Mathematical Modeling Of Fuel Cell.pdf

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Anode : H2

2H+ + 2e-

Cathode: 1/2O2+ 2H+ + 2eOverall : H2 + 1/2O2

H2O

H2O Courtesy from Rina Lum (IHPC)

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H2 + 1/2O2

H2O (l); ∆H = -286 kJ/mol - HHV

H2 + 1/2O2

H2O (g); ∆H = -242 kJ/mol - LHV

Parameter

#

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Higher Heating Value (HHV)

Lower Heating Value (LHV)

Enthalpy ∆H

-286 kJ/mol

-242 kJ/mol

Gibbs Free Energy ∆G

-237 kJ/mol

-229 kJ/mol

Reversible Voltage, Er

1.229 V

1.185 V

Theoretical Efficiency, εr

83%

94.5%

H2 + 1/2O2

H2O

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' Garcia, et al, 2004, JFSCT

Accounts for concentration variations of methanol across the anode backing layer (ABL), anode catalyst layer (ACL), and membrane

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Mass transport equations are combined with kinetic equations to calculate the cell voltage Anode and cathode over-potential

δ I Vcell = U oO2 − U oMeOH − η a + η c − m cell κ

Thermodynamic equilibrium potentials of oxygen reduction and methanol oxidation

ηa, ηc and Icell are dependent on

Ohmic drop across the membrane

concentration profile of methanol

1

&

Anode Backing Layer (ABL) B dN MeOH ,z

Differential mass balance Methanol flux (Fick’s Law of Diffusion) Governing Equation

dz

B N MeOH , z = − DB B d 2 c: MeOH ,z

dz Boundary Conditions:

z= : 0 z = zI

Analytical solution

=0

2

B dc MeOH ,z

dz

=0

B cMeOH = cb B cMeOH = c IB = K I c IA

B cMeOH =

K I c IA − cb

δB

z + cb

'

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'

Membrane Layer (M) Differential mass balance

M dN MeOH ,z

dz

Methanol flux (diffusion and electro-osmotic drag ) Governing Equation

M N MeOH , z = − DM

M d 2: cMeOH ,z

dz Boundary Conditions:

Analytical solution

=0

2

M dc MeOH ,z

dz

=0

M A z = z II c M = c = K c MeOH II II II z = z III c M MeOH ≈ 0

M c MeOH = K II c IIA

δB + δA − z +1 δM

I + ξ MeOH cell F Electro-osmosis drag coefficient, number of methanol molecules dragged by hydrogen ion in the membrane

1

&

'

Anode Catalyst Layer (ACL) Differential mass balance

A dN MeOH ,z

dz

r j = MeOH = − M MeOH 6F

Homogenous reaction Volumetric current density (A/cm3)

j = aI 0MeOH , ref

Methanol flux (Fick’s Law of Diffusion) Governing Equation

Boundary Conditions

A kcMeOH

A cMeOH + λeα Aη A F / RT

:

A N MeOH , z = −DA

DA

M d 2 c MeOH ,z

dz 2

=

j 6F

z = zI

A cMeOH = c IA

z = z II

A cMeOH = c IIA

eα Aη A F / RT

A dc MeOH ,z

dz

1

&

'

Anode Catalyst Layer (ACL) A Analytical solution cMeOH =

I cell z 2 + C1z + C2 12 Fδ A D A A A I cell δ B (δ B + δ A ) A (c II − c I )δ B + C2 = c I − δA 12 Fδ A D A A B N MeOH = N MeOH , z ,z A M N MeOH = N MeOH , z ,z

c IIA − c IA I cell (2δ B + δ A ) − C1 = δA 12 Fδ A D A Jump mass balance at interface of ACL

c IA =

c IIA =

z = z II

I cell δ B : I δ ) + δ M D A DB cb − (1 + 6ξ MeOH ) cell B 12 F 6F DB K I (δ A DM K II + δ M D A ) + δ B D A DM K II I I D A DB cb − δ A DB K I (1 + 12ξ MeOH ) cell − δ B D A (1 + 6ξ MeOH ) cell 2F 6F DB K I (δ A DM K II + δ M D A ) + δ B D A DM K II

δ A DM K II ( DB cb −

δM

z = z: I

' Cell current density

I cell =

δ B +δ A

δ B +δ A

δB

δB

j dz =

aI 0MeOH , ref

A kcMeOH

A cMeOH + λeα Aη A F / RT

eα Aη A F / RT dz

Obtain ηa for given value of Icell

Cathode Oxygen reduction

I cell + I leakage =

I 0O, 2ref

co 2 co 2 , ref

M eα cη c F / RT I leakage = 6 FN MeOH ,z

Obtain ηc for given value of Icell

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