Design and optimization of polymer electrolyte membrane …cecas.clemson.edu/~mica/Publications/Ref_112.pdf · · 2004-04-02Design and optimization of polymer electrolyte membrane

Design and optimization of polymer electrolytemembrane (PEM) fuel cells

M. Grujicic*, K.M. ChittajalluDepartment of Mechanical Engineering, Clemson University, 241 Engineering Innovation Building, Clemson, SC 29634-0921, USA

Received 27 April 2003; received in revised form 27 April 2003; accepted 30 October 2003

Abstract

The performance of polymer electrolyte membrane (PEM) fuel cells is studied using a single-phase two-dimensional

electrochemical model. The model is coupled with a nonlinear constrained optimization algorithm to determine an optimum

design of the fuel cell with respect to the operation and the geometrical parameters of cathode such as the air inlet pressure, the

cathode thickness and length and the width of shoulders in the interdigitated air distributor. In addition, the robustness of the

optimum design of the fuel cell with respect to uncertainties in several electrochemical reaction and species transport parameters

(e.g., gas diffusivity, agglomerate particle size, etc.) is tested using a statistical sensitivity analysis. The results of the

optimization analysis show that higher current densities at a constant cell voltage are obtained as the inlet air pressure and the

fraction of the cathode length associated with a shoulder of the interdigitated air distributor are increased, and as the cathode

thickness and the length of the cathode per one interdigitated gas distributor shoulder are decreased. The statistical sensitivity

analysis results, on the other hand, show that the equilibrium cathode/membrane potential difference has the largest effect on the

predicted polarization curve of the fuel cell. However, the optimal design of the cathode side of the fuel cell is found not to be

affected by the uncertainties in the model parameters such as the equilibrium cathode/membrane potential difference. The results

obtained are rationalized in terms of the effect of the fuel-cell design on the air flow fields and the competition between the rates

of species transport to and from the cathode active layer and the kinetics of the oxygen reduction half-reaction.

# 2003 Elsevier B.V. All rights reserved.

PACS: 82.47.-a; 82.47.Gh

Keywords: Polymer electrolyte membrane (PEM) fuel cells; Design; Optimization; Robustness

1. Introduction

A fuel cell is an electrochemical energy conver-

sion device which is typically two to three times

more efficient than an internal combustion engine in

converting fuel to power. In a fuel cell, fuel (e.g.,

hydrogen gas) and an oxidant (e.g., oxygen gas

from the air) are used to generate electricity, while

heat and water are typical byproducts of the fuel-

cell operation. A fuel cell typically works on the

following principle: as the hydrogen gas flows into

the fuel cell on the anode side, a platinum catalyst

facilitates oxidation of the hydrogen gas which pro-

duces protons (hydrogen ions) and electrons

(Fig. 1a). The hydrogen ions diffuse through a

membrane (the center of the fuel cell which separates

Applied Surface Science 227 (2004) 56–72

* Corresponding author. Tel.: þ1-864-656-5639;

fax: þ1-864-656-4435.

E-mail address: [email protected] (M. Grujicic).

0169-4332/$ – see front matter # 2003 Elsevier B.V. All rights reserved.

doi:10.1016/j.apsusc.2003.10.035

the anode and the cathode) and, again with the help

of a platinum catalyst, combine with oxygen and

electrons on the cathode side, producing water. The

electrons, which cannot pass through the membrane,

flow from the anode to the cathode through an

external electrical circuit containing a motor or other

electric load, which consumes the power generated

by the cell. The resulting voltage from one single

fuel cell is typically around 0.7 V. This voltage can

be increased by stacking the fuel cells in series, in

which case the operating voltage of the stack is

simply equal to the product of the operating voltage

of a single cell and the number of cells in the stack.

Fuel cells are generally classified according to the

type of membrane (polymer electrolyte membrane

fuel cells (PEMFC), molten carbonate fuel cells

(MCFC), etc.) they use. One of the most promising

fuel cells are the so-called polymer electrolytic mem-

brane or proton exchange membrane (PEM) fuel cells.

The polymer electrolyte membrane is a solid, organic

polymer, usually poly(perfluorosulfonic) acid. The

most frequently used PEM is made of NafionTM

produced by DuPont, whose chemical structure con-

sists of three regions:

(1) A Teflon-like, fluorocarbon backbone containing

hundreds of repeating –CF2–CF–CF2– units.

