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Non-Equilibrium Thermodynamics: Foundations and Applications.
Lecture 9: Modelling the polymer electrolyte fuel cell
Signe Kjelstrup
Department of Chemistry, Norwegian University of Science and Technology,
Trondheim, Norway
and
Engineering ThermodynamicsDepartment of Process and Energy, TU Delft
http://www.chem.ntnu.no/nonequilibrium-thermodynamics/
Non-Equilibrium Thermodynamics: Foundations and Applications
Lecture 9. Transport heat, mass and charge in fuel cells
Chapter 19
Lecture aims
Scetch solution procedure for one-dimensional modeland present some results
Present an idea for more energy- and material-efficient design
PEM fuel cell challenges• Too expensive• Lifetime lower than competing technology• Efficiency high, but can be higher
P. Mock, J. Power Sources (2009)
PEMFC Description
Anode Flow Channel
Cathode Flow Channel
•O2
•H2
e
e
H+
•Design must consider
•Catalyst nanoporous layer
•Microporous support layer
•Gas supply system
•Water removal system
• Fuel cell models must consider
•Heat transport
•Electron production
•Proton and water transport
•Gas access
•Water removal
Working procedure for one-dimensional model1. The entropy production of five separate layers2. The fluxes and conjugate forces3. The coefficients from experiments4. Electric work from chemical energy in the cell5. The dissipation of energy; lost work
Limitations in this study:One-dimensional transportsNo pressure gradients
Transport of heat, mass and charge acrossfive parallell layers
Nernst equation is: ΔG = -nFEWhat is the local potential profile?
Is there any temperature profile?
In which contexts are concentration profilesimportant?
What is the entropy production?
Possible questions:
A systematic way to describetransport processes
• 2.law, local formulation
• Onsager’s symmetry relationsmust be obeyed for anysubsystem:
0i ii
J Xs= >å
Example: The PEM fuel cell
1 1 1 1 1 2 2 . . . . . .J L X L X= +
2 2 1 1 2 2 2 . . . . . . . ..J L X L X= +
Three ways to find the lost work:
• The cell potential as function of the current density shows 3 distinct regimes (Eg. Weber and Newman, review, 2004)
1 '
1 '
out ins s
out outq i iout
in inq i iin
dx J J
J J ST
J J ST
s = -
é ùê ú= +ê úë ûé ùê ú- +ê úë û
ò
å
å
1. By integrating over σ2. By calculating Js
in Jsout
3. By measuring the heatproduction
Energy is conserved!
Thermoneutralpotential (squares),
polarisation curve (triangles)
electric power (circles)
as functions of current density in a fuel cell operated at 323 K, 1 atm, with oxygen and hydrogen
Burheim et al. Electrochim. Acta. 55 (2010) 935-942
0.00
0.30
0.60
0.90
1.20
1.50
0.00 0.20 0.40 0.60 0.80 1.00j / Acm-2
E / V
0.00
0.30
0.60
P /
W c
m-2
E TNEcellP
b) Nafion 115
Excess properties of the electrode surfaceaccording to Gibbs
H(s)
HM
H 2
H 2
2
O
backing surface
Pt
membrane
s s sj jU TS dA dN= +g + måGibbs equation:
The entropy production for the cathode surface
• We use: Mass balances, first law, Gibbs equation
• One flux-forcepair for eachvariable
• Reversible conditions giveNernst equationfor the electrode!
' , ' ,, ,
, , , ,
,
1 1
1 1
1
m s s cq m s q s c
m cw m s w T w s c w T
c
m c
J JT T
J JT T
GjT F
æ ö æ ö÷ ÷ç çs= D + D÷ ÷ç ç÷ ÷ç çè ø è øæ ö æ ö÷ ÷ç ç+ - D m + - D m÷ ÷ç ç÷ ÷ç çè ø è ø
é ùæ öD ÷çê ú÷+ - D j+ç ÷ê úç ÷çè øë û
Part of the problem: Thermal osmosis(cf. Lecture 6)
' , '
' ' ,
m a aq q
c m cq q
J J J H
J J J H
- = D
- =- D( )
( )( )
, ' *
' *
, ' *
1`
1
1`
m a a a sqs
m mqm
c m c c sqs
T T J q J
dT J q Jdx
T T J q J
l
l
l
- = - -
= - -
- = - -
, *
, ,, ,
*
, *
, ,, ,
1
m a s
a m T a ms s a m a
mT
m
m c s
m c T m cs s c m c
RT qJ TD c T
d q c dTJdx D T RT dx
RT qJ TD c T
D m =- - D
m=- -
D m =- - D
Energy balanceT-profileConcentrationprofile
Flux-force relations - cathode surface
*,
, , ,,
1mm m
m s w T m s w wm c s
q jT J tT l Fmm
æ ö÷çD m =- D - - ÷ç ÷çè ø
' , *,,
1 m c m m m mm s q w ws
m
j jT J q J tF m
é ùæ ö÷çê úD =- - - -p÷ç ÷çê úè øl ë û
' , *,,
1 c m c c c cs c q w ws
c
j jT J q J tF m
é ùæ ö÷çê úD =- - - -p÷ç ÷çê úè øl ë û
*,
, , ,,
1cc c
s c w T s c w wc m s
q jT J tT l Fmm
æ ö÷çD m =- D - - ÷ç ÷çè ø
, , ,
, , , ,
m cc
m c m s s cm c
m csw w
m c w T s o w T
G T TFT FT
t t r jF F
p pD j+D =- D - D
- D m - D m -
Temperature jumps:
Changes in chemicalpotential of water:
The jump in electricpotential acrossthe surface
Thermal osmosis and electro-osmosis!
