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PEM Fuel Cell Using Equation-based Modelling
John Blackburn, Neil McCartney
Materials Division, NPL
COMSOL Conference Grenoble October 2015
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
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
18
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.
Wednesday, 21 October 2015
20