PEM Fuel Cell Modeling and Simulation Using Matlab || Fuel Cell Charge Transport

CHAPTER 4

Fuel Cell Charge Transport

4.1 Introduction

The electrochemical reactions that occur in the fuel cell catalyst layers are one of the most important concepts to understand when trying to model a fuel cell. In addition to the activation losses, there are also losses during the transport of charge through the fuel cell. Electronic charge transport describes the movement of charges from the electrode where they are pro- duced, to the load where they are consumed. The two major types of charged particles are electrons and ions, and both electronic and ionic losses occur in the fuel cell. The electronic loss between the bipolar, cooling, and contact plates is due to the degree of contact that the plates make with each other due to the compression of the fuel cell stack. Ionic transport is far more difficult to predict and model than fuel cell electron transport. The ionic charge losses occur in the fuel cell membrane when H § ions travel through the electrolyte. This chapter will cover the fuel cell electronic and ionic charge transport and voltage losses due to transport resistance. The specific topics to be covered are:

�9 Voltage loss due to charge transport �9 Electron conductivity of metals �9 Ionic conductivity of polymer electrolytes

Charge transport resistance results in a voltage loss for fuel cells called ohmic loss. Common methods of reducing ohmic losses include making electrolytes as thin as possible, and employing high conductivity materials that are well connected to each other.

4.2 Voltage Loss Due to Charge Transport

Every material has an intrinsic resistance to charge flow. The material's natural resistance to charge flow causes ohmic polarization, which results in a loss in cell voltage. All fuel cell components contribute to the total

78 PEM Fuel Cell Modeling and Simulation Using MATLAB |

electrical resistance in the fuel cell, including the electrolyte, the catalyst layer, the gas diffusion layer, bipolar plates, interface contacts, and terminal connections. The reduction in voltage is called "ohmic loss," and includes the electronic (Reler and ionic (Rionic) contributions to fuel cell resistance. This can be written as:

Dohmic = iRohmic = i(Relec + Rionic) (4-1)

Rionic dominates the reaction in Equation 4-1 because ionic transport is more difficult than electronic charge transport. Rionic represents the ionic resistance of the electrolyte, and RebeC includes the total electrical resistance of all other conductive components, including the bipolar plates, cell interconnects, and contacts.

The material's ability to support the flow of charge through the material is its conductivity. The electrical resistance of the fuel cell components is often expressed in the literature as conductance(a), which is the recipro- cal of resistance:

i o" = (4 -2 )

Rohmic

where the total cell resistance (Rohmic) is the sum of the electronic and ionic resistance. Resistance is characteristic of the size, shape, and properties of the material, as expressed by Equation 4-3:

R = Lc~ (4-3) O'Acond

w h e r e Lcond is the length (cm)of the conductor, Acond is the cross-sectional area (cm ~) of the conductor, and cr is the electrical conductivity (ohm -1 cm-1). The current density, j, (A/cm2), can be defined as:

i j = (4 -4 )

Acen

or

j = ncarriersqVdrift = 0"~ (4-5)

where Acell is the active area of the fuel cell, ncarriers is the number of charge carriers (carriers/cm3), q is the charge on each carrier (1.6 x 10 -19 C), Vd~ift is the average drift velocity (cm/s)where the charge carriers move, and ~ is the electric field. The general equation for conductivity is:

V o" = nq-~ (4-6)

Fuel Cell Charge Transport 79

v The term -~ can be defined as the mobility, U i. A more specific

equation for material conductivity can be characterized by two major factors: the number of carriers available, and (2) the mobility of those carriers in the material, which can be written as:

O'i = (]Zi] * F)* c i * u i (4-7)

where G i is the number of moles of charge carriers per unit volume, u~ is the mobility of the charge carriers within the material, z~ is the charge number (valence electrons) for the carrier, and F is Faraday's constant.

