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1 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
Introduction to Basic Electrochemistry for
Fuel Cells and Electrolysis
Claude LAMY, Professor
Institut Européen des Membranes, CNRS - GDR n°3339 (PACS),
University of Montpellier 2,2 Place Eugène Bataillon, 34095 Montpellier, France,
E-mail: claude.lamy@iemm.univ-montp2.fr
2 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
Introduction♦ Thermodynamics and theoretical energy efficiency♦ Reaction mechanisms and kinetics limitations
The Polymer Electrolyte Fuel Cell♦ Principle of a Polymer Electrolyte Fuel Cell ♦ Cell components (membrane, catalysts, etc.) ♦ Survey of various applications of PEFC
The Direct Methanol Fuel Cell
♦ Reaction mechanisms and kinetics limitations♦ The fuel crossover problem
Conclusions
3 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
C.F. Schönbein, W.R. Grove, Philosophical Magazine, (1839)
1801 : Discovery of the Volta Cell1839 : Birth of Georges Leclanché1859 : Gaston Planté discovered the Pb/PbO2 battery1867 : Invention of the Leclanché cell
H2 + ½ O2 → H2O + electricity (+ heat)with ∆G°r = - 237 kJ/mole ; ∆H°r = - 286 kJ/mole
It looks like the reverse of water electrolysis : H2O + electricity → H2 + ½ O2
Electrical Energy :We = - ∆∆∆∆G° ≈≈≈≈ 118 MJ/kg ≈≈≈≈ 33 kWh/kg
Standard e.m.f. : E ° = - ∆∆∆∆G/nF = 1.229 VTheoretical efficiency : εεεε = ∆∆∆∆G/∆∆∆∆H = 83% (25°C)
The Hydrogen/Oxygen Fuel Cell
4 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
H2/O2 Fuel Cell
Water electrolysis
j = I/S = n F v i (A/cm 2)
E / V(SHE)
Eelectrolysis = 2,1 V
EFC = 0,7 V
H2O →→→→ O2H2 →→→→ 2 H+
2 H+ →→→→ H2 O2 →→→→ H2O
Overall efficiency (1.23/2.1)x(0.7/1.23 )
= 0.7/2.1 ≈ 33%
Electrolysis efficiency 1.23/2.1 ≈ 59%
Fuel Cell efficiency0.7/1.23 ≈ 57%
Practical energy efficiency of water electrolysis and fuel cells
5 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
Leclanché CellZn/MnO2
Internal Combustion Engine: Heat (Q 1, T1)
Fuel CellH2/O2 (air)
BatteryPb/PbO2
FuelTank
Oxidant Heat (Q)
Electricity
Electricity (W e)
Mechanical work (W m)
Heat(Q2, T2)
Thermal Engine (Carnot)εεεεr = Wm/(-∆H) = (Q1-Q2)/Q1
= 1 – Q2/Q1 = 1 – T2/T1
e.g. εεεε = 1 – T2/T1 ≈ 30 to 45 %Practical efficiency ≈ 18 to 30 %
Electrochemical Sourceεεεε = We/(-∆H) = ∆G/∆H
≈ 80 to 97 %Practical efficiency ≈ 40 to 60 %
Energetic Efficiency
Electricity (W e)
Electricity (W e)
FuelTank
Oxidant
How does work a fuel cell : it is an electrochemica l device which transforms directly the combustion energy of a fuel (- ∆G) into electricity (and heat) Comparison with other electric source and Internal Combustion Engine (ICE)
6 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
Thermodynamic Relations
A key variable in Electrochemistry: the Electrochemical Potential
Definition Thermodynamic relations
♦ Equilibrium between 2 charged phases α and β♦ Nernst potential ♦ The Standard Hydrogen Electrode (SHE)
Theoretical energy efficiency ♦ Under standard conditions (j = 0) ♦ As a function of temperature
7 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
Definition of the Electrochemical potentialConsider a charged phase α containing different species (i) : solvent
molecules (H2O), chemical species Ai of charge zi (zi = 0 for a neutral species, zi > 0 for a cation, < 0 for an anion).
The Electrochemical Potential of one mole of species (i) in phase αis the total work to bring species (i) from a standard reference point (with no interaction), usually Infinity (∞), to the bulk phase α :
µiα = chemical potential; F = Faraday = NNNNeo = 96485 C
Φ = inner potential; Ψ = outer potential; Χ = surface potential
µα
i
ααααi
α
iN,P,Tiαα
i χ with F z )N/G( µµj
++++ΨΨΨΨ====ΦΦΦΦΦΦΦΦ++++====∂∂∂∂∆∆∆∆∂∂∂∂====
A i
µα
i
µiα
χααααPhase α Ψα∞
++
+
+ +
+
+
--
-
- -
+
+
+
+
-+ -
∑∑∑∑
∑∑∑∑
====ΧΧΧΧ
====ΨΨΨΨ
i3iro
iii
ii ro
i
r εε π4
r.p N
r εε π4q
rr
8 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
Equilibrium at the electrode-electrolyte Interface
0 distance to electrode x
Inner potential Φ
ΦM
ΦS
E = ΦM - ΦS
αααα β
Si
S
i
S
i
Mi
M
i
M
i F z F z ΦΦΦΦ++++========ΦΦΦΦ++++==== µµµµ
0 - i.e. µµµµµβ
iα
iiβ
iα
i ========∆∆∆∆====
F z / ) - - E iM
i
S
iSM µµ(====ΦΦΦΦΦΦΦΦ====
E: electrode potential; [A i]: activity
A izA + n i e- ←→←→←→←→ B i
zB with µµµµi = µµµµio + RT Ln[A i] and n i = zi
A - ziB
→→→→ E = ΦM – ΦS = (µAS - µB
S + µeM)/nF
E = (µAo - µB
o)/nF +RT/nF Ln([A]/[B]) + µeM/F
E = EoA/B +RT/nF Ln([A]/[B]) + µe
M/F
which looks like a Nernst law, but E is not directly measu rable
µµµSB
Mei
SA n ====++++
9 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
Only the potential difference between 2 electrodes, the potential of each is defined by 2 electrochemical systems in equilibrium, is measurable :
A1zA1 + n1 e- ←→←→←→←→ B1
zB1 and A 2zA2 + n2 e- ←→←→←→←→ B2
zB2
Eeq = E2 – E1 = EoA2/B2 – Eo
A1/B1 + RT/nF Ln([A 2] [B 1]/[B 2] [A 1])
since the electron chemical potential (µeM/F) cancels out with the same
metal (Cu) as connecting leads at the extremity of the measuring device.