(2) –O–CF2–CF–O–CF2–CF2–side chains which

connect the molecular backbone to the third

region.

(3) Ionic clusters consisting of sulfonic acid ions,

SO3� Hþ. The negative SO3

� ions are perma-

nently attached to the side chains and are

immobile. On the other hand, when the mem-

brane is hydrated by absorbing water, the

hydrogen ions combine with water molecules to

form hydronium ions which are quite mobile.

Nomenclature

cg total molar concentration of the gas-

phase (mol/m3)

caggH2

hydrogen concentration at the surface of

agglomerates (mol/m3)

caggO2

oxygen concentration at the surface of

agglomerates (mol/m3)

DaggH2

diffusion coefficient of hydrogen inside

the agglomerate (m2/s)

DaggO2

diffusion coefficient of oxygen inside

the agglomerate (m2/s)

ia exchange current density in the anodic

active layer (A/m2)

ic exchange current density in the cathodic

active layer (A/m2)

p pressure (Pa)

u gas velocity (m/s)

yH2molar fraction of hydrogen

yO2molar fraction of oxygen

Greek letters

fm electrolytic potential in the membrane

(V)

fs electronic potential in the electrodes (V)

Fig. 1. A schematic of: (a) a polymer electrolyte membrane (PEM)

fuel cell and (b) an interdigitated fuel/air distributor.

M. Grujicic, K.M. Chittajallu / Applied Surface Science 227 (2004) 56–72 57

Hydronium ions hop from one SO3� site to

another within the membrane and, thus, give rise

to the diffusion of protons making the hydrated

solid electrolytes like NafionTM excellent con-

ductors of hydrogen ions.

A schematic of the PEM fuel cell is given in Fig. 1a.

The anode and the cathode (the electrodes) are porous

and made of an electrically conductive material, typi-

cally carbon. The faces of the electrodes in contact with

the membrane (generally referred to as the active layers)

contain, in addition to carbon, polymer electrolyte and a

platinum-based catalyst. Each active layer is denoted by

athickvertical line inFig.1a.AsalsoindicatedinFig.1a,

oxidation and reduction fuel-cell half reactions take

place in the anode and the cathode active layer, respec-

tively. The PEM electrodes are of gas-diffusion type and

generally designed for maximum surface area per unit

material volume (the specific surface area) available for

the reactions, for minimum transport resistance of the

hydrogen and the oxygen to the active layers, for an easy

removal of the water from the cathodic active layer and

for theminimumtransport resistanceof theprotons from

theactive sites in theanodic layer to theactive sites in the

cathodic active layer.

As shown in Fig. 1a and b, a PEM fuel cell also

typically contains an interdigitated fuel distributor on

the anode side and an interdigitated air distributor on

the cathode side. The use of the interdigitated fuel/air

distributors imposes a pressure gradient between the

inlet and the outlet channels, forcing the convective

flow of the gaseous species through the electrodes.

Consequently, a 50–100% increase in the fuel-cell

performance is typically obtained as a result of the

use of interdigitated fuel/air distributors. The regions

of the interdigitated fuel/air distributors separating the

inlet and the outlet channels, generally referred to as

the shoulders, serve as the anode and cathode electric

current collectors.

Due to their high-energy efficiency, a low tempera-

ture (60–80 8C) operation, a pollution-free character,

and a relatively simple design, the PEM fuel cells are

currently being considered as an alternative source of

power in the electric vehicles. However, further

improvements in the efficiency and the cost are needed

before the PEM fuel cells can begin to successfully

compete with the traditional internal combustion

engines. The development of the PEM fuel cells is

generally quite costly and the use of mathematical

modeling and simulations has become an important

tool in the fuel-cell development. Over the last dec-

ades a number of fuel-cell models have been devel-

oped. Some of these models are single-phase (e.g.,

[1,2]) while the others are two-phase (e.g., [3]), i.e.,

they consider the effect of the liquid water supplied to

the anode and the one formed in the cathodic active

layer. Due to the slow kinetics of oxygen reduction,

some of these models focus only on the cathode end of

the fuel cell (e.g., [1,3]) while the others deal with the

entire fuel cell (e.g., [2]). Most of the models like the

ones cited above are used to carry out parametric

studies of the effect of various fuel-cell design para-

meters (such as the cathodic and anodic thicknesses,

the geometrical parameters of the interdigitated fuel/

air distributors, etc.). However, a comprehensive opti-

mization analysis of the PEM fuel-cell design is still

lacking. Hence, the objective of the present work is to

combine the single-phase two-dimensional PEM fuel-

cell model as presented in Ref. [2], with a mechanical

design optimization and design robustness analysis in

order to suggest the optimum fuel-cell design.

