Losses in Fuel Cells

Losses in Fuel Cells

Fuel Cell Thermodynamics

Thermodynamics

Nobel laureate Richard Feynmann said,

“ In physics today, we have no knowledge of what energy is.”

Self-consisting “laws” from fundamental assumptions based on human experience.

Q: How to “prove” Newton’s Law or Schrodinger Equation?

Internal energy (U)

No movement in macroscopic view, but microscopic movement exist.

• Kinetic energy

– Molecular movement &

vibration

• Chemical (potential) energy

Laws of Thermodynamics

First law: conservation of energy

d(Energy)

= d(Energy)

+ d(Energy)

= 0 d(Energy)

= -d(Energy)

dU = dQ – dW (dW)

= pdV dU = dQ – pdV

Second law: Entropy of a system increases (dS

≥0)

Microscopic: S = k log Ω

k: boltzmann’s constant

Ω: No. of possible microstates accessible to the system

* Two-property rule (Gibbs rule)

Two independent properties are sufficient to define the state of simple systems of constant mass (e.g. ideal gas law : PV=nRT)

Entropy Example I

a) S = k log Ω

b) Entropy is an extensive property (i.e. amount dependent).

S = S

+ S

= k log Ω

+ k log Ω

= k log (Ω

* Ω

)

Ω₁ Ω₁ Ω₁

a) S = S₁, Ω = Ω₁ b) S=2S₁, Ω = Ω₁ * Ω₁

Entropy Example II

a) A perfect crystal of 100 atoms: only 1 microstate (configuration)

S = k log Ω = k log 1 = 0

b) A crystal of 97 atoms with 100 lattice space:

Ω=

C

=1.6 * 10

Thermodynamic Potentials

Rules describe energy transfer form one form to another Let’s define four potentials U(S,V), H(S,p), G(T,p),F(T,V) for the

variables S, V, T, p

Starting from dU = dQ – pdV = TdS – pdV ⇔ U = U(S, V) or

(T, p) (S, p) (T, V)

Thermodynamics Potential

Enthalpy of Reaction

Heat of reaction dH = TdS + Vdp

⇔

dH = TdS = dQ (heat of reaction)

= dU + dW for p=const

For general reactions (molar based) aA + bB -> mM + nN

Δh

= { mΔh(M) + nΔh(N) } - { aΔh(A) + bΔh(B) }

similar for entropy: Δs

= {mΔs(M)+nΔs(N)}-{aΔs(A)+bΔs(B)}

Temperature dependency

RTP (room temperature & pressure) is defined as thermodynamic quantity at 298.15 K (so-called room temperature) and 1 bar (or 1 atm). All the

thermodynamic potentials defined at RTP are tabulated as reference, often denoted with superscript 0. (e.g. ΔH⁰)

Gibbs Free Energy

Gibbs Energy: potential under constant p & T

⇔ dG = dH – TdS – SdT

⇔ Δg = Δ h – TΔs (can be found from tables.)

Electrical work

dG = d(U+pV) – TdS – SdT

= dU – TdS – SdT + pdV + Vdp use dU = dQ-dW = d(TS) – dW

= SdT + TdS – (pdV + dW_elec)

⇔ dG = – dW_elec

⇔ W_elec = -Δg_rxn

This is valid for any T & p not-changing through reaction

Voltage of electric work

F: Faraday number, n: number of electrons involved in reaction

⇔ E = -Δg/nF

Gibbs Free Energy

Dimensional analysis

Standard Electrode Potential

Standard Electrode Potential

E

= ΣE

Cell potentials of most

reaction are tabulated

from reaction energy.

Standard Electrode Potential

Hydrogen fuel cell

Direct methanol fuel cell

* Note that half cell voltages of cathode for hydrogen fuel cell

and direct methanol fuel are identical even though

stoichiometry is different. (1/2 vs 3/2)

Standard Electrode Potential

Maximum Possible Efficiency

90 80 70 60 50 40 30 20 10 0

0 200 400

Carnot Limit Fuel Cell

600 800

Temperature / Celsius

Efficiency limit / %

1000

ε = 1 − T

T

ε= ΔG

ΔH =1−TΔS ΔH

Fuel Cell Efficiency

Temperature Effect

( )

(

)

) (

p const for

T nF T

E s E

nF s dT

dE

nFE g

dT s g d

dT S dG

Vdp SdT

dG

T

p p p

D - +

=

= D

÷ ø ç ö

è æ

-

= D D

-

÷ = ø ç ö

è

æ D

-

÷ = ø ç ö

è æ

+ -

=

As Δs is almost constant vs. temperature, E decrease with increasing temperature

Gibbs Free Energy

Lab environment (p=const, T=const)

Gibbs energy difference drives the spontaneous chemical reaction

Reaction:

A ⇔ B

Extent of a reaction ξ: amount of substahce that reacts from A to B and vice versa dξ=-dn_A=dn_B

