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

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

Slide title

The catalyst – a nanoporous agglomerate

PEMFC Description

The microporous layers

Edge View

PEMFC Description

Another common design

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:

PEMFC

• T(x)

• Jq’(x)

• E(x)

Temperature profiles

Potential profiles

Lost work and accumulatedentropy production

The cell potential as a function of j Accumulated entropy production

cell

dSirr /dt

The problem:

Losses are unevenly distributed across the membrane! Cf. Lecture 10

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).

On a road towards the hydrogen society….

Where does H2 come from?• H2 from natural gas• Electrolysis of water• Geothermal sources

wind hydro geo