The organization of the paper is as following: In

Section 2, the single-phase two-dimensional model for

a PEM fuel cell and a solution method for the resulting

set of partial differential equations are briefly dis-

cussed. An overview of the optimization and the

statistical sensitivity methods is presented in Section

3. The main results obtained in the present work are

presented and discussed in Section 4. The main con-

clusions resulting from the present work are summar-

ized in Section 5.

2. The model

As indicated in Fig. 1a, the PEM fuel cell works on

the principle of separation of the oxidation of hydro-

gen (taking place in the anodic active layer) and the

reduction of oxygen (taking place in the cathodic

active layer). The oxidation and the reduction half-

reactions are given in Fig. 1a. Electrons liberated in

the anodic active layer via the oxidation half-reaction

travel through the anode, an anodic current collector,

an outer circuit (containing an external load, typically

a power conditioner connected to an electric motor), a

cathodic current collector, the cathode until they reach

58 M. Grujicic, K.M. Chittajallu / Applied Surface Science 227 (2004) 56–72

the cathodic active layer. Simultaneously, protons

(Hþ) generated in the anodic active layer diffuse

through the polymer electrolyte membrane until they

reach the cathodic active layer, where the oxygen

reduction half-reaction takes place.

2.1. Assumptions and simplifications

In this section a simple single-phase two-dimen-

sional steady-state model of a PEM fuel cell is pre-

sented. The model is developed under the following

simplifications and assumptions:

� The computational domain denoted with dashed

lines in Fig. 1a consists of three regions: (a) a

porous anode; (b) a polymer electrolyte membrane;

and (c) a porous cathode.

� The anode/membrane and the membrane/cathode

interfaces are considered as zero-thickness anodic

and cathodic active layers, respectively.

� A portion of the left edge of the anode and a portion

of the right edge of the cathode are designated as the

current collector surfaces.

� Humidified hydrogen of a fixed concentration and

pressure is supplied at the anode inlet while dry air

of fixed composition and pressure is supplied at the

cathode inlet.

� The water droplets generated in the anodic active

layer and transported through the porous cathode

are assumed to be of a negligible volume (i.e., no

two-phase flow is considered).

� The behavior of the H2 þ H2O gas mixture in

the anode and the O2 þ N2 gas mixture in the

cathode are considered to be governed by the ideal

gas law.

� The electrodes are considered as homogeneous

media in which the porosity and permeability are

distributed uniformly.

� Within the electrodes’ pores, the gas-phase is con-

sidered as a continuous phase and, hence, its

momentum conservation equation can be repre-

sented using the Darcy’s law.

� The electrochemical half-reactions taking place in

the active layers are assumed to be represented by

an agglomerate model based on the analytical solu-

tion of a diffusion-reaction problem in a spherical

porous agglomerate particle [4,5]. As indicated in

Fig. 2, the agglomerates consist of carbon particles

with embedded Pt-catalyst nano-particles and a thin

layer of the polymer electrolyte.

According to the agglomerate model [4,5], the

current density in the anodic active layer is defined as:

ia ¼ �K1ðcaggH2

� crefH2

expð�K2ðfs � fm � Dfeq;aÞÞÞ� ð1 � K3 coth K3Þ (1)

where

K1 ¼6dlð1 � eÞFD

aggH2

ðRaggÞ2(2)

K2 ¼ 2F

RT(3)

K3 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

i0;aS

2FcrefH2

DaggH2

sRagg (4)

Fig. 2. A schematic of the agglomerate model for the transport of gaseous species and charged particles in the anodic and cathodic active

layer.


An explanation of the quantities appearing in Eqs. (1)–

(4) as well as of those appearing in the subsequent

equations and the values of these quantities are given

in section entitled ‘Nomenclature’ and in Tables 1–4.

The quantities are listed in one of the four tables

depending on the regions (the anode, the anodic active

layer, the membrane, etc.) in which the quantity is

used.

