PEM Fuel Cell Using Equation-based Modelling

John Blackburn, Neil McCartney

Materials Division, NPL

COMSOL Conference Grenoble October 2015

Channels of inter-digitated fuel cell

Anode side

Anode side

H2

H2O

Bipolar plateBipolar plate

GDL GDL

Anode Cathode

Membrane

O2, N2

H2O

H2

H2OO2, N2

H2O

3D diagram of

fuel cell model

4

2

1

1

H k k k k H H

k

pD x x R

p

u

p F

0ele ele eleV Q

. k kk k mem k

F z cD c V S

RT

Poisson’s equation

Darcy’s Law

Maxwell-Stephan equation

Nernst-Planck equation

Equations to solve

Complex Source terms

Wednesday, 21 October 2015

5

2 21 ano ano

H H H H H H O H OR m S m S

2

3coth 1 0ano agg aggano

H H H H H

ano

fS D c

r

22 exp exp

2

ref ano ano

ano ano ano ox ano red anoH agg ref

H H

i s r F F

F D c RT RT

0eq

ano ele mem anoV V V

Non-linear source terms (agglomerate model)

COMSOL Module based FE setup

Coupling dangerous

Cannot generalise PDEs

Slightly inconsistent approach

Specifying a finite element problem

• Pick from a list of predefined systems

– Good for standard systems like electrostatic, elastic, basic

heat flow problems

– Solver optimised for particular physics system

– User shielded from mathematics details

– Not so good for bespoke problems or complex multiphysics

systems and non-linearity.

• Enter equations yourself

– Good for unusual, bespoke equation sets. Can have

multiphysics and/or non-linearity

– General purpose solver: convergence not guaranteed

– Good if you like mathematics!

– Total control over system you are solving

– Traceability back to fundamental physics

McCartney’s “Theory of Everything”

k

n

k

k

n

i

iikkk

n

k

n

i

ii

n

i

ikkkk

n

k

kk

kkk

n

i

iikkkk

mm

mmxTsvvt

t

t

nkmmt

bvJJ

vσJIΣ

JJevv

bσvvv

v

vJJ

1

1

1

1

1

1

01

1

22

1

1

1

ˆˆ

ˆ)(ˆˆ2

1

2

1

)(

0)(

,,2,1ˆˆ

Mass conservation

(n or n-1 issue…)

(chemical potentials assume ideal gas)

Momentum conservation

(Darcy law may be inadequate => Navier Stokes)

Energy conservation

(ignored!)

Our current simulation comes from simplified mass and

momentum conservation laws. Energy balance not even

considered yet and this is the most complex eq’n.

Unlikely any packaged FE does justice to

this theory.

In my opinion…

• Either go for fully packaged system (then

you are restricted to basic problems) or set

up system completely by hand.

• Comsol allows PDEs to be input manually in

three different ways

– Coefficient mode: specify coefficients of

generalised PDE.

– Weak mode: use variational principles. Not PDE.

(very general purpose but hard to use)

– General form. Note sure: coefficient mode on

steroids…

Coefficient mode

• Some terms omitted. C-matrix is most

important, corresponds to diffusion

(permittivity, stiffness etc)

• f-matrix encodes source terms.

• Pattern in Cijkl tensor describes simulation.

Only difference between quantum transistor

and fuel cell is this tensor

• Like DNA of simulation.

Nit

ufua

x

uC

x

iii

j

ij

jkl l

kijkl

j

,2,1

C-matrix for fuel cell problem

OHOH

OHOHmemOH

OHmemmemmem

OHOHOOHOH

OHOOOO

HHH

eleele

cc

ccVc

cVVV

xxxxpx

xxxxpx

xxpx

pp

VV

c

cc

cc

ccc

ccc

cc

c

c

c

22

333

3

2222

2

,

,,

,,

,,,

,,,

,,

,

,

0000000

000000

000000

00000

00000

000000

0000000

0000000

(2,3,4)

(2,3,4)

(2,3,4)

(2,3,4) I

(2,3,4)

(4,5)

(4,5)

(4,5) )()(

(4,5)

(4,5)

(4,5) )()(

(1,2)

(1,2) )(

5) 4, 2, (1,

5) 4, 2, (1,

2,

33

3,

3,

333,

3,22,22,

3,21,22,

3,22,2223,21,22

,

3,12,1,

3,11,1,

3,12,1223,11,1,

2,11,1,

2,11,1,

,

,

22

,3

33

3

22

2

2

2

IDc

czFuc

Iz

zDc

Dc

IczFuc

IDDc

IDDc

IDDxDDxp

c

IDDc

IDDc

IDDxDDxp

c

IDDc

IDDxp

c

Ic

Ic

OHcc

OHOHMVc

M

OHMcc

OHcV

OHOHOHVV

OHxx

OHxx

OHOHOOOH

px

Oxx

Oxx

OHOHOOO

px

Hxx

HHH

px

pp

eleVV

OHOH

memOH

OHOH

OHmem

memmem

OHOH

OOH

OH

OHO

OO

O

HH

H

eleele

OHOH

OHOHmemOH

OHmemmemmem

OHOHOOHOH

OHOOOO

HHH

eleele

cc

ccVc

cVVV

xxxxpx

xxxxpx

xxpx

pp

VV

c

cc

cc

ccc

ccc

cc

c

c

c

22

333

3

2222

2

,

,,

,,

,,,

,,,

,,

,

,

0000000

000000

000000

00000

00000

000000

0000000

0000000

OH

OH

mem

OH

O

H

ele

c

c

V

x

x

x

p

V

2

3

2

fc u

sourceVeleele )(

sourcep )/(

HH

HHHHH

R

xDDp

pDDx

u

])())(([ 2,11,12,11,1

Nernst Planck in coefficient mode

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Region dependent variables

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14

Oxygen mass fraction variation

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15

Built-in module vs. PDE coefficient mode

• Good agreement after some effort

• Here V=0.7 V,

• I=0.3716 A/m (coeff), I=0.406 A/m original

• Careful! E-field not defined on boundary

Wednesday, 21 October 2015

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5Ce

ll P

ote

nti

al (

V)

Current density (A/cm2)

Factor = 1.0

Factor = 0.6

Factor = 0.2

Factor = 0.1

Effect of varying oxygen mass fraction at inlet

Wednesday, 21 October 2015

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Experimental polarisation curves

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0C

ell

Po

ten

tia

l (V

)

Current density (A/cm2)

Unmodified cell

RE array on cathode

RE array on anode

No perturbation

at low and intermediate

current densities

Some perturbation

in mass transport limited

regime due to impregnated

Nafion in GDL

Nafion impregnation procedure can be further optimised but in

any case degradation studies are focused on high potentials

The tip of the iceberg…

• Now extended to 3D (previously 2D model)

• Added additional fluid layers outside of the

GDL layers for channel flow

• Replaced Darcy with more realistic Navier

Stokes

• Unfortunately the last does not converge

yet. Equation based modelling allows any

equations to be specified but no guarantee

you can solve them!

• Work is ongoing.

Wednesday, 21 October 2015

19

Summary

• Derived PDEs for fuel cells which are more

advanced than in normal literature

• Solved these equations using Comsol’s

coefficient mode (equation based modelling)

• Good initial agreement with experimental

results (but theory very simple cf experiment)

• Eq’n based modelling is better for complex,

multiphysics, non-linear systems. Allows new

equations to be solved without waiting for

built-in module

• Gives better traceabilty.

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20