A Mathematical Model for Hydrogen Production of a Proton ...s defense.pdf · Hydrogen PhotoelectrochemicalCells Equations Results Conclusions A Mathematical Model for Hydrogen Production

HydrogenPhotoelectrochemical Cells

EquationsResults

Conclusions

A Mathematical Model for Hydrogen Productionof a Proton Exchange Membrane

Photoelectrochemical Cell

Bryan Van Scoy, Josh Adams, Robert MoserDr. Young, Dr. Clemons, Dr. Kreider

University of Akron

April 7, 2011

Bryan Van Scoy, Josh Adams, Robert Moser Dr. Young, Dr. Clemons, Dr. KreiderA Mathematical Model for Hydrogen Production of a Proton Exchange


EquationsResults

Conclusions

Benefits of Hydrogen

Little or no emissions

Hydrogen engines more efficient than gasoline

Fuel cells available

Many ways to produce



EquationsResults

Conclusions

Ways to Produce Hydrogen

Natural gas

Coal

Biomass

Waste

Wind

Nuclear power

Sunlight



EquationsResults

Conclusions

Basic Cell Operation

Anode Water ChannelMembrane (PEM)Proton Exchange

+ −

Anode CL Cathode CL

Cathode Water Channel

H2OH2O

H+

hν

H2O2

e−

Vcell



EquationsResults

Conclusions



+ −

Anode CL Cathode CL


H2OH2O

H+

hν

H2O2

e−

Vcell

&%'$



EquationsResults

Conclusions

Nafion Membrane

Hydration Shell

Water Region

Polymer Backbone

x

SO−3 Charge



EquationsResults

Conclusions

Delta Functions

-800

-600

-400

-200

0

200

400

600

800

0 10 20 30 40 50 60

15 45C

harg

e D

ensi

ty (

C/m

3 )

Length (µm)



EquationsResults

Conclusions



+ −

Anode CL Cathode CL


H2OH2O

H+

hν

H2O2

e−

Vcell



EquationsResults

Conclusions



+ −

Anode CL Cathode CL


H2OH2O

H+

hν

H2O2

e−

Vcell

&%'$

&%'$



EquationsResults

Conclusions

Electrode Nanowire Array Assembly

Current ConductingSupport Scaffold

SiliconGermanium

Proton ExchangeMembrane (PEM)Silicon

Electrocatalyst (Pt)

Germanium

Electrocatalyst (Pt)

Anode Catalyst Layer Cathode Catalyst Layer

-

e−

Vcell

+



EquationsResults

Conclusions

Photograph of Nanowire Arrays



EquationsResults

Conclusions

Electrode Nanowire Array Assembly

Cathode

1 cm

1 cm

Mem

bran

e

Pitch (P)

ScaffoldThickness

Anode

(Pscaffold)

Default Lengths:LA = 15 µmLM = 30 µmLC = 15 µm

LM LCLA

y

x



EquationsResults

Conclusions

Symbol Description Symbol Description

A Surface area/volume ratio [m−1] pos Position of point-chargesc Speed of light [m/s] q Charge of a proton [C]D Diffusivity of protons [m2/s] R Gas constant [J/K·mol]Dw Diffusivity of water [m2/s] S Source/Sink termE Activation energy [J/mol] T Temperature [K]EW Equivalent weight of electrolyte

[kg/mol]W Molecular weight [kg/mol]

F Faraday constant [C/mol] V Volume [m3]h Planck constant [m2

·kg/s] V0 Equilibrium potential [V]Iν Radiant intensity [W/m2] η Overpotential [V]j Current density [A/m3] µ Mobility of protons [m2/V·s]J Flux ρ Density [kg/m3]kB Boltzmann constant [J/K] κ Thermal conductivity [W/m·K]L Length [m] σ Ionic conductivity [S/m]m Mass of an electron [kg] ǫ Permittivity [F/m]NA Avogadro constant [mol−1] ν Frequency of sunlight [Hz]

NSO

−

3

Number of SO−

3 charges χ Surface potential difference [J]

n Concentration of protons [mol/m3] φmetal Work function of metal [J]



EquationsResults

Conclusions

Governing Equation

Concentration of H+ 0 = ∇ · (D ∇n + µn ∇Φ) + S

Potential (CLs) 0 = ∇ · (σ ∇Φ) + S

Potential (Membrane) 0 = ∇ · (ǫ ∇Φ) + S

Water Content 0 = ∇ ·

(

ρmem

EWDmemw ∇λ

)

−∇ ·

(

ndjF

)