The agglomerate model further provides the follow-

ing equation for the current density in the cathodic

active layer:

ic ¼ K4caggO2

ð1 �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK5 expð�K6ðfs � fm � Dfeq;cÞÞ

qÞ

� cothffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK5 expð�K6ðfs � fm � Dfeq;cÞÞ

q(5)

where

K4 ¼12dlð1 � eÞFD

aggO2

ðRaggÞ2(6)

K5 ¼ i0;cSðRaggÞ2

4FcrefO2

DaggO2

(7)

K6 ¼ 0:5F

RT(8)

Concentrations of the dissolved hydrogen and oxygen

at the surface of the agglomerate particles are defined

by the following equations:

caggH2

¼ pyH2

HH2

and caggO2

¼ pyO2

HO2

(9)

2.2. The dependent variables

The following dependent variables are used in the

present model.

In the anodic domain:

(a) the electronic potential, fs;

(b) the gas-phase pressure, p; and

(c) the hydrogen mole fraction in the gas-phase, yH2.

Table 1

General parameters used for modeling the PEM fuel cell

Parameter Symbol SI units Value

Faraday’s constant F A s/mol 96,487

Universal gas constant R J/mol K 8.314

Temperature T K 353

Atmospheric (reference) pressure p0 Pa 1.013 � 105

Table 2

Reference-case electrodes’ parameters used for modeling the PEM fuel cell


Potential at the anode current collector fcca V 0

Potential at the cathode current collector Vcell V 0.7

Dry porosity of the electrodes e No units 0.4

Electronic conductivity of the electrodes ks S/m 1000

Specific surface area of the electrodes S m2/m3 1.0 � 107

Permeability of the electrodes kP m2 1.0 � 10�13

Gas viscosity in the electrode’s pores Z kg/m/s 2.1 � 10�5

Molar fraction of hydrogen at the anode inlet yH2 ;in No units 0.6

Molar fraction of hydrogen at the anode outlet yH2 ;out No units 0.5

Molar fraction of oxygen at the cathode inlet yO2 ;in No units 0.21

Molar fraction of oxygen at the cathode outlet yO2 ;out No units 0.17

Henry’s concentration coefficient for hydrogen HH2Pa m3/mol 3.9 � 104

Henry’s concentration coefficient for oxygen HO2Pa m3/mol 3.2 � 104

Gas pressure at the anode inlet pa,in Pa p0 � 1.03

Gas pressure at the cathode inlet pc,in Pa p0 � 1.03

Gas pressure at the anode outlet pa,out Pa p0

Gas pressure at the cathode outlet pc,out Pa p0

Gas diffusive coefficient inside the electrode’s pores Dgas m2/s 1.0 � 10�5 � (e)1.5

Anode thickness ta m 0.00025

Cathode thickness tc m 0.00025

Electrode’s height he m 0.002


In the membrane:

(a) the electrolytic potential, fm.

In the cathodic domain:

(a) the electronic potential, fs;

(b) the gas-phase pressure, p; and

(c) the oxygen mole fraction in the gas-phase, yO2.

2.3. The governing equations

The governing equations for the anode, the mem-

brane and the cathode are given in Figs. 3–5, respec-

tively. They include:

For the anode:

(a) an electronic charge conservation equation;

(b) a gas-phase continuity equation; and

(c) a hydrogen mass balance equation.

For the membrane:

(a) an electrolytic charge conservation equation.

For the cathode:

(a) an electronic charge conservation equation;

(b) a gas-phase continuity equation; and

(c) an oxygen mass balance equation.

Table 3

Reference-case membrane parameters used for modeling the PEM

fuel cell


Electrolytic con-

ductivity

km S/m 9

Thickness tm m 0.0001

Height hm m 0.002

Fig. 3. Dependent variables, governing equations and boundary conditions for the anode in a PEM fuel cell. Please see the text for explanation

of the symbols.


2.4. The boundary conditions

The boundary conditions used are also summarized

in Figs. 3–5. The boundary conditions can be briefly

summarized as following.