This corresponds to the overall chemical reaction :
a2 A2zA2 + b1 B1
zB1 ←→←→←→←→ a1 A1zA1 + b2 B2
zB2
with equilibrium constant Ke = ([A1]a1[B2]b2/ [A2]a2[B1]b1) = exp(- ∆Gr/RT)where the reaction free energy is related to the chemical potentials:
(j = products, i = reactants)
so that: Eeq = E2 – E1 = - ∆Gr / nF or
∑∑∑∑∑∑∑∑ ∆∆∆∆∆∆∆∆====∆∆∆∆i
iij
jjr µ N - µ N G
0 G E nF G reqr ====∆∆∆∆====++++∆∆∆∆
10 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
Important remarks : Activity and Fugacity
For an electrolytic solution the activity [Ai] of species (i) is related to its concentration Ci = Ni/V (Ni moles in solution of volume V) by:
[Ai] = γi Ci where γi is the activity coefficient(γi → 0 if Ci → 0 ∼ ideal solution)
For a gaseous phase the fugacity of a gaseous species (i) is related to its partial pressure through its chemical potential µµµµi = µµµµi
o + RT Ln (f i /Po) where Po is the pressure of a reference state (Po = 1 bar usually) and f i is proportional to the partial pressure pi, i.e. f i = γi p iwith an activity coefficient γi → 0 if pi → 0 ∼ ideal gas)
In the following, activity and fugacity will be assimilated to concentration and partial pressure, respectively.
11 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
Thermodynamic Relations∆G < 0 for spontaneous reaction ; Ecell = Ec – Ea
∆G = - S∆T + V∆P + Σ µi ∆Ni with µi = µio + RT Ln[Ai]
Then : We = - nF Ecell = ∆FT + P ∆V = ∆GT,P = ∆H – T ∆S
For an half cell whose electrode potential is controlled by an electrochemical reaction :
Ai + ni e- ←→←→←→←→ Bi with µi = µio + RT Ln[Ai]
Eeq = EoA/B + (RT/nF) Ln ([Ai] / [Bi]) with
EoA/B = (µA
o - µBo ) / nF = - ∆Go/nF (standard state)
one may define a standard Redox potential EoA/B in the
potential scale of the Standard Hydrogen Electrode (SHE). The Redox potential of all electrochemical systems can
thus be classified in the SHE scale (electrochemical series).
12 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
2 H+ + 2 e- ←→←→←→←→ H2 Pt/H2 (p=1 bar) / ([H +]=1), 25°CA i + n i e- ←→←→←→←→ B i____________________
A i + n i/2 H2 ←→←→←→←→ B i + n i H+ with ∆Gr < 0 or > 0
which gives:
Thus Eeq = EoA/B + (RT/nF) Ln ([Ai] / [Bi]) ≈≈≈≈ Nernst law
E Fn - )µ 2/n µ( - µ n µ µ Nj - µ N G eqio
2HiiAoHiiB
jj
iiir ====++++++++====∆∆∆∆∆∆∆∆====∆∆∆∆ ++++∑∑∑∑∑∑∑∑
definition by T 0 F/)µ 1/2 - µ( E since
nF/)µ - µ( F/)µ 1/2 - µ( - nF/)µ - µ( E- E Eo
2HoHSHE
BAo
2HoHBASHEA/Beq
∀∀∀∀========
============
++++
++++
Reference electrode: the Standard Hydrogen Electrode (SHE)
Alkali
0.50.340 1 1.23
SHE
-0.76
Zn2+/Zn Cu2+/Cu
-1 E/V
O2/H2O
0.77
Fe3+/Fe2+ Halogen≈≈ ≈≈ ≈≈ ≈≈
13 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
Reference electrodes:
Standard Hydrogen Electrode (SHE) Saturated Calomel Electrode (SCE)
1 M H2SO4
H2 gaspH2 = 1 bar
Pt wire
Pt/Pt foil
Platinum wire
Hg
Hg2Cl2
Frit
KClsaturated solution
KClcrystals
Porous Glass Frit
Platinum wire
Hg
Hg2Cl2
Frit
KClsaturated solution
KClcrystals
Porous Glass Frit
14 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
Scheme of a 3-electrode cell for electrochemical measurements with the potential control of the working electrode by a
potentiostat
Cooling water
Working electrode
Reference electrode
Luggin capillary
Salt bridge
15 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
Electrochemical half-cell reactions:H2 —–> 2 H+ + 2 e-
1/2 O2 + 2 H+ + 2 e- —–> H2O
Overall reaction : H2 + ½ O2 —–> H2O Then :
Eeq = - ∆Gr/nF = -∆Go/2F + (RT/2F) Ln[(pH2) (pO2
)½ / pH 2O]
Eeq = Eo + (RT/2F) Ln[(pH2) (pO2
)½ / pH 2O]
with Eo = -∆Go/2F is the standard cell voltage, i.e. Eo = 1.229 V at 25°C,
where pH2, pO2
, and pH 2O are the partial pressure of hydrogen, oxygen and
water, respectively.