+ S

Temperature 0 = ∇ · (κ ∇T ) + S

D - Diffusivity of protons Dmemw - Diffusivity of water

n - Concentration of protons λ - Water contentµ - Mobility of protons nd - Electro-osmotic dragσ - Electrical conductivity j - Current densityΦ - Electric potential F - Faraday constantǫ - Permittivity κ - Thermal conductivityρmem - Density of membrane T - TemperatureEW - Equiv. weight of dry membrane



EquationsResults

Conclusions

Governing Equation

Concentration of H+ 0 = ddx(D dn

dx+ µn dΦ

dx) + S

Potential (CLs) 0 = ddx(σ dΦ

dx) + S

Potential (Membrane) 0 = ddx(ǫdΦ

dx) + S

Water Content 0 = ddx

(

ρmem

EWDmemw

dλdx

)

−ddx

(

ndjF

)

+ S

Temperature 0 = ddx(κdT

dx) + S

D - Diffusivity of protons Dmemw - Diffusivity of water

n - Concentration of protons λ - Water contentµ - Mobility of protons nd - Electro-osmotic dragσ - Electrical conductivity j - Current densityΦ - Electric potential F - Faraday constantǫ - Permittivity κ - Thermal conductivityρmem - Density of membrane T - TemperatureEW - Equiv. weight of dry membrane



EquationsResults

Conclusions

Current Density

jν =FIν

NA

mc2

h2ν2

(

1−φmetal + χ

hν

)

(Light)

japplied = iA0

[

exp

(

FηA

RT

)

− exp

(

−FηA

RT

)]

(Anode)

japplied = iC0

[

n

nrefexp

(

−FηC

RT

)

−n

nrefexp

(

FηC

RT

)]

(Cathode)



EquationsResults

Conclusions

Overpotentials

ηA =RT

Fsinh−1

(

japplied

2iA0

)

(Anode)

ηC = −RT

Fsinh−1

(

japplied

2iC0

nref

nC

)

(Cathode)

ηM =LM

σj (Membrane)

ηI = .05V0 (Interface)

V0 = 1.23− .9× 10−3(T − 298.15) (Equilibrium Potential)

φ0 = V0 + ηA − ηC + ηM + ηI (Cell Voltage)



EquationsResults

Conclusions

Other Equations

σ = (.5139λ− .326) exp

[

1268

(

1

303−

1

T

)]

(Conductivity)

D = 8× 10−10λ− 3.1× 10−9 (Diffusivity)

µ =Dq

kBT(Mobility)

RH2 =nC

nref

j

F

[

mol

m2 s

]

=nC

nref

j

F

WH2

ρH2

VC

P+ Pscaffold

[

L

s

]



EquationsResults

Conclusions

Electric Potential - Governing Equation

0 = (σΦx)x + S

0 = σΦxx + σxΦx + S

0 =σ

∆x2[Φi−1 − 2Φi +Φi+1] +

1

4∆x2[σi+1 − σi−1] [Φi+1 − Φi−1] + S



EquationsResults

Conclusions

Electric Potential - Matrix Equation

[1] Φi−1

+ [−2] Φi = −∆x2

σiS −

1

4σi(σi+1 − σi−1)(Φi+1 − Φi−1)

+ [1] Φi+1



EquationsResults

Conclusions

Electric Potential - Boundary Conditions

Left Boundary Anode/Membrane Membrane/Cathode Right Boundaryx = xA = 0 x = xAM x = xMC x = xC

ΦA = V0 + ηA − ηC ΦA = ΦM + ηI2 ΦM = ΦC + ηI

2 ΦC = 0+ηM + ηI ǫA∇ΦA · n̂ ǫM∇ΦM · n̂

= ǫM∇ΦM · n̂ = ǫC∇ΦC · n̂



EquationsResults

Conclusions


ǫ1dΦ1

dx= ǫ2

dΦ2

dx

ǫ1

2∆x[Φi−2 − 4Φi−1 + 3Φi ] =

ǫ2

2∆x[−3Φi + 4Φi+1 − Φi+2]



EquationsResults

Conclusions


[ǫ1] Φi−2

+ [−4ǫ1] Φi−1

+ [3(ǫ1 + ǫ2)] Φi = 0

+ [−4ǫ2] Φi+1

+ [ǫ1] Φi+2



EquationsResults

Conclusions

Concentration of Hydrogen - Governing Equation

nt = (Dnx + µnΦx)x + S

1

∆t[nk+1

i − nki ] =Di

2∆x2[(nki−1 − 2nki + nki−1) + (nk+1

i−1 − 2nk+1i + nk+1

i−1 )]