For the anode:

(a) the composition of the H2 þ H2O gas and its

pressure and a (zero) current density are pre-

scribed at the anode input and the output;

(b) the electronic potential is set (arbitrarily) to zero

at the current collector (shoulder) where zero-

flux conditions are applied for the gas-phase and

the hydrogen;

(c) at the anodic active layer, the electronic current

density is defined by Eq. (1), while the H2 þ H2O

gas and the hydrogen fluxes are related with

the anodic current density as indicated in Fig. 3;

and

Table 4

Reference-case active layer parameters used for modeling the PEM fuel cell


Anodic exchange current density i0,a A/m2 1.0 � 105

Cathodic exchange current density i0,c A/m2 1.0

Reference oxygen concentration in the active layer crefO2

mol/m3 yO2 ;in � p0=HO2

Reference hydrogen concentration in the active layer crefH2

mol/m3 yH2 ;in � p0=HH2

Anode/membrane equilibrium potential difference Dfeq;a V 0

Cathode/membrane equilibrium potential difference Dfeq;c V 1

Dry porosity of electrode e No units 0.4

Volume fraction of polymer em No units 0.2

Agglomerate particle radius Ragg m 1.0 � 10�7

Active layer thickness dl m 1.0 � 10�5

Gas diffusive coefficient inside the agglomerate Dagg m2/s 1.2 � 10�10 � ((1 � e) � em)1.5

Fig. 4. Dependent variables, governing equations and boundary conditions for the membrane in a PEM fuel cell. Please see the text for

explanation of the symbols.


(d) at the remaining surfaces, zero-flux boundary

conditions are prescribed for the electronic

current density, the gas concentration and the

hydrogen mole fraction.

For the membrane:

(a) the current densities at the anodic active layer and

the cathodic active layer side are given by Eqs. (1)

and (5), respectively; and

(b) zero-flux current density conditions are applied at

the remaining sides of the membrane.

For the cathode:

(a) the composition of the O2 þ N2 gas and its

pressure and a (zero) current density are pre-

scribed at the cathode input and the output;

(b) the electronic potential is set equal to the cell

voltage at the current collector where zero-flux

conditions are applied for the gas-phase and the

oxygen;

(c) at the cathodic active layer, the electronic current

density is defined by Eq. (5), while the O2 þ N2

gas and oxygen fluxes are related with the

cathodic current density as indicated in Fig. 5.

At the remaining surfaces, zero-flux boundary con-

ditions are prescribed for the electronic current den-

sity, the gas concentration and the oxygen mole

fraction.

2.5. Computational method

The stationary, nonlinear two-dimensional system

of governing partial differential equations (discussed

in Section 2.3 and in Figs. 3–5) subjected to the

boundary conditions (discussed in Section 2.4 and

in Figs. 3–5) are implemented in the commercial

mathematical package FEMLAB [6] and solved (for

the dependent variables discussed in Section 2.2 and

in Figs. 3–5) using the finite element method.

Fig. 5. Dependent variables, governing equations and boundary conditions for the cathode in a PEM fuel cell. Please see the text for

explanation of the symbols.


The FEMLAB provides a powerful interactive envir-

onment for modeling various scientific and engineer-

ing problems and for obtaining the solution for the

associated (stationary and transient, both linear and

nonlinear) systems of governing partial differential

equations. The FEMLAB is fully integrated with the

MATLAB, a commercial mathematical and visualiza-

tion package [7]. As a result, the models developed in

the FEMLAB can be saved as MATLAB programs for

parametric studies or iterative design optimization.

3. Design optimization and robustness

There are many PEM fuel-cell parameters which

affect its performance. Some of these parameters such

as electrodes permeability and porosity are controlled

by the micro structure of the porous electrode mate-

rial, the others such as gas diffusivity inside the

agglomerate and the agglomerate radius are function

of the microstructure and morphology of the porous

material of the anodic and cathodic active layers.

Since these microstructure sensitive parameters are

mutually interdependent in a complex and currently

not well understood way, they will not be treated as

design parameters within the fuel-cell optimization

procedure described below. Instead, operation para-

meters (e.g., fuel and air inlet pressure, gas species

inlet concentrations) and the geometrical parameters

(e.g., electrodes and membrane thicknesses) will be

considered. As explained earlier, the kinetics of reduc-

tion half-reaction in the cathodic active layer is very

slow and, it is generally recognized that significant

improvements in the fuel-cell performance can be

obtained by optimizing the operation and the geome-

try of its cathode side. This approach is adopted in the

present work and following He et al. [3], the following

four cathode design parameters have been identified as

the most important:

(a) the inlet pressure of the air (1:03 atm ¼ 104,

365 Pa);

(b) the cathode thickness (0.00025 m);

(c) the cathode length per one shoulder segment of

the interdigitated air distributor (0.002 m); and

(d) the fraction of cathode length associated with the

shoulder of the interdigitated gas distributor

(0.5).

The numbers given within the parentheses corre-

spond to the values of the four design parameters in the

initial (reference) design of the PEM fuel cell.