Therefore by increasing the pressure of reactants (H2 and O2) and decreasing the pressure of the product (water) one can increase the cell voltage. The variation of Eeq with T is contained mainly in the entropic term, i.e.∆Sr
o = nF (dEo/dT) : temperature coefficient.
Nernst equation for the H 2/O2 system
16 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
Thermodynamics of Hydrogen Oxidation
H2 —–> 2 H+ + 2 e- acid mediumAnode
H2 + 2 OH- —–> 2 H2O + 2 e- alkaline medium
1/2 O2 + 2 H+ + 2 e- —–> H2O acid mediumCathode
1/2 O2 + H2O + 2 e- —–> 2 OH- alkaline mediumH2 + 1/2 O2 —–> H2O overall reaction
with ∆G° = - 237 kJ and ∆H° = - 286 kJ/mole H2 (standard state)
Energy efficiency under standard conditions: εrcell = ∆G° / ∆H° = 0.829
Volt1,23 2x96500
10237x
nFG E - E E
3aceq ========∆∆∆∆−−−−========
°°°°−−−−++++°°°°
kWhr/kg 33 32.9 3600x2237x
3600xM
W 103Ge ≈≈≈≈========∆∆∆∆−−−−====
°°°°
17 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
Reversible energy efficiency (I = 0)
83% 286237
H
S T 1
HG
H)(
E F n
H)(W
eqecellr ==
∆∆−=
∆∆=
∆−=
∆−=ε
For a Fuel Cell working under standard conditions (25°C, 1 bar) :
43% C)(350 623C)(80 353
- 1 TT 1
Q
Q 1
H)(W
1
2
1
2rthermalr =
°°=−=−=
∆−=ε
For a Thermal Engine, such as an ICE, working between an hot source Q1 at 350°C and a cold source Q 2 at 80°C :
18 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
Thermodynamic data for the H2/O2 combustion reaction (kJ/mol) as a function of temperature
Physical state of water
T/K ∆H ∆G EeqEfficiency (ε=∆G/∆H)
Standard state 298 -285.83 -237.17 1.229 0.830
Liquid 328 -284.88 -232.32 1.203 0.816
Liquid 348 -284.25 -229.14 1.187 0.806
Liquid 368 -283.61 -225.99 1.171 0.797
Gaseous 298 -241.82 -228.58 1.185 0.945Gaseous 400 -242.84 -223.90 1.160 0.922Gaseous 500 -243.82 -219.05 1.135 0.898Gaseous 600 -244.75 -214.01 1.109 0.874Gaseous 700 -245.62 -208.81 1.082 0.850Gaseous 800 -246.42 -203.50 1.055 0.826Gaseous 900 -247.16 -198.09 1.027 0.801Gaseous 1000 -247.82 -192.60 0.998 0.777
19 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
Thermodynamic data for the H2/O2 combustion reaction (kJ/mol): H2 + 1/2 O2 —–> H2O
In the liquid state of water (273 K < T < 373 K) :
∆H = ∆G + T ∆S = ∆HHHV = ∆HLHV + ∆QCond
so that in the standard state (T = 298.15 K, p =1 bar) :∆QCond = ∆HHHV – ∆HLHV = - 285.8 + 241.8 = - 44 kJ/mol
εrHHV = ∆∆∆∆G° / ∆∆∆∆H°HHV = 237/286 = 0.83
In the gaseous state of water (T > 373 K) :
∆H = ∆G + T ∆S = ∆HLHV
so that at e.g. T = 400 K (≈≈≈≈ 127°C, p =1 bar) :∆H = ∆HLHV = - 242.84 kJ/mol
εrLHV = ∆∆∆∆G° / ∆∆∆∆H°LHV = 224/243 = 0.92
HHV = High Heating Value ; LHV = Low Heating Value ; ∆∆∆∆QCond = Heat of Condensation
20 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
Theoretical energy efficiency (FC vs. Carnot)
0,00
0,10
0,20
0,30
0,40
0,50
0,60
0,70
0,80
0,90
1,00
300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500Hot source temperature / K
The
oret
ical
effi
cien
cy
FC efficiency
Carnot efficiency
εεεεcell = εεεεCarnot 80°C ≈≈≈≈ 0,72 at T ≈≈≈≈ 1250 K (977°C)
εεεεcell = εεεεCarnot 25°C ≈≈≈≈ 0,74 at T ≈≈≈≈ 1150 K (877°C)
εεεε Cellεεεε Carnot (T cold = 25°C)εεεε Carnot (T cold = 80°C)
Linear (εεεε Cell)Polynomial (εεεε Carnot (T cold = 25°C)Polynomial (εεεε Carnot (T cold = 80°C)
21 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
Kinetics Relations
A typical kinetics law in Electrochemistry: the Butler-Volmer equation
Kinetics of a simple electrochemical reaction
The Butler-Volmer law ♦ Establishment of the Butler-Volmer law ♦ Mass transfer limitations ♦ Cell voltage vs. current density curves♦ Energy efficiency under working conditions (j ≠ 0)
22 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
mass transfer
mass transfer
Electrolyte
I ions
electron transfer
