaches seek to relate pressure drop and flow rate, adopting a volume‐averaged perspective

• It has led to local volume‐averaging (REV) “continuum approach”

• Averaging results in new model parameters

Effective diffusion coefficients

• The effective diffusion coefficients Deff (for mass, ion, electrons) are related to the bulk diffusion coefficients Dbulk by

𝐷𝑒𝑓𝑓 =𝜀

𝜏2𝐷𝑏𝑢𝑙𝑘

Assumptions

• Representative elementary volume (REV)

• Solid phase rigid and fixed

• Closely packed arrangement

• REV is larger than the pore volume

• Look for solutions at a scale much larger than the REV

• Porous continuum

Dimensions

• Pore scale and particle diameter 1‐10 microns but for FCs and batteries down to nm

• REV 0.1‐1 mm

Overview

• Porous media applications are quite a few.

• Transport equations can be set up.

• Simulation tools of CFD and related areas can be used.

• Number of parameters is large.

• Parameter estimation plays a central role in modeling and points towards need for careful experiments.

• Dependence on parameters can be reduced by carrying out multi‐scale simulations.

Highly Porous Anode for Application in High‐Temperature Electrochemical Devices

NiO-CGO foam after sintering

Foam after cutting to the desired anode size and polishing to achieve a smooth surface upon which the electrolyte could be screen printed

High resolution-SEM (scanning electron microscope) of the porous anode before (A–C) and after (D–F) reduction

Illustration Porosity and Tortuosity

Momentum EquationVelocity field

Porosity = 0.76

Tortuosity = 1.17

𝜀 =𝑣𝑜𝑖𝑑 𝑣𝑜𝑙𝑢𝑚𝑒

𝑡𝑜𝑡𝑎𝑙 𝑣𝑜𝑙𝑢𝑚𝑒

Porous media- SOFC cathode

SOFC cathode

Equations to be considered in the analysis

+ = Si ii

u

t x

Overall Mass Conservation Equation

Equations to be considered in the analysis

Darcy’s law

= ii

K pu

x

Equations to be considered in the analysis

Brinkmann’s equation

2

2 = - + ii e

i i

upu

x K x

Equations to be considered in the analysis

Momentum Equation

2

2[ + u ] = g - + - u i i ij i i

j i j

u u up

t x x x K

The Forchheimer extended Darcy's law is applied at high velocities in the porous media

air air Fi i

i

Cpu u u

x

− = +

There are two major models for the heat transfer of the foam or porous medium

1) The thermal equilibrium model

2) The non-thermal equilibrium model (two-equation model).

( )f pf jeff

j j j

c u T T

x x x

=

Thermal Equilibrium Model

Non-thermal equilibrium model

( )( )

f pf j f f

fe sf sf s f

j j j

c u T Th a T T

x x x

= + − 0 ( )sse sf sf s f

j j

Th a T T

x x

= − −

Fluid domain Solid domain