Non-equilibrium thermodynamics (NET):
A tool for in fuel cell design
Thermal osmosis and thermoelectric potentials
in polymer electrolyte fuel cell materials
Signe Kjelstrup, NTNU and TU Delft
Collaborators:
D. Bedeaux, K. Glavatskiy, J. Pharoah, O. Burheim, L. Akyalcin, P. Zefaniya
Water phenomena in PEM Fuel Cells
Water •Absorbs in the membrane
•Diffuses across membranes, electrode layers
•is carried along with protons and produced at the cathode
•is transported by pressure or temperature gradients
HOW CAN WE DESCRIBE TRANSPORT IN HETEROGENEOUS MATERIALS? Outline
 Non-equilibrium thermodynamics for
heterogeneous systems
 Application to transport phenomena in PEMFC
-Electro-osmosis
-Thermal osmosis (Soret effect)
-Seebeck coefficients and Peltier heats
 Conclusions
A systematic theory of transport
• The second law is obeyed locally and globally,
   Ji X i   J j X j  0
n
n
dSirr / dt    dx  J s
out
 Js  0
in
• Linear laws and Onsager’s symmetry relations
for each layer:
J1  L11 X 1  L12 X 2
J 2  L21 X 1  L22 X 2
L21  L12
The cell: Transport of heat, water and charge through
• 2 microporous layers
• 2 catalytic layers
• 1 membrane
For surface layer no 2 at steady state:
s 
J
'A,i
q
1
 1
 A,o  A,i
T
T
  wA,o   wA,i (T A,o ) 


  Jw  
A,o
T



  A,o   A,i  r G / F 
+j  


A,o
A,o
T
T


A,i
A,o
Equilibrium surfaces described by
Gibbs excess densities
• The excess density cAs is equal to the
integral over cA minus extrapolated
densities.
• The value of cAs depends on the choice of
the dividing surface.
• The position of the equimolar surface d is
found by taking cAs=0.
• Experimental results do not depend on the
choice of the dividing surface.
Scientific papers of J.W. Gibbs, Dover, New York, 1961
Surface tension
Equilibrium and non-equilibrium values are the same
Local equilibrium
A. Røsjorde, D.W. Fossmo, D. Bedeaux, S. Kjelstrup and B. Hafskjold,
J. of Colloid and Int. Science 232 (2000) 178-185; 240 (2001) 355-364.
Heat and mass transfer through surfaces
Excess entropy production:
s
i
o
s
n
n








1
1
1
1




s
i
o
i
o
m
m
m
m
  Jq  s  i   Jq  o  s    Jm  s  i    Jm  o  s 
T  m1  T
T 
T T 
 T T  m1  T
For stationary states the total heat flux is continuous and can
be expressed in the measurable heat fluxes on either side of
the surface
n
n
J  J   r and J q  J  J  J   H J  J   H mo J mo
i
m
o
m
s
m
i
q
o
q
'i
q
m 1
i
m
i
m
'o
q
m 1
Substitution gives:
1
 1
 J  o i
T
T
s
'i
q
o
i
o
 n i  m  m (T ) 

   Jm 
To
 m 1 

1  n o  mo (T i )  mi 
 1
 J  o  i    Jm 

T  m 1 
Ti
T

'o
q
Possible sets of force-flux equations:
Heat and mass transport
n
1
1
s,i 'i
s,i


r
J

r

qq q
qm J m
o
i
T
T
m 1
 oj   ij (T o )
T
o
n
when we use the measurable
heat flux on the i side, and
 r J   r Jm
s,i
jq
'i
q
m 1
s,i
jm
n
1
1
s,o 'o
s,o


r
J

r

qq q
qm J m
o
i
T
T
m 1
 oj (T i )   ij
Ti
n
s,o
 rjqs,o J q'o   rjm
Jm
when we use the measurable
heat flux on the o side
m 1
Using the relation between the measurable heat fluxes gives:
s,o
s,o
s,i
rqqs,o  rqqs,i  rqqs , rqm
 rmq
 rqm
 H m rqqs
r
s,o
jm
r
s,o
mj
 r  H r  H r  H j H r
s,i
jm
s,i
m jq
s,i
j qm
s
m qq
where H m  H m
are the latent heats
o
 H mi
The measurable heats of transfer are given for the interface by:
qm*s,i
s,i
'o
s,o
 J q'i 