A schematic of the cathode is given in Fig. 6 to

explain the four design parameters (denoted as x(i),

i ¼ 1�4) defined above.

The objective function f[x(1), x(2), x(3), x(4)] is

next defined as the electric current per unit fuel-cell

width (at a cell voltage of 0.7 V), which is a measure

of the degree of utilization of the expensive Pt-based

catalyst in the active layers of the electrodes. The

electric current per unit fuel-cell width is determined

as an integral of the current density over a distance

associated with one shoulder of the interdigitated air

distributor along the membrane/cathode interface in

the cathode length direction.

Thus, the fuel-cell design optimization problem can

be defined as:

minimize1

f ½xð1Þ; xð2Þ; xð3Þ; xð4Þ� with respect to xð1Þ; xð2Þ; xð3Þ and xð4Þ

subject to :

102; 313 Pa xð1Þ 111; 430 Pa

0:0002 m xð2Þ 0:001 m

0:001 m xð3Þ 0:004 m

0:3 xð4Þ 0:7

Fig. 6. Definition of the four cathode-base design parameters used

in the fuel-cell optimization procedure.


The lower and upper limits for the four design para-

meters are selected based on the parametric study of

PEM fuel cells carried out in the work of He et al. [3].

3.1. Optimization

The optimization problem formulated above

is solved using the MATLAB optimization toolbox

[7] which contains an extensive library of computa-

tional algorithms for solving different optimization

problems such as: unconstrained and constrained non-

linear minimization, quadratic and linear program-

ming and the constrained linear least-squares

method. The problem under consideration in the pre-

sent work belongs to the class of multidimensional

constrained nonlinear minimization problems which

can be solved using the MATLAB fmincon() optimi-

zation function of the following syntax:

fminconðfun; x0;A;B;Aeq;Beq;LB;UB; confun; optionsÞ;(10)

where fun denotes the scalar objective function of a

multidimensional design vector x, while confun con-

tains nonlinear non-equality (cðxÞ 0) and equality

(ceqðxÞ ¼ 0) constrained functions. The matrix A and

the vector b are used to define linear non-equality

constrains of the type Ax b, while the matrix Aeq and

the vector beq are used to define linear equality con-

straining equations of the type Aeqx ¼ beq. LB and UB

are vectors containing the lower and the upper bounds

of the design valuables and x0 is the initial design

point. The MATLAB fmincon( ) function implements

the Sequential Quadratic Programming (SQP) method

[8] within which the original problem is approximated

with a quadratic programming subproblem which is

then solved successively until convergence is achieved

for the original problem. The method has the advan-

tage of finding the optimum design from an arbitrary

initial design point and typically requires fewer func-

tion and gradient evaluations compared to other meth-

ods for constrained nonlinear optimization. The main

disadvantages of this method are that it can be used

only in problems in which both the objective function,

func, and the constrained equations, confun, are con-

tinuous and, for a given initial design point, the

method can find only a local minimum in the vicinity

of the initial design point.

3.2. Statistical sensitivity analysis

Once the optimum combination of the design para-

meter is determined using the optimization procedure

described above, it is important to determine how

sensitive is the optimal fuel-cell design to the variation

in the model parameters whose magnitudes are asso-

ciated with considerable uncertainty. In the field of

mechanical design, these parameters are generally

referred to as factors, and this term will be used

throughout this paper. To determine the robustness

of the optimum design with respect to variations in the

factors the method commonly referred to as the sta-

tistical sensitivity analysis [9] will be used in the

present work.

The first step in the statistical sensitivity analysis

is to identify the factors and their ranges of variation.

The selection of the factors and their ranges is

subjective and depends on engineering experience

and judgment and it is essential for proper formula-

tion of the problem. Typically two to four values

(generally referred to as levels) are selected for each

factor.

The next step in the statistical sensitivity analysis is

to identify the analyses (finite element computational

analyses in the present work) which need to be per-

formed in order to quantify the effect of the selected

factors. In general, a factorial design approach can be

used in which all possible combinations of the factor

levels are used. However, the number of the analyses

to be carried out can quickly become unacceptably

large as the number of factors and levels increases. To

overcome this problem, i.e., to reduce the number of

analyses which should be performed, the orthogonal

matrix method [9] will be used. The orthogonal matrix

method contains column for each factor, while each

row represents a particular combination of the levels

for each factor to be used in an analysis. Thus, the

number of analyses which needs to be performed is

equal to the number of rows of the corresponding

orthogonal matrix. The columns of the matrix are

mutually orthogonal, that is, for any pair of columns,

all combinations of the levels of the two factors appear

and each combination appear an equal number of

times. A limited number of standard orthogonal

matrices [10] is available to accommodate specific

numbers of the factors with various number of levels

per factor.