+1

8∆x2[Di+1 − Di−1][(n

ki+1 − nki−1) + (nk+1

i+1 − nk+1i−1 )]

+µi

2∆x2[nk+1

i − nki ][Φi−1 − 2Φi +Φi+1]

+µi

8∆x2[(nki+1 − nki−1) + (nk+1

i+1 − nk+1i−1 )][Φi+1 − Φi−1]

+1

8∆x2[µi+1 − µi−1][n

k+1i − nki ][Φi+1 − Φi−1]

+ Ski



EquationsResults

Conclusions

Concentration of Hydrogen - Matrix Equation

[

−r̃

2Di +

r̃

8(Di+1 + Di−1) +

r̃

8µi (Φi+1 − Φi−1)

]

nk+1i−1

+

[

1 + r̃Di −r̃

2µi (Φi+1 − 2Φi +Φi−1)−

r̃

8(µi+1 − µi−1)(Φi+1 − Φi−1)

]

nk+1i

+

[

−r̃

2Di −

r̃

8(Di+1 − Di−1)−

r̃

8µi (Φi+1 − Φi−1)

]

nk+1i+1

= nki +r̃

2Di (n

ki−1 − 2nki + nki−1) +

r̃

8(Di+1 − Di−1)(n

ki+1 − nki−1)

+r̃

2µin

ki (Φi−1 − 2Φi +Φi+1) +

r̃

8µi (Φi+1 − Φi−1)(n

ki+1 − nki−1)

−r̃

8nki (µi+1 − µi−1)(Φi+1 − Φi−1) + ∆t Sk

i



EquationsResults

Conclusions

Concentration of Hydrogen - Boundary Conditions

Left Boundary Anode/Membrane Membrane/Cathode Right Boundaryx = xA = 0 x = xAM x = xMC x = xCnA = n0 nA = nM nM = nC

~JA · n̂ = ~JM · n̂ ~JM · n̂ = ~JC · n̂ ~JC · n̂ = KMT [nC − n0]

~J = D ∇n − µn ∇Φ



EquationsResults

Conclusions


D1dn1

dx+ µ1n1

dΦ1

dx= D2

dn2

dx+ µ2n2

dΦ2

dx

D1

2∆x[ni−2 − 4ni−1 + 3ni ] +

µ1ni

2∆x[Φi−2 − 4Φi−1 + 3Φi ]

=D2

2∆x[−3ni + 4ni+1 − ni+2] +

µ2ni

2∆x[−3Φi + 4Φi+1 − Φi+2]



EquationsResults

Conclusions


[D1] ni−2

+ [−4D1] ni−1

+ [3(D1 + D2) + µ1(Φi−2 − 4Φi−1 + 3Φi )

−µ2(−3Φi + 4Φi+1 − Φi+2)] ni = 0

+ [−4D2] ni+1

+ [D2] ni+2

[Di ] ni−2

+ [−4Di ] ni−1

+ [3Di + µi (Φi−2 − 4Φi−1 + 3Φi )− 2KMT∆x ] ni = −2KMTn0∆x

+ [−4D2] ni+1

+ [D2] ni+2



EquationsResults

Conclusions

Temperature - Governing Equation

0 = (κTx)x + S

0 = κTxx + S

0 =κ

∆x2[Ti−1 − 2Ti + Ti+1] + S



EquationsResults

Conclusions

Temperature - Matrix Equation

[κ] Ti−1

+ [−2κ] Ti = −∆x2 S

+ [κ] Ti+1



EquationsResults

Conclusions

Temperature - Boundary Conditions

Left Boundary Anode/Membrane Membrane/Cathode Right Boundaryx = xA = 0 x = xAM x = xMC x = xCTA = T0 TA = TM TM = TC TC = T0

∇TA · n̂ = ∇TM · n̂ ∇TM · n̂ = ∇TC · n̂



EquationsResults

Conclusions


κ1dT1

dx= κ2

dT2

dx

κ1

2∆x2[Ti−2 − 4Ti−1 + 3Ti ] =

κ2

2∆x2[−3Ti + 4Ti+1 − Ti+2]



EquationsResults

Conclusions


[κ1] Ti−2

+ [−4κ1] Ti−1

+ [3(κ1 + κ2)] Ti = 0

+ [−4κ2] Ti+1

+ [κ2] Ti+2



EquationsResults

Conclusions

Water Content - Governing Equation

0 =

(

ρmem

EWDw λx

)

x

−

(

ndj

F

)

x

+ S , nd =2.5

22λ

0 =ρmem

EW

Dwi

∆x2[λi−1 − 2λi + λi+1]