rqm
J
r
q
qm
 
  s,i , qm*s,o   
  s,o
rqq
rqq
 J m  T 0, J k 0
 J m  T 0, J k 0
The energy balance relates the measurable heat fluxes and gives:
qm*s,o  qm*s,i    H mo  H mi   H
s,i
r
qq
qm*s,o   s,i s,o H   k H
rqq  rqq
The common assumption that
qm*s,i and qm*s,o
are zero violates thermodynamics!
NEMD simulations of a butane–silicalite surface
Whole surface
Difference in heats of transfer
= -55 KJ/mol
In agreement with independent MD calculation of ∆adsH
I. Inzoli, S. Kjelstrup, D. Bedeaux and J-M. Simon, Microporous Mesoporous Materials (2009)
Evaporation of water
Kinetic theory k = 0.2
G. Fang and C. Ward Phys. Rev. E 59 (1999) 417-428
D. Bedeaux and S. Kjelstrup, Transfer Coefficients for Evaporation, Physica A, 270 (1999) 413
For a bulk membrane:
Dufour effect
Peltier heat
Fourier’s law
dT
*m
m j
m j
J  
 q ( J w  tw )  
dx
F
F
q*m dT
m dcw
m j
Jw  
 Dw
 tw
T dx
dx
F
d
 m dT
m d w

 tw
 rm j
dx
T dx
dx
Electro'
q
Thermal osmosis,
Soret effect
m
osmosis
Seebeck coefficient
Fick’s and Ohm’s laws
Electro-osmosis of water in the membrane
Constant temperature:
j
m j
J  q (Jw  t
) 
F
F
m dcw
m j
J w   Dw
 tw  0
dx
F
d
m d w
 tw
 rm j
dx
dx
'
q
500
2500
5000 A/m2
Teranishi et al. JES, 2006
*m
m
w
Thermal osmosis water(i)| membrane | water(i)
Kim, Mench, JMS, 2009
• Why is water transported from the cold to the hot reservoir?
• What will happen if the material of the surface layers changes?
Thermal osmosis. Model with vapor at a and c.
Constant energy flux through the layers
J q  J q'a  H wa J wa  J q'm  H wm J wm
Influence of interfaces studied through k and  , 
 sa   m / d m ,
qw*sa  k H w
Dwsa   Dwm / d m
104 <  ,  < 104 ,
H w  45 kJ/mol
0  | k | 1
Reucroft, Rivin, Schneider, Polymer, 2002
Data from Fuel Cell Handbook
Thermal osmosis. Predicted water flux, driven by 10 K
Vapour pressure at both boundaries: 104 Pa
The interfaces
determine
the flux
The membrane
determines
the flux
Scaling factor for interface thermal conductivity,
Scaling factor for interface diffusivity: 1
θ
qw*s A  k H
H  45 kJ/mol
0  | k | 1
Thermal osmosis model.
Predicted water flux
driven by 10 K
Vapour pressure at boundaries: 104 Pa
Water flux as function of k and
- scaling factor for thermal conductivity
(top)
- scaling factor for diffusivity (bottom)
Thermal osmosis apparatus
water(l)| PTL | membrane| PTL |water(l)
Thermal osmosis. Measured water liquid flux driven by ∆T
Water flux, / kg.m‐2.s‐1
0,0006
0,0005
0,0004
0,0003
0,0002
0,0001
Nafion 117 layered between: 30C
45C
1) GDL 10BA ‐ 5% Teflonized Carbon Papers
60C
75C
T‐30C
T‐45C
T‐60C
T‐75C
Flux from hot ‐‐> cold
Temperature difference across the three layers / K
0
‐0,0001
0
2
4
‐0,0002
‐0,0003
‐0,0004
2) GDL 10AA ‐ 0% Teflonized b
Flux from cold ‐‐> hot
6
8
10
12
Thermal osmosis across membrane plus
porous transport layers:
Equation set for one interface.
Thermal conductivity
T  
 wA,o   wA,i (T A,o )
T A,o
 qw*s,Ai 
  0
Measurable heat of transfer?
1
 sa
J
'Ai
q
 qw*s,Ai J w 
Mass resistivity
Majsztrik et al. J Mem.Sci, 2007
T
sa
 q

R
ww J w
2
T
J w RT 2
5
8.31x3002

 (  4 )
 60 kJ/mol
i i
8
T c D
10  55.6 / 2  x1x10
*s,Ai
w
Diffusion, not self-diffusion may explain a reasonable heat of transfer
Seebeck coefficient and Peltier heats. Theory
From Onsager's relations:
 EF 
 T 
j 0
 J q' 
 H2
  

T
 j  T 0
Measure cell with hydrogen electrodes.
Calculate oxygen electrode performance from:
Relation of properties, formation cell (fc) and concentration cells:
 EF 
 EF 