I
Kinetic relations for a simple electrochemical reaction : A i + n i e- ←→←→←→←→ B i
I = dQ/dt = nF ( dNA/dt) = nF ( dNB/dt)
I = nF v with the reaction rate v = ( dNA/dt)
I < 0 for a reduction reaction ; I > 0 for an oxidati on reaction
Reaction mechanism
23 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
AizA
B izB
Kinetics of a charge transfer reaction (heterogeneous reaction) : A i
zA + n i e- ←→←→←→←→ B izB
Activation barrier for an electrochemical reaction (K is the decrease in activation energy due to the electrode catalyst)
0 I 0 E nF G G ≠≠≠≠⇒⇒⇒⇒≠≠≠≠++++∆∆∆∆====∆∆∆∆aa v nF I (anodic) reaction oxidation : 0 I 0 G ====≥≥≥≥⇒⇒⇒⇒≥≥≥≥∆∆∆∆
cc v nF - I (cathodic) reaction reduction :0 I 0 G ====≤≤≤≤⇒⇒⇒⇒≤≤≤≤∆∆∆∆
24 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
Expression of the reaction rates with the Arrhenius law
E nF α G G with )RT/Gexp(- C S k C S T)(E,k v cccelecA
oc
elecAcc ++++∆∆∆∆====∆∆∆∆∆∆∆∆======== ++++++++++++
E nF )α -(1 - G G with )RT/Gexp(- C S k C S T)(E,k v aaaelecB
oa
elecBaa
++++++++++++ ∆∆∆∆====∆∆∆∆∆∆∆∆========where αααα is the charge transfer coefficient (0 < αααα < 1), i.e. the fraction of electrical energy which activates the reduction reaction (A → B), and (1- αααα) nFE activating the oxidation reaction (B → A). Thus:
[[[[ ]]]] ]RT/nFEα-exp[ )RT/G-exp(C k-]RT/nFE)α-1exp[( )RT/G-exp(C k nFS/Ij celecA
oca
elecB
oa
++++++++ ∆∆∆∆∆∆∆∆========
At equilibrium va = vc and j = 0, so that ja = -jc, E = Eeq and Cielec = Ci
o :
Then Eeq=EoA/B + RT/nF Ln(CA
o/CBo) with Eo
A/B = exp(-∆Go/RT) ≈≈≈≈ Nernst law
and jo = nF (k co)1-αααα (ka
o)αααα (CAo)1-αααα (CB
o)αααα = nF k so (CA
o)1-αααα (CBo)αααα
Out of equilibrium j = ja - jc ≠ 0. Dividing j by jo one obtains:j(η) = jo [(CB
elec/CBo) exp(1-α)(nF/RT)η - (CA
elec/CAo) exp-α(nF/RT)η] or
j(η) = jo [exp(1- αααα)(nF/RT)η – exp- αααα(nF/RT)η] (no mass transfer limitation)
)RT/nFEα-exp( )RT/G-exp(CnFk ]RT/nFE)α-1exp[( )RTG-exp(CknFj-j j eqo
coA
oceq
oa
oB
oaoca
++++++++ ∆∆∆∆====∆∆∆∆============
Butler-Volmer law with ηηηη = E - Eeq the overvoltage and j o the exchange current density
25 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
Expression of the Butler-Volmer law in some limitin g cases
Very often α = ½ so that j(η) = 2 jo sh[(nF/2RT) η] ∼∼∼∼ hyperbolic sinus curveIf α = 0 or 1 then j(η) = jo [exp(nF/RT η) - 1] or j(η) = jo [1 - exp(nF/RT) η]
For |η| << RT/nF (≈ 25/n mV at 25°C) then j(η) = jo (nF/RT) η = η/Rt or η = Rt j with Rt = RT/nFjo the charge transfer resistance (∼∼∼∼ Ohm’s law )
For η >> RT/nF then j(η) = jo exp[(1- α )nF/RT η] or η = RT/(1-α)nF Ln(j/jo) and for η << - RT/nF then j(η) = - jo exp[- α nF/RT η] or η = -RT/αnF Ln(-j/jo)i.e. η = a ± b log |j| ∼∼∼∼ Tafel law with the Tafel slopes bi = 2.3 RT/αinF in V dec-1
For the Hydrogen Oxidation Reaction (HOR) the exchange current density is very high (jo ≈ 1 mA cm-2) so that the reaction is reversible and the Ohm’s law does apply (linear relationship between overvoltage and current density)
For the Oxygen Reduction Reaction (ORR) the exchange current density is small (jo ≈ 1 µA cm-2 ) so that the reaction is irreversible (one may neglect the reverse reaction, i.e. water oxidation) and one may write:j(η) = - jo exp(-α nF/RT η) or ηc
act = - RT/ααααcnF Ln(|j|/j o) ∼∼∼∼ activation overvoltage
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GDR n°3339Piles A Combustible, Systèmes
Mass transfer limitations : concentration overvolta ge
The reaction kinetics can be controlled by the mass transfer (diffusion, migration, convection) of the reacting species inside the electrolytic phase, so that the current density is proportional to the mass flux density Ji of species (i): ji = |zi|F Ji with Ji = 1/S ∆Ni/∆t = Ci vi (vi is the velocity).