Fuel cell performance will improve if the fuel cell resistance is decreased. The fuel cell resistance changes with area. When studying ohmic losses, it is helpful to compare resistances on a per-area basis using current density. Ohmic losses can be calculated from current density using Equa- tion 4-8:

1)ohmic = j(ASRohmic) = j(AcellRohmic) (4-8)

where ASRohmic is area-specific resistance of the fuel cell. The conduction mechanisms are different for electronic versus ionic conduction. In a metal- lic conductor, valence electrons associated with the atoms of the metal become detached and are free to move around in the metal. In a typical ionic conductor, the ions move from site to site, hopping to ionic charge sites in the material. The number of charge carriers in an electronic conductor is much higher than an ionic conductor. Electron and ionic transport is shown in Figures 4-1 and 4-2.

FIGURE 4-1. Electron transport in a metal.


[ Rest of Polymer Membrane I

Sulfonic Acid Group

+

/ i _ Water Molecule

FIGURE 4-2. Ionic transport in a polymer membrane.

Therefore, with increasing land area, or decreasing channel area, the contact resistance losses will decrease and the voltage for a given current will be higher. This concept is illustrated in Figure 4-3.

As mentioned previously, one of the most effective ways for reducing ohmic loss is to either use a better ionic conductor for the electrolyte layer, or a thinner electrolyte layer. Thinner membranes are advantageous for PEM fuel cells because they keep the anode electrode saturated through "back" diffusion of water from the cathode. At very high current densities (fast fluid flows), mass transport causes a rapid dropoff in the voltage, because oxygen and hydrogen simply cannot diffuse through the electrode and ionize quickly enough, therefore, products cannot be moved out at the necessary speed ~.

Since the ohmic overpotential for the fuel cell is mainly due to ionic resistance in the electrolyte, this can be expressed as:

•ohmic = iRohmic = jAcell( 6thick ] = j~thick GAfuelcell (7

(4-9)


where Acell is the active area of the fuel cell, ~thick is the thickness of the electrolyte layer, and rr is the conductivity. As seen from Equation 4-9 and Figure 4-4, the ohmic potential can be reduced by using a th inner electrolyte layer, or using a higher ionic conduct ivi ty electrolyte.

Table 4-1 shows a summary and comparison of electronic and ionic conductors and the fuel cell components that are classified under each type.

TABLE 4-1 Comparison of Electronic and Ionic Conduction for Fuel Cell Components

Materials Conductivity Fuel Cell Components

Electronic Conductors Metals 10 3 to 10 7

Semiconductors Ionic Conductors Solid/polymer electrolytes

10 -3 to 10 4

10 -1 to 103

Bipolar plates, gas diffusion layer, contacts, interconnects, end plates

Bipolar plates, end plates

PEMFC Nation electrolyte

Cell Voltage & Current Density Based Upon Land to Channel 1.05 . . . . .

I I I I I I I I I I I I

I I t I I 1 . . . . . . . . . 1 - - I . . . . . T 1 . . . . . 1 - I I I I I I

I I I

t ' lQ~ . . . . . . - . . . . . . r l . . . . . r I I I I I I I I 1

. . . . . . . . . . . . . . . . f- 7 . . . . . T

I I I I I I I I I

Lanc~ to Cha~nnel R ~ i o , 1 " I I

" ~ 1 I I I

0.25 " - ' - " - - 0 .50 . . . . . .

'" 0.75

0.7 1.0

1 25 �9 t I

0 .65 1 .50 . . . . . . �9 . . . . . �9 I I I I I 1.75 ,

I I I I

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Cell Current (A/cm 2)

0.9 A g}

o > 0 .85

O} ~ 0.8

0 > 0 .75

FIGURE 4-3. Cell voltage and current density based upon land to channel.


Cell Voltage & Current Densit, 1.051 ,,

~ . . . . . . . . . . . . . . . . . . . . . .

0.9

I

I

I

I- . . . .