F

dT
 T  H2  T  O2
dE fc
S fc  
H
T
2

O
T
2
Seebeck coefficient. Experiment
Half cells with
thermostatted jackets
To potentiometer
Gas outlets
Thermocouples
Pneumatic cylinder
3 Nafion 1110 membranes used
Seebeck coefficient. Experimental results
Electromotive force and temperature difference across the electrodes
Kjelstrup et al. Electrochim. Acta 2013
Seebeck coefficient. Experimental results.
Hydrogen, 1 bar electrodes
Stationary state for water
Nafion membrane
ETEK porous layers
0.67 ± 0.05 mV/K
1 0
 EF 
*
*



S
(
S
S

Pt )  67 J/K mol
H2
H
 T 
2
H 2  electrodes
SH*   1  5 J/K mol
Seebeck coefficient and Peltier heat
• Reaction entropy for formation of water (liquid)
∆S = -86 J/K mol
• Entropy removed from left hydrogen electrode:
H
1
 EF 
*
*
 



S
(
S
  S Pt )  67 J/K mol
H
H
T
2 2
 T  H2 electrodes
2
• Entropy removed from left oxygen electrode:
O
1
1
 EF 
*
*
 




S
S
(
S
  S Pt )
w
O
H
T
4 2 2
 T  O2 electrodes
2
 S 
H
T
2
 86  67  19 J/K mol
• Anode heat source: 350x(-67) J/mol = - 23.5 kJ/mol
• Cathode heat source : 350x(-19) J/mol = - 6.6 kJ/mol
Peltier heat effect under cell operation
Vie and Kjelstrup, Electrochim. Acta., 2005
Calculated profiles of heat flux, potential profile (example)
Kjelstrup, Røsjorde, J. Phys.Chem., 2005, Kjelstrup Bedeaux, World Scientific, 2008
Electric current densities:
500, 2500 and 5000 A/m2
Conclusions
• Gibbs excess variables and NET offer a systematic way to
approach transport phenomena in electrochemical cells
• The transported heat and interface is a fraction of the
enthalpy of the phase change. This can explain a large
thermal osmosis effect.
• The Peltier heat of the anode is much larger than that of
the cathode. Their difference amounts to the entropy
change of the cell reaction times the temperature.
Thank you for the attention!
Thanks to financial support from the Research Council of
Norway through programs: FRIENERGI, RENERGI,
NANOMAT, STORFORSK.
Non-equilibrium surfaces
• Time-dependent position and curvature of the surface
• One can still define excess densities
• One can also define excess fluxes along moving
surfaces
D. Bedeaux, A.M. Albano and P. Mazur, Physica A 82 (1976) 438-462
A.M. Albano, D. Bedeaux, J. Vlieger, Physica A 99 (1979) 293; 102 (1980) 105
D. Bedeaux, Adv. Chem. Phys. 64 (1986) 47-109
A.M. Albano and D. Bedeaux, Physica A 147 (1987) 407-435
S. Kjelstrup and D. Bedeaux, Non-Equilibrium Thermodynamics of Heterogeneous
Systems; World Scientific, 2008; Series on Advances in Statistical Mechanics, Vol. 16
Basic postulate: local equilibrium
• Each excess density is the same function of Ts,
μAs , μBs , R1 and R2 as in equilibrium
Local equilibrium of a Gibbs surface
under non-equilibrium conditions
Molecular dynamics simulations
A. Røsjorde, D.W. Fossmo, D. Bedeaux, S. Kjelstrup and B. Hafskjold, J. of Colloid and Int.
Science 232 (2000) 178-185; 240 (2001) 355-364
J-M. Simon, S. Kjelstrup, D. Bedeaux, and B. Hafskjold, J. Phys. Chem. B 108 (2004) 7186
J. Xu, S. Kjelstrup and D. Bedeaux, Phys. Chem. Chem. Phys. 8 (2006) 2017-2027
J. Ge, S. Kjelstrup, D. Bedeaux, J-M. Simon, B. Rousseaux, Phys. Rev. E 75 (2007) 061604
S. Kjelstrup, D. Bedeaux, I. Inzoli and J-M. Simon, Energy 33 (2008) 1185-1196.
The square gradient model
E. Johannessen and D. Bedeaux, Physica A 330 (2003) 354-372
The square gradient model for mixtures
K.S. Glavatskiy and D. Bedeaux, Phys. Rev. E 77 (2008) 061101, 1-15.
K.S. Glavatskiy and D. Bedeaux, Phys. Rev. E 79 (2009) 021608, 1-19.