+ρmem

EW

1

4∆x2

[

Dwi+1 − Dwi−1

]

[λi+1 − λi−1]

−2.5

22

i

F

1

2∆x[λi+1 − λi−1]



EquationsResults

Conclusions

Water Content - Matrix Equation

[

ρmem

EW

(

Dwi−

Dwi+1 − Dwi−1

4

)

+∆x2.5

22

i

F

]

λi−1

+

[

−2ρmem

EWDwi

]

λi = −∆x2 S

+

[

ρmem

EW

(

Dwi+

Dwi+1 − Dwi−1

4

)

−∆x2.5

22

i

F

]

λi+1



EquationsResults

Conclusions

Water Content - Boundary Conditions

Left Boundary Anode/Membrane Membrane/Cathode Right Boundaryx = xA = 0 x = xAM x = xMC x = xCλA = λ0 λA = λM λM = λC λC = λ0

DwA∇λA · n̂ DwM

∇λM · n̂

= DwM∇λM · n̂ = DwC

∇λC · n̂



EquationsResults

Conclusions


Dw1

dλ1

dx= Dw2

dλ2

dx

Dw1

4 ∆x[λi−2 − 4λi−1 + 3λi ] =

Dw2

4 ∆x[−3λi + 4λi − λi+2]



EquationsResults

Conclusions


[Dw1 ] λi−2

+ [−4Dw1 ] λi−1

+ [3(Dw1 + Dw2)] λi = 0

+ [−4Dw2 ] λi+1

+ [Dw2 ] λi+2



EquationsResults

Conclusions

BC3 BC4 BC5

A2 A3 A4 A5

A1 A2 A3 A4 A5

A1 A2 A3 A4 A5 0

. . .. . .

. . .. . .

. . .

A1 A2 A3 A4 A5

BC1 BC2 BC3 BC4 BC5

A1 A2 A3 A4 A5

. . .. . .

. . .. . .

. . .

A1 A2 A3 A4 A5

BC1 BC2 BC3 BC4 BC5

0 A1 A2 A3 A4 A5

. . .. . .

. . .. . .

. . .

A1 A2 A3 A4

BC1 BC2 BC3

x1

x2

x3

x4

...

xA−1

xA

xA+1

...

xM−1

xM

xM+1

...

xC−1

xC

=

b1

b2

b3

b4

...

bA−1

bA

bA+1

...

bM−1

bM

bM+1

...

bC−1

bC



EquationsResults

Conclusions

Default Electric Potential

0

0.5

1

1.5

2

2.5

0 10 20 30 40 50 60

15 45P

oten

tial (

V)

Length (µm)



EquationsResults

Conclusions

Default Hydrogen Concentration

0

5

10

15

20

25

30

0 10 20 30 40 50 60

15 45H

+ C

once

ntra

tion

(mol

/m3 )

Length (µm)



EquationsResults

Conclusions

Default Temperature

352.98

353

353.02

353.04

353.06

353.08

353.1

353.12

353.14

353.16

353.18

0 10 20 30 40 50 60

15 45T

empe

ratu

re (

K)

Length (µm)



EquationsResults

Conclusions

Default Water Content

19.5

20

20.5

21

21.5

22

0 10 20 30 40 50 60

15 45W

ater

Con

tent

(m

ol S

O3- /m

ol H

20)

Length (µm)



EquationsResults

Conclusions

Effects of Temperature

0

0.5

1

1.5

2

2.5

0 10 20 30 40 50 60

15 45P

oten

tial (

V)

Length (µm)

T = 353KT = 333KT = 303K



EquationsResults

Conclusions

Effects of Temperature

16

17

18

19

20

21

22

0 10 20 30 40 50 60

15 45W

ater

Con

tent

(m

ol S

O3- /m

ol H

20)

Length (µm)

T = 353KT = 333KT = 303K



EquationsResults

Conclusions

Effects of Charges in Membrane

25.8

26

26.2

26.4

26.6

26.8

27

27.2

27.4

15 20 25 30 35 40 45

H+ C

once

ntra

tion

(mol

/m3 )

Length (µm)

NSO3- = 30

NSO3- = 100

NSO3- = 300



EquationsResults

Conclusions

Effects of Charges in Membrane

25

25.5

26

26.5

27

27.5

28

15 20 25 30 35 40 45

H+ C

once

ntra

tion

(mol

/m3 )