I

I I

I I

I I

. . . . . . i . . . . . . L . . . . . I I

I I

I I

I I

I I

. . . . . k- . . . . . ~- . . . . . i- . . . . . . .

I

1

I

I [ . . . . . . . . . . . .

I I I I

. . . . . I - . . . . . I - . . . . . L . . . . .

A = 0.85 . . . . . . ,~

0 , I I u.o . . . . . . ,- . . . . .

I

I

t

f

. . . . . . I -

1

1

I

I

I 0.7 f

G)

o 0.75 >

0.65

0.6

f Due to Electrolyte Thickness i I I I I

. . . . . . . . . . . L _ I

I

I

I

I

. . . . . t - . . . . . t - - -

2 0

3 0

- - - - - - - 4 0

.......... 5 0

........ 6 0

- - - ................. 7 0

8 0

I I I

! Increasing Thiokness

. . . . . t . . . . . ,~ . . . . . I I

t . . . . . . . . . ' i . . . . . " . . . . . I I I

1 P . . . . . 1- I I I I I I I I I I I I L- . . . . . I- . . . . . L . . . . . I-

I I I I

I I I I

I I I I

I I I I

I I I I I

. . . . . . . . . . . . I - . . . . . r . . . . . . F . . . . . F . . . . . F - -

I I I I I

I I I I I

I I I t I

I I I I I

I I I I I 0.55' 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Cell Current (AJcm 2)

FIGURE 4-4. Cell voltage and current density due to electrolyte thickness.

The total fuel cell ohmic losses can be written as:

Dohmi c j A ~ R iA[ 1~ le lc ] - - + + ( 4 - 1 0 ) O ' a A O ' e A GA

where 1 is the length or thickness of the material. The first term in Equa- tion 4-9 applies to the anode, the second to the electrolyte, and the third to the cathode. In the bipolar plates, the "land area" can vary depending upon flow channel area. As the land area is decreased, the contact resistance increases since the land area is the term in the denominator of the contact resistance:

Rcontact and Acontact = Land Area (4-11) Rcontact = acontact


EXAMPLE 4-1" Calculating the Ohmic Voltage Loss

Determine the ohmic voltage loss for a 100 cm 2 PEMFC that has an electrolyte membrane with a conductivity of 0.20 f~-l cm-~ and a thickness of 50 microns (pm). The current density is 0.7 A/cm 2 and Relec for the fuel is 0.005 f2. Plot the ohmic voltage losses for electrolyte thick- nesses of 25, 50, 75, 100, and 150 microns (pm).

First, calculate Rionic based upon electrolyte dimensions to calculate l)ohmi c. The current of the fuel cell is

I = iA = 0.7A/cm 2 x 100cm 2 = 70A

L 0.0050cm R . . . . 5 X 10-4~

o'A (0.10f2-1cm-1)*(100cm 2)

l)ohmic = I(Relec + R i o n i c ) - - 70A, (0.005f~ + 5 x 10-4f~)= 0.385V

If this equation is calculated for thinner and thicker membranes, one will notice that the ohmic loss is reduced with thinner membranes.

Using MATLAB to solve"


iiii! iiil;iii!! !;== . . . . . . : ill I!N

!!. ii

!~i!i~i~q~i~ii::i;::i::!i~ii:~iii!ii;i~iii:.! ::::ili::ii ~i~iii~i:!!ii::::i:::ii:i~::i:: :~ :.::: i:..: .i .... ~:i : : : . . : : ~ :~: ~. . :" : . ; ; . : : i i~i;!:;i::,:iii.i::iiiiii:iTi:.i::iiiii:!!i~:i::~i!~i~iiiii!~!