A computational finite element analysis is next

performed for each combination of the factor levels

as defined in the appropriate row of the orthogonal

matrix. The results of all the analyses are next

tabulated and the overall mean value of the objective

function calculated. The mean values of the objective

function associated with each level of each factor are

also calculated. As discussed earlier, each level of a

factor appears an equal number of times within its

column in the orthogonal matrix. The objective

function results associated with each level of a factor

are averaged to obtain the associated mean values.

The effect of a level of a factor is then defined as the

deviation it causes from the overall mean value and is

thus obtained by subtracting the overall mean value

from the mean value associated with the particular

level of that factor. This process of estimating the

effect of factor levels is generally referred to as the

analysis of means (ANOM). The ANOM allows

determination of the main effect of each factor,

however, using this procedure it is not possible to

identify possible interactions between the factors. In

other words, the ANOM is based on the principle of

linear superposition according to which the system

response Z (the objective function in the present

case) is given by:

Z ¼ overall mean þX

ðfactor effectÞ þ error (11)

where error denotes the error associated with the linear

superposition approximation.

To obtain a more accurate indication of the relative

importance of the factors and their interactions, the

analysis of variance (ANOVA) can be used. The

ANOVA allows determination of the contribution

of each factor to the total variation from the overall

mean value. This contribution is computed in the

following way: First, the sum of squares of the

differences from the mean value for all the levels

of each factor is calculated. The percentage that this

sum for a given factor contributes to the cumulative

sum for all factors is a measure of the relative

importance of that factor.

The ANOVA also allows estimation of the error

associated with the linear superposition assumption.

The method used for the error estimation generally

depends on the number of factors and factor levels as

well as on the type of the orthogonal matrix used in the

statistical sensitivity analysis. The method described

below is generally referred to as the sum of squares

method.

The sum of squares due to error, SSerror, can be

calculated using the following relationship:

SSerror ¼ SSgrand � SSmean � SSfactors (12)

where SSgrand is the sum of the squares of the results of

all the analyses, SSmean value is equal to the overall

mean squared multiplied by the number of analyses

and SSfactors is equal to the sum of squares of all the

factor effects. Each quantity in Eq. (12) is associated

with a specific number of degrees of freedom. The

number of degrees of freedom for the grand total sum

of squares, DOFgrand, is equal to the number of

analysis (i.e., the number of rows in the orthogonal

matrix). The number of degrees of freedom associated

with the mean value, DOFmean, is one. The number of

degrees of freedom for each factor, DOFfactor, is one

less than the number of levels for that factor. The

number of degrees of freedom for the error can hence

be calculated as:

DOFerror ¼ DOFgrand � 1 �X

ðDOFfactorÞ (13)

For Eq. (12) to be applicable, the number of degrees of

freedom for the error must be greater than zero. If the

number of degrees of freedom for the error is zero, a

different method must be used to estimate the linear

superposition error. An approximate estimate of the

sum of the squares due to error can be obtained using

the sum of squares and the corresponding number of

degrees of freedom associated with the half of the

factors with the lowest mean square.

Once the sum of squares due to the error and

the corresponding number of degrees of freedom for

the error have been calculated, the error variance,

VARerror, and the variance ratio, F, can be computed

as:

VARerror ¼SSerror

DOFerror

; (14)

and

F ¼ ðMEANfactorÞ2

VARerror

(15)

where MEANfactor is a mean value of the objective

function for a given factor. The F ratio is used to


quantify the relative magnitude of the effect of each

factor. A value of F less than one normally implies

that the effect of the corresponding given factor is

smaller than the error associated with the linear

superposition approximation and hence can be

ignored. A value of F above four, on the other hand,

generally suggests that the effect of the factor at hand

is quite significant.

4. Results and discussion

4.1. The reference case

The voltage versus the current density Tafel plot

(also known as the polarization curve) for the refer-

ence-case design of the fuel cell is shown in Fig. 7a.