Length (µm)

pos = 0.5 pos = 0.25pos = 0.1



EquationsResults

Conclusions

Effects of Water Content

0

10

20

30

40

50

60

70

0 10 20 30 40 50 60

15 45

H+ C

once

ntra

tion

(mol

/m3 )

Length (µm)

λ0 = 22λ0 = 18λ0 = 12



EquationsResults

Conclusions

Effects of Mobility

0

20

40

60

80

100

120

140

160

180

0 10 20 30 40 50 60

15 45

H+ C

once

ntra

tion

(mol

/m3 )

Length (µm)

Full MobilityHalf MobilityNo Mobility



EquationsResults

Conclusions

Effect of Mass Transfer Coefficient

0

50

100

150

200

250

300

350

400

46 48 50 52 54 56 58 60

H+ C

once

ntra

tion

(mol

/m3 )

Length (µm)

KMT = .01KMT = 0KMT = ∞



EquationsResults

Conclusions

Hydrogen Production (Part I)

Test Case H2 Production[ml/min] % of Default

Default 5.9531 100.0 %Low Temperature (T = 333K) 5.8859 98.9 %Low Temperature (T = 303K) 6.0412 101.5 %λ0 = 18 6.8881 115.7 %λ0 = 12 9.4656 159.0 %100 SO−

3 and H+ Charges 5.9714 100.3 %300 SO−

3 and H+ Charges 6.0240 101.2 %pos = .25 6.0044 100.9 %pos = .1 6.0345 101.4 %KMT = 0 12.1445 204.0 %KMT = ∞ 5.5605 93.4 %



EquationsResults

Conclusions

Hydrogen Production (Part I)


Default 5.9531 100.0 %Low Temperature (T = 333K) 5.8859 98.9 %Low Temperature (T = 303K) 6.0412 101.5 %λ0 = 18 6.8881 115.7 %λ0 = 12 9.4656 159.0 %100 SO−

3 and H+ Charges 5.9714 100.3 %300 SO−

3 and H+ Charges 6.0240 101.2 %pos = .25 6.0044 100.9 %pos = .1 6.0345 101.4 %KMT = 0 12.1445 204.0 %KMT = ∞ 5.5605 93.4 %



EquationsResults

Conclusions

Hydrogen Production (Part II)


Default 5.9531 100.0 %Half Mobility 7.5597 127.0 %No Mobility 1.6420 27.6 %Iν = 0.6 mW/cm2 5.9869 100.6 %Iν = 1.2 mW/cm2 6.0556 101.7 %P = 5µm 9.6349 161.8 %P = 3µm 17.9736 301.9 %LA = LC = 10µm 2.4909 41.8 %LA = LC = 30µm 22.5301 378.5 %LM = 20µm 5.3653 90.1 %LM = 40µm 6.4290 108.0 %



EquationsResults

Conclusions

Hydrogen Production (Part II)


Default 5.9531 100.0 %Half Mobility 7.5597 127.0 %No Mobility 1.6420 27.6 %Iν = 0.6 mW/cm2 5.9869 100.6 %Iν = 1.2 mW/cm2 6.0556 101.7 %P = 5µm 9.6349 161.8 %P = 3µm 17.9736 301.9 %LA = LC = 10µm 2.4909 41.8 %LA = LC = 30µm 22.5301 378.5 %LM = 20µm 5.3653 90.1 %LM = 40µm 6.4290 108.0 %



EquationsResults

Conclusions

Significant Factors

Electrode surface area

Mass transfer coefficient between cathode and water channel

Input water concentration



EquationsResults

Conclusions

Significant Factors






EquationsResults

Conclusions

Significant Factors






EquationsResults

Conclusions

Future Work

Inclusion of water channels

Multi-dimensional

Non-linear channel flow

Optimal mobility and diffusivity

Transient response



EquationsResults

Conclusions

Future Work


Multi-dimensional



Transient response



EquationsResults

Conclusions

Future Work


Multi-dimensional



Transient response



EquationsResults

Conclusions

Future Work


Multi-dimensional



Transient response



EquationsResults

Conclusions

Future Work


Multi-dimensional



Transient response



EquationsResults

Conclusions

References

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cells, Journal of Membrane Science 185 (2001), no. 1, Sp. Iss. SI, 29–39 (English).

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J.M. Spurgeon, S.W. Boettcher, M.D. Kelzenberg, B.S. Brunschwig, H.A. Atwater, and N.S. Lewis,

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A Mathematical Model for Hydrogen Production of a Proton ...s defense.pdf · Hydrogen PhotoelectrochemicalCells Equations Results Conclusions A Mathematical Model for Hydrogen Production

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