!~i!::ii?!i!!~!!:!i!%!i~!!~!:.~i~!!!:.:::i:,:,iiTi:.:: " : " . ':~-::: : : . . . . . " :. : i -:?:::~:;:'::!:~::~::.::~'.~.::.III.G;.:.::.~:~,::I.I-~

.... < ~ . . . . . . , . . . . . . . . . . . ~ . . . . . . . :<<.....,~::~: :.. :~;, :. . . . . ~ : . .: .:.. :.: :~..~: . . . . . . . . . .!,~,,, :i~::~i.i:::':::

~ . . . : - : . . . . . . . ~ . . . . . . ::::. : :: . . : : :~ : : ~ : : : ~, .... : . . . . . . . . . . ............. : : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..... . . . . . . . . . . . . . . . ======================~:..~......::~.::::~.::~:~

Figure 4-5 illustrates the ohmic loss as a function of electrolyte thickness when the current density is 0.8 A / c m 2 and the active area is 25 cm 2.

E X A M P L E 4-2: C a l c u l a t i n g the O h m i c V o l t a g e Loss

Calculate the ohmic voltage losses for two fuel cell sizes at a current density of 0.7 A/cm~: (a) A1 = 16 cm 2, R1 = 0.05 f2; (b) A2 = 49 cm 2, R~. = 0.02 f2; (c) for A = 1 to 100 cm 2, R~ = 0.05. Plot the ohmic loss as a function of fuel cell area.

(a)

ASR1 = R1A1 = ( 0 . 0 5 ~ ) ( 1 6 c m 2) = 0.8f2cm ~

The ohmic loss can be calculated as follows:


0.35 Ohmic Loss as a Function of Electrolyte Thickness

I I I I I I I I

L = 0 . 0 0 2 5

O. 3 I- - - - L=O. 0050

. . . . . . . . . . L = 0 . 0 0 7 5

............................ L=O.O01

0.25 - L=0 .015

A > v

r 0.2 t~ 0

- I

o

E 0. 5

0

T . . . . . - f

0.1t -

O. 051 _ ~ ~ ~ ~ _ j ~ c _ . . . . .

I

I t

. . . . . _L . . . . .

I

I I I I l I

- - I . . . . . . I - . . . . . / I

I I I I t I I I I I I i

J I

I I I

-1 . . . . . -"1 . . . . . . I I I I i I , \ : I I

Increasing, Electrol~/te I Thickness ,

4- . . . . . -I- . . . . . -1 . . . . . . I I I I I I I I I I I I I 1

O IP ~ ( t I

0 0.1 0.2 0 .3 0.4 0.5 0 .6 0.7 0 .8 0.9

Current Density (A/cm 2)

F I G U R E 4-5. O h m i c loss as a f u n c t i o n of e l e c t r o l y t e t h i c k n e s s .

1 ) o h m i c t = j(ASR,) = 10.7cm ~S )0.8~cm2 = 0.56V

C o n v e r t the current dens i t i e s into fuel ce l l s w i t h currents:

A il = jA] - 0.7 * 1 6 c m z = 11 .2A

c m 2

T h e o h m i c vo l tage lo s ses are

l ) o h m i c 1 " - il(R1) - 11.2A* 0.05f~ = 0.56V


(bl

ASR2 = R2A2 = (0.02f~)(49cm ~) = 0.98f~cm 2

The o h m i c loss can be ca lcula ted as follows:

1)~ = j(ASR1)= ( 0"7 cm 2 A ) 0.98f~cm~ - 0 .686V

Conve r t the cur ren t densi t ies in to fuel cells w i t h currents :

A il = jA1 = 0.7 * 49cm 2 = 34.3A

cm 2

The o h m i c vol tage losses are

Dohmic 2 = i2(R2) = 34 .3A* 0.02f~ = 0.686V

Us ing MATLAB to solve:


......... iliiiiiiiii

U:i.:;,i. : . : . i : . . . . . . . . . . . . . . . " . <

!iiii:;;; iiiiii!ii!i~i',i! ~: i'::i::~ i Y ? ;:ii~,i i iii iili iil ii!iii~iiiiii~iq::i!;i!i!i~iiiii:i~!i ~i: i,;,: ~i~!;~i!~,~i~;:: !: :i;ii~i i~! ~iii~ iii'~ill iiiiiii ii~!;~!i!ii~i

The plot of the ohmic losses as a function of fuel cell area for Example 4-2 is shown in Figure 4-6.