Ji can be expressed by the Nernst-Planck equation :
In the following we will neglect the contribution of the migration currentjim = |zi| Fui Ci EEEE = |zi| λi Ci EEEE = κi EEEE and the convection current jicv = |zi|F Ci vi
ext
so that we will only consider the diffusion process (Fick’s laws ) of species (i):
with Di = diffusion coefficient; ui = ziFDi/RT = electric mobility; λi = Fui = molar conductivity; Κi = |zi| λi Ci = ionic conductivity;
EEEE = - ∇Φ = electrical field
extiiii
iii
extiii
iii vC CD
RTFz
- CD- vC µRT
DC - J
rrrrrr++++ΦΦΦΦ∇∇∇∇∇∇∇∇====++++∇∇∇∇====
on)conservati mass law 2 s(Fick' C D Jdiv - t
CJ Fz j and law) 1 s(Fick' C D - J
ndii
di
i
dii
di
stii
di
≈≈≈≈∆∆∆∆========∂∂∂∂
∂∂∂∂====∇∇∇∇====
r
rrrr
27 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
)δ x (0 (0)C xδ
)0(C - C (0)C x )x/C( )x(C and
δ
)0(C - C )x/C( to leading 0 x/)t,x(C D t/)t,x(C
x/C D Fz j and 0 t/C ,x/C D - J
iiio
i0xii
iioi
2i
2ii
0xiiidiiii
di
≤≤≤≤≤≤≤≤++++====++++∂∂∂∂∂∂∂∂====
====∂∂∂∂∂∂∂∂====∂∂∂∂∂∂∂∂====∂∂∂∂∂∂∂∂
∂∂∂∂∂∂∂∂±±±±========∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂====
====
====
Resolution of Fick’s equations (in the approximation of a semi-infinite linear diffusion, under steady state conditions):
Cio
xδ
Ci(x)
Ci(0)
00
Electrode distance
Electrode Electrolyte
δ = thickness of the
diffusion layer
28 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
For the electrochemical reaction, A i + n i e- ←→←→←→←→ B i, the rate of which is controlled only by diffusion, the current density can be written as:
ljl = |jid| = nF Di |cio – ci(0)| / δ, with n = zA – zB
with ci(0) the concentration at the electrode surface.For high rate, i.e. high current densities, all the reacting species
arriving at the electrode surface are immediately consumed and their activity (concentration) becomes 0. Thus the concentration gradient reaches a maximum value, leading to a limiting current density:
jli = nF Di cio / δwith Di (in m2/s or cm2/s) the diffusion coefficient of species (i), cio its concentration in the bulk of electrolyte and δ the thickness of the diffusion layer (in the approximation of a semi-infinite linear diffusion).
Mass transfer limitations : concentration overvolta ge
29 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
For a very fast transfer reaction, only limited by mass transfer, the Nernst equation does apply at the surface, leading to concentration overvoltages:
Eeq = EoA/B + (RT/nF) Ln (cA(0)/cAo / cB(0)/cBo) = E1/2 + (RT/nF) Ln[(j-jlc)/(jla-j)]
(equation of a Polarographic Wave assuming the same diffusion coefficients)
anode theat jj
1lnFn
RTη
ala
conca
−−−−====
cathode theat jj
1lnFn
RTη
clc
concc
−−−−====
By doing the ratio of j with jli one may obtain:ljl/jli = [cio – ci(0)] / cio = 1 - ci(0)/cio = 1 - pi(0)/pio
The superficial concentration (or partial pressure for gaseous species) is :ci(0)/cio = pi(0)/pio = 1 - ljl / jli
Mass transfer limitations: concentration overvoltag e Mass transfer limitations : concentration overvolta ge
30 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
E(j) curves limited by mass transfer (diffusion)
0,00,20,40,60,81,01,21,4
00,
20,
40,
60,
81
1,2
1,4
Cur
rent
den
sity
/ A
cm
-2
Cell voltage / V
0,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4
0 0,2 0,4 0,6 0,8 1 1,2 1,4
Current density / A cm -2C
ell v
olta
ge /
V
Tafel slope bc = 60 mV/dec.
Limiting current jlc = 1.4 A/cm2
31 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
The cell voltage vs. current density characteristics for the H2/O2 fuel cell, taking into account charge transfer overvoltages, concentration overvoltages and ohmic drop can thus be written as:
E(j) = Ec(j) – Ea(j) – Rej = Eeq – (|ηηηηaact(j)| + |ηηηηa
conc (j)| + |ηηηηcact(j)| + |ηηηηC
conc (j)|) – Rej
jRj
j1
j
j1ln
nFRT
j
jjln
nFα
RTE)j(E e
ll2oo
eqca
ca −
−×
−+
×+=
Cell voltage vs. current density E(j) curves
32 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
IjI = n F v i = n F(1/S) dN i/dt
j/mA
cm
-2
300
≈ 0.8 V
≈ 0.3 V
E eq ( ) ≈ 1.21 V
100
η a η c200
CH 3OH
CHCH 33OHOHH2HH22 O 2OO 22
PtPtPtPt -Ru50 -50
PtPt --RuRu5050 --5050
Pt -Ru80 -20
PtPt --RuRu8080 --2020Pt -Ru -
XPtPt --RuRu --XX
E = Ec – Ea ≈≈≈≈ 0.8 V
≈ 0.3 V
Eeq(CH3OH) ≈ 1.21 V
η a η c
H2
PtPtPtPt -Ru50 -50
PtPt --RuRu5050 --5050
Pt -Ru80 -20
PtPt --RuRu8080 --2020Pt -Ru - XPtPt --RuRu --- X
O2CH3OH
0.5 1.23E/V vs. RHE00
EcEa
Current density vs. electrode potential for electrochemical reactions involved in fuel cells
33 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
0,000
0,200
0,400
0,600
0,800
1,000
1,200
0 0,25 0,5 0,75 1 1,25 1,5
j/A cm -2
E/ V
Ohmic losses R e jCharge transfer
overvoltage
Eeq = 1.23 V
Mass transferovervoltage
0,7
0,4 0,9
Po = 0,175 W/cm2 P/Po=1,6 P/Po= 3,6 P/Po= 6
Cell voltage vs. current
density E(j) curves j0 = 10-8 ; Re = 0,15 ; j l = 1,3
j0 = 10-6 ; Re = 0,10 ; j l = 2,2j0 = 10-6 ; Re = 0,15 ; j l = 1,4
j0 = 10-8 ; Re = 0,30 ; j l = 1,2
E(j) = Ec(j) – Ea(j) – Re j = Eeq – (|η|η|η|ηc(j) |||| + |η|η|η|ηa(j) |||| + Re j)
34 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
Eeq Eeq Eeq )jeR )j(cη )j(aη( Eeq jeR )j(Ea )j(Ec )j(E −−+=⟨++−=−−−+=
Energy efficiency under working conditions (j ≠ 0)
n
nexpjm
j ε and % 57
1.230.7
eqE
)jeR (j)cη (j)aη( 1 eqE
E(j) ε FE ==≈=
++−==
FEcellr
exp
eq
eqexpcell εεε
n
n
EE(j)
H)(
E F n
H)(
E(j) F n ε ××××××××====××××××××
∆∆∆∆−−−−====
∆∆∆∆−−−−====
ηa = E-(j) – E-eq with ηa > 0 (fuel oxidation)
ηc = E+(j) – E+eq with ηc < 0 (oxygen reduction)
Re = electrolyte and interface resistance
εεεεcell = 0.83 x 0.57 x 0.99 ≈≈≈≈ 47% and εεεεCHP ≈≈≈≈ 85 to 95%
35 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
The Polymer Electrolyte Fuel Cell (PEFC)
♦ Principle of a Polymer Electrolyte Fuel Cell
♦ Cell components (membrane, catalysts, etc.)