ΔG=0

ΔG<0

ΔG>0

Concentration Effect: Nernst Equation

Concentration Effect: Nernst Equation

Example: Available Work at Constant T

1

0

1 1

0 0

0 0

0

0

0

0 1 0

0 1

0 1 0

0

or

ln ln

ln ln

V

V

V V

V V

W pdV pV nRT const p nRT

V

nRT dV

dV nRT

V V

V p

nRT nRT

V p

W g nRT p nRT p

p

= = = =

= =

= =

= -D = =

ò

ò ò

In general

Nernst Equation

Equation Nernst

: ln

) (

ln

ln )

( ) (

1 0

1 0

1

b B A

n N m M

b B A

n N m M

b B A

n N m M B

A N

M

p p

p p nF

E RT E

nFE g

p p

p RT p

g g

p p

p RT p

b m

m g

-

=

-

= D

+ D

= D

+ +

- +

=

D µ µ µ µ

nN mM

bB

A + Û + 1

÷÷ ø çç ö

è æ

P - P

P - P

=

i i

i

v v v

v

a a nF

E RT p or

p nF

E RT E

Reactants Product 0

Reactants Product

0

ln ln

Consider the following electrochemical reaction

Using the definition of the chemical potential

In general, Nernst Equation can be expressed as:

Nernst Equation: Examples

) ( 2 2

2 2

1

O g

H O

H + Û

(

)

nF E s

E_T = + D - Consider the hydrogen-oxygen reaction at 120C

Using the Nernst Equation

Q: At STP, how to calculate the voltage? (when water is liquid)

E^120C is calculated considering

temperature effect

) ( 2 2

2

2

1

O

H O

H + Û

A: For pure component, activity = 1 (read the textbook). Thus

Q: What about G when the product is water or vapor?

A: G will change by condensation energy (E⁰ from Lower Heating Value)

Concentration Cell

V

p p nF

E RT

E _x

H y H

296 . 0

100 ln10 96400

2

15 . 298 314

. 0 8

ln

8 0

2 2

=

× - ×

=

-

=

-

From Nernst Equation:

Q: How a fuel cell without chemical reaction is possible?

What is the driving force?

A: Electrochemical potential will answer your question.

.) . ( 2 .)

. (

2 h p

H

H Û

Electrochemical Potential

In concentration cell, concentration gradient will drive the migration (or diffusion) of hydrogen.

Previously, we found chemical potential is:

Thus, high pressure hydrogen has higher chemical potential. The difference in chemical potentials generates the voltages in the cell.

More conveniently, we may use “electrochemical potential” defined as

i i

i i

i i

i

µ z F f µ RT p z F f

µ ~ = + =

+ ln +

Electrochemical Potential

i i

i

i i i

i i

i

i i i

i i

F z f µ

n µ

n

µ n µ

n

D -

÷ = ø ç ö

è - æ

÷ ø ç ö

è Û æ

÷ = ø ç ö

è - æ

÷ ø ç ö

è æ

å å

å å

reactans products

reactans products

~ 0

~

At electrochemical equilibrium, the net change in the electrochemical potential is 0. (Remember Δg=0 at chemical equilibrium)

From this fact, you may re-derived the Nernst Equation as:

i i

i i

v v

v v e

p p nF

E RT

p p nF

RT nF

E g

Reactants Product 0

Reactants Product 0

ln

ln

P - P

=

P - P

- D

=

=

D f

1. Entropy loss

2. Internal loss in fuel cells: activation, ohmic, concentration losses

Heat Dissipation by Fuel Cells

(for HHV)

Mass Balance in Fuel Cells

If air at 20 mol/s is supplied to a hydrogen-air fuel cell that generates 1000 kA, we can find the oxygen output flux from the fuel cell:

Here, w_O2 represents the volume ratio of oxygen in air (=0.21)

Voltage (V)

iin

Current (A)

i E0

EH

V E_H-V

iout

Maximum Possible Efficiency Real Fuel Cell Efficiency

l i i V

E

E

Maximum Possible Efficiency Real Fuel Cell Efficiency

Example

If we A fuel cell jV is given as:

When what is heat dissipation from the fuel cell?

Assume HHV:

(For LHV, )

* No Stoichiometry number is considered.

Heat Dissipation in Fuel Cells

Alternatively,

Assume HHV:

Maximum Possible Efficiency

Reversible (Regenerative) Fuel Cells

2 2 2

2 2 2

Fuel Cell Mode: H + O1 H O + Electricity 2

Electrolyzer Mode: H O + Electricity H + O 1 2

®

®

,

ˆ

thermo electrolyzer

ˆ

h

h = D g

D

ˆ0 237.17

Dg = Dh^ˆ_HHV⁰ = 286

thermo,electrolyzer

286 1.2

e = 237 =

Fuel Cells Stacks

V

= V

× number of cells I

= I

P

= I

× V

= I

V

× number of cells

= P

× number of cells