3.5

2.5

>

r 2 m o _J 0 "~ 1.5

@

0.5

,

o

Ohmic Loss as a Function of Fuel Cell Area i I I I I I I / I I I i I i I ~ " t I I I I I I f " I I I I I I I ~ " I I I I I I ~ t i I i i i ~ ' l 4- 4- 4 -4 - 4 - - I - - I - t-- - - ~ - - t-- i I I I I I I I ~ I I I I I I i I I f I t I I I I i i ~ I I I I I i I I ~ " i I I I I i I i I / I I I I I I I I I ~ " I I

t 1- - t -1 -1 - - I - - I ~ - t - t-- I I I I I I ~ I I I I I i I I ~ " I I I I I I i I I ~ " I I i I I I i I I f " I I I 1 I t t t ~ t 1 t I I I I I ~ ' 1 i I I

. . . . . . T T -I" - r - 1 - - ~ ' - - I - I I - I - I I I I I ~ " I I I I I I I f " I I I I I I ~ I I I I I I . / I I I I I I I ~ " I I I I I I ~ i I

. . . . . . T T . . . . . -T . . . . . - 1 ~ " - - I r - i i i i i i i i i i i I i i i i

. . . . . . T ]- . . . . I I I I I I

.

/[ 10 20 30 40 50 60 70 80 90 100

Fue l Cel l Area ( c m 2)

FIGURE 4-6. Ohmic loss as a function of fuel cell area.

4.3 E lec tron C o n d u c t i v i t y of M e t a l s

The electronic conductivity of the metals used in a fuel cell is important because it affects the charge transfer of electrons. Fuel cell components that are typically made of metal include the flow field plates, current collectors, and interconnects. A c o m m o n expression for the mobil i ty of free electrons in a metal conductor can be written as:

q17 u - (4-12)

me

where �9 gives the mean free t ime between scattering events, m~ is the mass of the electron (m = 9.11 x ] 0 TM kg), and q is the elementary electron charge in coulombs (q = 1.68 x 10 -~9 C).

Inserting Equation 4-12 into the equation for conductivity (4-7):

0 " - ] Z e ] c ' e q ~ " ( 4 - 1 3 )

me


Carrier concentration in a metal can be calculated from the density of free electrons 2. Each metal atom contributes approximately one electron.

4.4 Ionic Conductivity of Polymer Electrolytes

Ionic transport in polymer electrolytes follows the exponential relationship:

oT = o'0 e-EdkT (4-14)

where r~0 represents the conductivity at a reference state, and E~ is the activation energy (eV/mol). As seen in Equation 4-14, the conductivity increases exponentially with increasing temperature.

A good conductive polymer should have a fixed number of charge sites and open space. The charged sites have a negative charge, and provide a temporary resting place for the positive ion. Increasing the number of charged sites raises the ionic conductivity, but an excessive number of charged side chains may reduce the stability of the polymer. In addition, increasing the free volume in the polymer allows more space for the ions to move. In polytetrafluoroethylene (PTFE)-based polymer membranes like Nation, ions are transported through the polymer membrane by hitching onto water molecules that move through the membrane. This type of membrane has high conductivity and is the most popular membrane used for PEM fuel cells. Nation has a similar structure to Teflon, but includes sulfonic acid groups (SO3-H +) that provide sites for proton transport. Figure 4-7 shows the chemical molecule of Nation.

[(C F2~C F2)m--'C F-'-'C F2]n I

0 I CF= I CF---CF3 I

0 I CFz I CF2 I SO3H

FIGURE 4-7. Chemical structure of Nation.