36 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
2 H+
I4 OH
Anodic catalyst
Cathodiccatalyst
H2O
½ O2
2e-
2e-
PEM
H2
2e-
+ We
Overall reaction: H 2 + ½ O2 →→→→ H2O with ∆G < 0
Schematic representation of a PEFC elementary cell
37 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
Schematic representation of a PEFC elementary cell
End Plate
BipolarPlate End Plate
Bipolarplate
Teflon ® gasket
MEA
Polymer Electrolyte Fuel Cell (PEFC)
38 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
Schematic representation of the Membrane Electrode Assembly (MEA)
AnodeVent
CathodeVent
AnodeFeed, H2
CathodeFeed, O2
0.5 to 0.9 V
Gas DiffusionBacking
Proton Exchange Membrane (≈≈≈≈ 100 µm)
CatalystlayerH+
Membrane Electrode Assembly (MEA)Thickness e ≈≈≈≈ 5 mm
Thickness e ≈≈≈≈ 500 µm
(10 µm)
(200 µm)
, Alcohol
39 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
jo = 10-8 A cm -2; Re = 0,15 ΩΩΩΩ cm 2; j l = 1,3 A cm -2
jo = 10-8 A cm -2; Re = 0,30 ΩΩΩΩ cm 2; j l = 1,2 A cm -2
0,000
0,200
0,400
0,600
0,800
1,000
1,200
0 0,25 0,5 0,75 1,0 1,25 1,5Current density j/A cm -2
Ohmic drop R e j
Charge transfer activation
Mass transferactivation
0,7
Eeq = 1,23 V
0,4
Cel
l vol
tage
E/V
Cell voltage E(j) = E +(j) – E–(j) - Re j
Influence of the membrane specific resistance (Re = e/σ) on the cell voltage-current density characteristics E(j)
40 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
E(j) characteristics of a PEFC elementary cell with different membranes : Dow (e = 125 µm) ; o Nafion® 115 (e = 125 µm) ; ∆ Nafion® 117 (e = 175 µm)
After K. Prater (Ballard), J. Power Sources, 29 (19 90) 239
Current density (j/A cm -2)
Cel
l vol
tage
(E/V
)
Specific resistance (R e = e / σσσσ)e.g. for Nafion ® 115 :
0.0125 cm / 0.125 S cm -1
≈≈≈≈ 0.1 ΩΩΩΩ cm 2
><><><><
DUPONT NAFION®
O
CF2
CF
O
CF2CF2CF2CFCF2CF2
H+
CF2
S=OO=
O
CF2
– CF3
––
Dow
41 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
j0 = 10-6 A cm -2 ; Re = 0,10 ΩΩΩΩ cm 2 ; j l = 2,2 A cm -2
j0 = 10-6 A cm -2 ; Re = 0,15 ΩΩΩΩ cm 2 ; j l = 1,4 A cm -2
0,000
0,200
0,400
0,600
0,800
1,000
1,200
0 0,25 0,5 0,75 1,0 1,25 1,5
Current density j/A cm -2
Cel
l vol
tage
E/ V
Ohmic losses R e jCharge transfer
overvoltage
Eeq = 1.23 V
Mass transferovervoltage
0,7
0,4 0,9
Cell voltage E(j) = E +(j) – E–(j) - Re j
Influence of the mass transfer limitations (limiting current density jl) on the cell voltage-current density characteristics E(j)
42 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
Current density / mA cm -2
Cell voltage vs. current density curves for an H 2/air FC with MEAs using three different types of carbon cathode GDL (80°C, 0.22 mg cm −2 Pt loading) : () microporous layer-coated and 30 wt.% FEP-impregna ted; () 30 wt.% FEP-impregnated; ( ) 10 wt.% FEP-impregnated.