Results1:
To do for each of the five subsystems:Solve J = L X for fluxes of heat, water and electric currentBoundary conditions used: Constant j, Jw,, Ta= Tc=370 K
1. S.Kjelstrup and A. Røsjorde, J. Phys. Chem. B, 109 (2005) 9020
Mole fractions of water (left) and oxygen (right)
Results for water concentration profile in the membrane
0 0.2 0.4 0.6 0.8 12
4
6
8
10
12
14
16
Dimensionless position in membrane
Wat
er c
onte
nt,
*,*
, , ,,
1/c
c cs c w T s c w wc m s
q jRT T J tT l Fmm
æ ö÷çD m = Dl l =- D - - ÷ç ÷çè øThe surface jump:
Lost work and accumulatedentropy production
The cell potential as a function of j Accumulated entropy production
cell
dSirr /dt
Entropy production in the human lung
Flow
regimeE. R. Weibel, Am. J. Physiol. (1991) 261
The entropy production (and the driving force) is constant in eachof the flow regimes
Diffusionalregime1
S. Gheorghiu, S. Kjelstrup, P. Pfeifer and M.-O. Coppens (2005),Fractals in Biology and Medicine, Vol. IV (Losa et al., ed. Birkhäuser Verlag, Basel), pp. 31-42.
Can we make more energy efficientdesigns?
A fractal injector in fluidized beds gives higher yield per unit time (Coppens, US patent) *US Patent
Highest thermodynamic efficiency with constant local entropy production and uniform feed over the area; or:
Right angles: OK when pressure losses are small!
A fractal injector gives a uniform 2D distribution*
Cf, Lectures 10,11
Tondeuret al. 2004
Effective 1-D diffusion2
2( ) (1 ) ( ) 0cD k c yy
dd w
The reaction can be limited by diffusion in the nanoporouspart
Reaction/diffusionStationary state
Porosity
d
w
E. Johannessen, G. Wang, M.-O. Coppens. Ind. Eng. Chem. Res. (2007)E. Johannessen, G. Wang, C.R. Kleijn, M.-O. Coppens. Ind. Eng. Chem. Res. (2007)
Given a constant production
with efficiency factor
Thiele modulus
6 2 35 10 m /s 10gas nanoD D
=c0
What is the optimal macroporosity ε and heigth H of the layer?
Find the minimum catalyst volume for a given current
0(1 )catV HLM J J LM
Kjelstrup, Coppens, Pharoah, Pfeifer, Energy and Fuels (2010)
Improving the material- and energy efficiency…
…. By uniform feed distributionand increasing access to the catalyst
Numerical example: Pure oxygen353 K, 1 bar
5.519.620.0Optimal layerheight/ µm
415117Rate coefficient /s-1
3.8Rate coefficient /s-1
0.3710.2960.192 Butler-Volmer Overpotential/ V
0.81.01.0Transfer factor
1500015000500Current density / Am-2
Pt cost reducedby a factor 8
Energy dissipationreduced by ≈100-200 mV10- 20 %
ETEK Elat: 0.5 mg Pt/cm2
Summary
• The coupling of transport phenomena in fuel cells is large
• The entropy production can be studied by calorimetryof fuel cells and by modeling
• The theory provides fundamental insight, and may help practical designs of experiments and equipments
• The entropy balance is not yet widely used in a systematic study of fuel cells. It can be used to find better designs (cf. Lectures 10,11).
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