The conductivity of Nation is dependent upon the amount of hydration, and can vary with the water content. Hydration can be achieved by humidifying the gases or by relying upon the water generated at the cathode. In the presence of water, the protons form hydronium complexes (HaO+), which transport the protons in the aqueous phase. When the Nation is fully hydrated, its conductivity is similar to liquid electrolytes.

The volume of Nation can increase up to 22% when fully hydrated 3. Since the conductivity and the hydration of the membrane are correlated, the water content can be determined through membrane conductivity. The humidity can be quantified through water vapor activity awater yap:

awater_vap = Pw (4-15) Psat

where Pw represents the partial pressure of water vapor in the system, and P~,t represents saturation water vapor pressure for the system at the temperature of operation 4.

The amount of water that the membrane can hold also depends upon the membrane pre-treatment. For example, at high temperatures, the water uptake by the Nation membrane is much lower due to changes in the polymer at high temperatures. The relationship between water activity on the faces of the membrane and water content can be described by:

= 0.043 + 17.18awater_vap -39.85(awater_vap) 2 + 36(awater_vap) 3 14-16)

Water uptake results in membrane swelling, which changes the membrane thickness along with its conductivity. Springer et al. s correlated the ionic conductivity (rr) (in S/cm) to water content and temperature with the following relation:

6 = (0.005139~- O.O0326)expI12681 1 303

1/1 ,41,, T

Since conductivity is proportional to resistance, the resistance of the membrane changes with water saturation and thickness. The total resistance of a membrane (Rm) is found by integrating the local resistance over the membrane thickness:

tm dz 14-18) t m = I O'[/~(z}]

o

where tm is the membrane thickness, ~, is the water content of the membrane, and rr is the conductivity (S/cm) of the membrane. Since the protons typically have one or more water molecules associated with them, the conductivity and hydration both change simultaneously. This phenomenon


of the number of water molecules that accompanies each proton is called the electroosmotic drag (ndrag), which is:

..Sat /~ (4-19) ndrag = lldrag 22

where ..s~t is the electroosmotic drag (usually between 2.5 + 0.2), and ;t is ttdrag the water content (which ranges from 0 to 22 water molecules per sulfonate group, and when X = 22, Nation is fully hydrated). The water drag flux from the anode to the cathode with a net current j is6:

~

JH20,drag = 2ndrag 2F (4-20)

where JH~o,d~ag is the molar flux of water due to the electroosmotic drag (mol/scm2), and j is the current density of the fuel cell (A/cm2).

The electroosmotic drag moves water in the fuel cell from the anode to the cathode. Since the reaction at the cathode produces water, it tends to build up at the cathode, and some water travels back through the membrane. This is known as "back diffusion," and it usually occurs because the amount of water at the cathode is many times greater than at the anode. The water back-diffusion flux can be determined by:

JHxO, backdiffusion = Pdry D;t d& (4-21 ) M I dx

where Pdry is the dry density (kg/m 3) of Nation, Mn is the Nation equivalent weight (kg/mol), Dz is the water diffusivity, and z is the direction through the membrane thickness.

The total amount of water in the membrane is a combination of the electroosmotic drag and back diffusion, and can be calculated using Equa- tion 4-22:

JH20,backdiffusion = 211drag"SAT iF 22/~ Pdry Dz(;t)d,K (4-22) Mm dz

The concepts introduced by Equations 4-15 to 4-22 are illustrated by Example 4-3.

EXAMPLE 4-3: Calculating the Ohmic Voltage Loss Due to the Membrane

A hydrogen fuel cell operates at 80~ at 1 atm. It has a Nation 112 membrane of 50/~m, and the following equation can be used for the water content across the membrane: )~(z)= 5 + 2exp(100z). This fuel cell has a current density of 0.8 A/cm ~, and the water activites at the anode and


cathode are 0.8 and 1.0, respectively. Estimate the ohmic overvoltage loss across the membrane.