After C. Lim, C.Y. Wang, Electrochim. Acta 49 (2004) 4149–4156
43 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
Influence of the catalytic properties of electrodes (exchange current density jo) on the cell voltage E(j)
0,000
0,200
0,400
0,600
0,800
1,000
1,200
0 0,25 0,5 0,75 1,0 1,25 1,5Current density j/A cm -2
Cel
l vol
tage
E/V
Ohmic drop R e jCharge transfer
overvoltage
Eeq = 1,23 V
Mass transfer overvoltage
0,7
E(j) = E+(j) - E–(j) - Re j = Eeq – (|η|η|η|ηc(j) |||| + |η|η|η|ηa(j) ||||) - Re j
0,4 0,9
jo = 10-8 A cm -2; Re = 0,15 ΩΩΩΩ cm 2; j l = 1,3 A cm -2
jo = 10-6 A cm -2; Re = 0,15 ΩΩΩΩ cm 2; j l = 1,4 A cm -2
44 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
Fuel cell characteristics of a DEFC recorded at 110 °C.Influence of the nature of the bimetallic catalyst (30% loading).
Anode catalyst : 1.5 mg.cm -2 ; Cathode catalyst : 2 mg.cm -2 (40% Pt/XC72 E-TEK)Membrane : Nafion ® 117; Ethanol concentration : 1 M
0 20 40 60 80 100 120 140 160
0100200300400500600700800
PtPt-Sn(80:20)Pt-Ru(80:20)Pt-Mo(80:20)
Cel
l vol
tage
/ m
V
Current density / mA cm -2
Polarization curves Power density curves
0 20 40 60 80 100 120 140 1600
5
10
15
20
25
30 PtPt-Sn(80:20)Pt-Ru(80:20)Pt-Mo(80:20)
Pow
er d
ensi
ty /
mW
cm
-2
Current density / mA cm -2
After C. Lamy et al., in ”Catalysis for Sustainable Energy Production”, edited by P. Barbaro and C. Bianchini, Wiley-VCH, Weinheim, 2009, Chap.1, pp. 3-46.
45 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
What are their uses ?? Production of electric energy(and eventually heat in CHP systems) in a wide range of power (1 W to about 10 MW) with a similar energy efficiency :
Electric Power Plant (1 to 10 MW) Electric Vehicle (10 to 200 kW) Auxiliary Power Unit (APU from 1 to 100 kW) Aerospace power supply (1 to 50 kW) Household Application (1 to 10 kW) Power supply for portable electronics (1 to 100 W)
Survey of various applications of PEM Fuel Cells
46 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
City Bus equipped with a 200 kW FC for the 2008 Olympic Games in Beijing
47 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
Fuel CellSystem
(SAFRAN Group)
APU for aircrafts and spacecrafts
48 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
Airbus 320 equipped with a PEFC APU as demonstrated by DLR (Germany)
49 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
Laptop Computer ~ 10-100 W Cell Phone ~ 1-10 W
Application to portable electronics (1 to 100 W)
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GDR n°3339Piles A Combustible, Systèmes
The Direct Methanol Fuel Cell (DMFC)
♦ Principle of a Direct Methanol Fuel Cell
♦ Reaction mechanisms and kinetics problems
51 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
6e-I
Electrolyte
Ionic conductore.g. a PEM
CathodeAnode
Electronic conductor+ catalyst
Electronic conductor+ catalyst
CH3OH
+ H2OCO2 + 6H++ 6e- 6H+
CO2
6e-+ 6H+ + 3/2O2
Oxygen
(Air)
3H2O
We= 6.1 kWhr/kg A
−−−− 6e-
I
++++
Direct Methanol Fuel Cell
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GDR n°3339Piles A Combustible, Systèmes
Pt + CH3OH → Pt - CH3OHads
Pt - CH3OHads → Pt - CHOads + 3 H+ + 3 e-
Pt - CHOads → Pt - COads + H+ + e-
Ru + H2O → Ru – OHads + H+ + e-
Pt - COads + Ru – OHads → Pt + Ru + CO2 + H+ + e-
Overall reaction CH 3OH + H2O → CO2 + 6 H+ + 6 e-
Reaction mechanism of the electrocatalytic oxidation of methanol
at platinum-based electrodes
53 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
Modification of platinum by Ruthenium using the colloidal method (after Bönnemann et al.)
carbon
Pt-Ru
Pt-Ru alloy
Pt-Ru/XC72
carbone
PtRu
Pt and Ruon the same support
carbon
Pt
Pt+Ru/XC72
Pt and Ru on different supports
carbone
Pt
carbon carbone
Ru
carbon
Pt
Pt
RuRu
Pt/XC72+Ru/XC72
This a complex 6-electron reaction mechanism, which needs the preparation of bimetallic and plur imetallic
electrocatalysts for methanol oxidation catalysts
54 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
Pt/XC72Pt+Ru/XC72 Pt/XC72+Ru/XC72Pt-Ru/XC72
0,3 0,4 0,50
20
40
60
80
100
E/V vs. RHE
I/mA
mg
Pt-1
Oxidation of 0.1M methanol (v=5mV s -1)
0,0 0,2 0,4 0,6 0,8 1,0 1,2
0
10
20
30
40
50
Pt/XC72Pt+Ru/XC72Pt/XC72+Ru/XC72Pt-Ru/XC72
I/ m
A m
g P
t-1
E/V vs. RHE
Alloy >>>>PlatinumOn same C particles
On different C particles>>>> >>>>On same
C particles AlloyOn different C particles Platinum>>>> >>>> >>>>
Oxidation of preadsorbed CO (v=50mV s -1)
Evaluation of the electrocatalytic activity of PtRu / C electrodes by cyclic voltammetry (room temperature)
55 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
0
20
40
60
80
100
120
140
160
180
0 2 4 6 8 10
Time / hours
MeO
H /
mm
ol
Nafion 117
NEM 1
NEM 6
NEM 20
NEM 21
Permeability to methanol of different Solvay membra nes in comparison with Nafion 117 at room temperature.