Convert the water activity on the Nation surfaces to water con- tents"

Xa - 0.043 + 17.18 '0 .8 - 39.85*0.82 + 36*0.83 = 7.2

,~ = 0.043 + 17.18 * 1 - 39.85 * 1 ~ + 36 * 13 - 14.0

Using these values as boundary conditions, Equation 4-23 can be arranged to create:

o / ~zz 2Fp-~Dz

E / 1 1)1 - "2 .563 - Dz 10 -6 exp 2416 303 353

0.33,10 + 0.0264,102 - 0.000671,10 3

11~, ( 0"7 A 1"0 kt ) + C exp cm 2 mol * 2.5

2.5 22 ,96500 C 0.00197 kg 3.81,10_6cm 2 m o l c m 3 s

;t(z) - 4 .4a + 2.30 �9 exp(109.8z)

C is determined from the boundary conditions where Z(0)= 7.2 and ~,(0.0125)- 14.0, and ~, varies across the membrane.

The conductivity profile of the membrane is

a(z) = 0.005193(5 + 2 exp (100z) - 0.003 26) x exp 1268( 1 303 333

o~z} - 0.04107 + 0.018 78exp(100z)

The resistance of the membrane is

tm d z 000S0

e m - ! - ! alZ(z))

dz 0.04107 + 0.018 78 exp (100z)

= 0.15 f~cm ~

The ohmic overvoltage due to the membrane resistance in this fuel cell is

Voh m - j x R m = 0.8A/cm 2 x 0.15f~cm 2 - 0.12V

Using MATLAB to solve:



4-9. The figures generated in Example 4-3 are illustrated by Figures 4-8 and

Chapter Summary

The transport of charges through, the fuel cell layers (except the membrane) occurs through conduction. Therefore, ohmic losses occur due to the lack of proper contact by the gas diffusion layer, bipolar plates, cooling plates, contacts, and interconnect. However, the largest ohmic loss occurs during the transport of ions through the membrane. To decrease the ionic losses through the membrane, either the membrane needs to become more conductive or the membrane needs to become thinner. It is usually easier to make the membrane thinner because developing high conductivity electrolytes is very challenging. The challenge occurs in creating a material that not only is highly conductive, but also stable in a chemical environment and able to withstand the required fuel cell temperatures. The electrolyte equations presented in this chapter are applicable for Nation, but if another type of electrolyte is employed, the equations may need to be altered to suit the chemistry.

16

15

~ " 14 0

~ lO

8

Membrane Thickness and Water Content I I I I I I I I I I I I I i I I I 1 i I

l t l i / l i l l i f l l i i i / l J l i i / I

L L - I ~ - - / .... -I i ~ I i i j i l I I l i / I i l i a ~ f I i I l i l#f" ' L L I_ _I j~r_ i i i i l i i i ~#" i i l i J j#" i i i i i j#" i i i i i ~#- l r- 1- T "1" ~ ' - -'r I J I I ~ " I I I I I ~ " I I I 1 I j # " I I I I / I I - 1-- 4- . ~ - I - 4 I I I ~ " I I I I I i I I t I i / I I i I I ~ " I I L L •163 Z _i I i ~ I l I I ~ I I l I I ~ l I I l I ~ l I I I I I- I I I I

_

l l I l I i l i l I

0.01 0.012 7 I t 1 I

0 0.002 0.004 0.006 0.008

Membrane Thickness (cm)

FIGURE 4-8. Membrane thickness and water content.

0.15

0.14

A

I::: 0.13

0.12 >, ,0- , I m e

>

:= 0.11 0

1 0 C 0 . 1 0 o �9 ~ 0.09 @

0 �9 - J 0 . 0 8

0.07

Membrane Thickness and Local Conductivity I I I I I I I I

t I , ,

t I I I

I I I I

- I - ..-I - I -1 J - - - - i ~ i i I j I i i i I = / I

i I i / I I I I 1 i I I

_ t _ _~ _0 . . . . . . . . . _1 . , f _ _ _L I I I I ~ " I I I I I i I I I I I i I i I I I ~ " I

_l_ _I J ......... J ~ Z I l i ~ I i l i il l i i i ~ i i l i l ~ l i l l l i l I

- - I - --I -3 - - - 7 . . . . . - f -T I I I ~ " I I I I ~ I i I I ~ I I I I I . ~ " I I I

- - I - - --t I I I I

I I I I

0.06 i i J 0 0.002 0.004 0.006 0.008

Membrane Thickness (r FIGURE 4-9. Membrane thickness and local conductivity.

I

0.01 0.012


Problems

�9 A 10-cm 2 fuel cell has Relec = 0.01 f~ and O'electrolyte = 0.10 ~"~-1 cm-1. If the electrolyte is 100/./m thick, predict the ohmic voltage losses for the fuel cell at j= 500 mA/cm 2.

�9 Estimate the ohmic overpotential for a fuel cell operating at 70 ~ The external load is 1 A/cm 2, and it uses a 50-pm-thick membrane. The humidity levels aw, a~ode and aw,r are 1.0 and 0.5, respectively.

�9 A fuel cell is operating at 0.8 A/cm 2 and 60~ Hydrogen gas at 30~ and 50% relative humidity is provided to the fuel cell at a rate of 2 A. The fuel cell area is 10 cm 2, and the drag ratio of water molecules to hydrogen is 0.7. The hydrogen exhaust exits the fuel cell at 60 ~ and p = 1 atm.

�9 In a PEMFC, the water activities on the anode and cathode sides of a Nation 115 membrane are 0.7 and 0.9, respectively. The fuel cell is operating at a temperature of 60~ and 1 atm with a current density of 0.8 A/cm 2. Estimate the ohmic overvoltage loss across the membrane.

Endnotes

[1] Lin, B. Conceptual design and modeling of a fuel cell scooter for urban Asia. 1999. Princeton University, masters thesis.

[2] O'Hayre, R., S.-W. Cha, W. Colella, and F.B. Prinz. Fuel Cell Fundamentals. 2006. New York: John Wiley & Sons.

[3] Ibid. [4] Ibid. [5] Springer et al. Polymer electrolyte fuel cell model. J. Electrochem. Soc. Vol.

138, No. 8, 1991, pp. 2334-2342. [6] O'Hayre, Fuel Cell Fundamentals.

Bibliography

Barbir, F. PEM Fuel Cells: Theory and Practice. 2005. Burlington, MA: Elsevier Academic Press.

Li, X. Principles of Fuel Cells. 2006. New York: Taylor & Francis Group. Mench, M.M., C.-Y. Wang, and S.T. Tynell. An Introduction to Fuel Cells and

Related Transport Phenomena. Department of Mechanical and Nuclear Engineer- ing, Pennsylvania State University. Draft. Available at: http://mtrll.mne.psu .edu/Document/jtpoverview.pdf Accessed March 4, 2007.

Mench, M.M., Z.H. Wang, K. Bhatia, and C.Y. Wang. Design of a Micro-Direct Methanol Fuel Cell. 2001. Electrochemical Engine Center, Department of Mechanical and Nuclear Engineering, Pennsylvania State University.

Rowe, A., and X. Li. Mathematical modeling of proton exchange membrane fuel cells. J. Power Sources. Vol. 102, 2001, pp. 82-96.

Sousa, R., Jr., and E. Gonzalez. Mathematical modeling of polymer electrolyte fuel cells. Journal of Power Sources. Vol. 147, 2005, pp. 32-45.

You, L., and H. Liu. A two-phase flow and transport model for PEM fuel cells. J. Power Sources. Vol. 155, 2006, pp. 219-230.

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