Effect of the crossover of the fuel through the pol ymer membrane
56 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
As a result of the crossover of the fuel to the cathode, the electrode potential Em will be determined by a mixed reaction resulting from the reduction of oxygen and the oxidation of methanol at the same potential Em. As both reactions are quite irreversible, a Tafel behavior is practically always observed. Under these conditions, the current density (jm) and mixed potential (Em) are given by the equations :
and :
where joi and bi are the exchange current density and the Tafel slopes, respectively, for both half-cell reactions.
cb)o
cEmE( -
ocab)o
aEmE(
oam 10 j 10j j
−−−−−−−−
========
oa
oc
ca
ca
ca
oca
oac
m jj
log bb
bb
b bE b E b
E++++
++++++++++++====
This will result in a negative shift, ∆Em, of the mixed potential at the oxygen cathode, as given by the following equation :
with jo’a = 103 joa and ba ≈ bc ≈ 120 mV/decade for both the oxygen reduction (at high current densities) and the methanol oxidation reactions.
mV 18010log2
120jj
logbb
bbE 3
a'o
oa
ca
cam −−−−≈≈≈≈≈≈≈≈
++++====∆∆∆∆ −−−−
57 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
Effect of methanol crossover through the membrane
0 50 100 150 200
0,0
0,2
0,4
0,6
0,8
1,0
emf MeOH/O2
Ean MeOH.
Ecat MeOH cross-over
Ean H2.
Ecat H2.
E /V
j / mA cm-2
0 50 100 150 200
0,0
0,2
0,4
0,6
0,8
1,0
emf MeOH/O2
Ean MeOH.
Ecat MeOH cross-over
Ean H2.
Ecat H2.
E /V
j / mA cm-2
Polarization curves of an H 2/O2 PEFC and a DMFC in the presence of methanol at the cathodic compartment : ( ) potential of the PEFC cathode;
(O) potential of the DMFC cathode ; ( ) DMFC cell voltage
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GDR n°3339Piles A Combustible, Systèmes
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 200 400 600 800 1000j / mA.cm-2
E /
V
0
20
40
60
80
100
120
P /m
W.cm
-2
Pt-Ru (1:1) Pt-Ru (2:1) Pt-Ru (4:1)
Anode 2 mg.cm -2 Pt-Ru / C, Nafion 117, Cathode 2 mg.cm -2 Pt / C, [MeOH] = 2 M, 2 mL/min., P MeOH = 2 bar ; Tcell = 110°C, O 2 = 120 mL/min., P O2
= 2.5 bar ; T MeOH = TO2= 95°C.
E(j) and P(j) curves of a single DMFC with Pt-Ru/C electrodes of different Pt-Ru atomic ratios
59 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
TOSHIBA DMFC for portable applications
Applications of DMFC to portable electronics: cell phone, laptop computer, cam recorder, etc.
60 2nd Joint European Summer School for FC & H2 Technology-Heraklion-September 2012
GDR n°3339Piles A Combustible, Systèmes
MCFCMCFC
PAFCPAFC
SOFCSOFC
PEMFCPEMFC
AFCAFC
MiniMiniPACPAC
1mW 0.1 W 1W 10 W 100W 1 kW 10 kW 1mW 0.1 W 1W 10 W 100W 1 kW 10 kW 100kW 1 MW100kW 1 MW
PortablePortable
TransportTransport
MCFCMCFC
PAFCPAFC
SOFCSOFC
PEFC
AFCAFC
MiniMiniPACPAC
1mW 0.1 W 1W 10 W 100W 1 kW 10 kW 1mW 0.1 W 1W 10 W 100W 1 kW 10 kW 100kW 1 MW100kW 1 MW1mW 0.1 W 1W 10 W 100W 1 kW 10 kW 1mW 0.1 W 1W 10 W 100W 1 kW 10 kW 100kW 1 MW100kW 1 MW
PortablePortable
TransportTransportation
Stationary
AerospaceGdR PACEMGdR PACEM
GdR ITSOFC
Bio-fuel Cells
Different kinds of applications of FC
GdR PACEMGdR PACEM
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[1] P.W. ATKINS, Physical Chemistry, Oxford Universi ty Press, Oxford, 1995.[2] A.J. BARD, L.R. FAULKNER, Electrochemical Meth ods - Fundamentals and
Applications, John Wiley & Sons, New York, 1980.[3] J. O’M BOCKRIS, S. SRINIVASAN, Fuel Cells : thei r electrochemistry,
McGraw Hill Book Co., New York, 1969.[4] A.J. APPLEBY, F.R. FOULKES, Fuel Cell Handbook, Van Nostrand Rheinhold,
New York, 1989.[5] K. KORDESH, G. SIMADER, Fuel cells and their app lications,
VCH, Weinheim, 1996.[6] K. PRATER, J. Power Sources, 29 (1990) 239.[7] C. LAMY, J.-M. LEGER, J. Phys.IV, Colloque C1, V ol.4 (1994) C1-253.[8] C. LAMY, J.M. LEGER, S. SRINIVASAN, Direct Metha nol Fuel Cells : From
Fundamental Aspects to Technology Development, in " Modern Aspects of Electrochemistry", J. O'M. Bockris, B.E. Conway (Ed s.), Plenum Press, New York, volume 34 (2001) pp. 53-117.
[9] Handbook of Fuel Cells, W. Vielstich, H. Gast eiger, A Lamm (Eds.), Wiley, Chichester (UK), Volume 1, "Fundamentals and Survey of Systems", (2003)pp.323-334. Volume 2 to 6, 2003-2009.
[10] F. BARBIR, PEM Fuel Cells: Theory and Practi ce, Elsevier/Academic Press, 2005.
References
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