The variation of the current density along the cath-

ode/membrane interface in the cathode-length direc-

tion as a function of the distance from the center of

the oxygen outlet at the cell voltage of 0.7 V is shown

in Fig. 7b. The results displayed in Fig. 7a and b are

as expected and show that: (a) at lower current

densities, the voltage losses are governed by the

sluggish kinetics of the oxygen reduction half-reac-

tion, while at larger current densities, the overall

process becomes controlled by the transport of spe-

cies to and from the cathode active layer. The onset

of the transport control regime is accompanied by an

exponential drop in the cell voltage; and (b) the

highest current densities are obtained near the oxy-

gen inlet where the concentration of oxygen in the

gas-phase (and hence at the cathode/membrane inter-

face) is the highest.

4.2. Fuel-cell design optimization

The optimum procedure described in Section 3.1

yielded the following optimal fuel-cell design para-

meters:

(a) the inlet pressure of the air: 111,458 Pa (the

upper bound);

(b) the cathode thickness: 0.0002 m (the lower

bound);

(c) the cathode length per one shoulder segment of

the interdigitated air distributor: 0.001 m (the

lower bound); and

(d) the fraction of the cathode length associated with

shoulders of the interdigitated gas distributor: 0.7

(the upper bound).

The polarization curve and the curve showing a

variation in the current density along the cathode/

membrane interface for the optimal fuel-cell design

are compared in Fig. 7a and b, respectively, with their

counterparts for the reference-case design. The results

displayed in Fig. 7a show that in the case of the

optimal design of the fuel cell, the onset of species

transport control regime is delayed. Consequently, as

the current density increases, the optimal design yields

increasingly higher cell voltage values, in comparison

to the ones obtained in the reference-case design, at

the same current density.

The results of the optimization analysis presented

above show that the optimum design point falls onto

the upper bounds of the air inlet pressure and the

fraction of the cathode length associated with

shoulders of the interdigitated gas distributor, and

onto the lower bounds of the cathode thickness and

the cathode length per one shoulder of the interdigi-

tated air distributor. A brief explanation of these

findings is given below.

As the air inlet pressure increases (i.e., as the

differential pressure within the cathode increases),

the convective air-flow rate through the cathode also

increases giving rise to a thinner diffusion (stagnation)

boundary layer at the cathode/membrane interface.

This, in turn, results in the enhanced transport of

oxygen to the reaction sites in the cathodic active

layer and, hence, to a higher current density at a given

cell potential.

In the cathode thickness range analyzed, it is

observed that as the cathode thickness increases, the

air begins to take the shortest route between the inlet

and the outlet and to flow mainly near the cathode/

shoulder interface. This causes the thickness of the

diffusion boundary layer adjacent to the cathode/

membrane interface to increase and, in turn, gives

rise to a decrease in the rate of species transport to the

cathodic active layer. It is hence justified that the

optimal design point associated with the largest elec-

tric current corresponds to the minimal cathode thick-

ness. If the gas flow through the cathode is analyzed

using an analogy with the gas flow through a pipe, then

a decrease in the cathode thickness is equivalent to a


decrease in the pipe diameter. Since as the pipe

diameter decreases, the resistance to the gas flow

increases, one should expect that below a critical

cathode thickness, the electric current would begin

to decrease with a decrease in the cathode thickness.

By keeping the other three design parameters fixed at

their optimal values, the critical cathode thickness was

found to be �0.00006 m.

The effect of cathode length per one shoulder of the

interdigitated air distributor can also be explained

using the gas flow through a pipe analogy. Hence, a

reduction of the cathode length per one shoulder of the

Fig. 7. The (a) polarization curve and the (b) current density at the cathode/membrane interface for the reference case and the optimal design

of the fuel cell.


interdigitated gas distributor is equivalent to a reduc-

tion in the pipe length and, due to a fixed pressure drop

over a shorter tube length, a higher gas flow rate is

attained. Consequently, the optimal fuel-cell design is

expected to be found at the lower bound of the cathode

length per one shoulder of the interdigitated gas

distributor.

As the fraction of the cathode length associated

with shoulders of the interdigitated gas distributor

decreases, it is found that the tendency for gas flow

via the shortest route is promoted and, consequently,

the thickness of the diffusion boundary layer at the

cathode/membrane is increased slowing down the

species transport to the cathode active layer. It is

hence justified to expect that the optimal design of

the fuel cell corresponds to the upper bound of the

fraction of cathode length associated with shoulders of

the interdigitated gas distributor.

A comparison in the x-component of the gas velo-

city and in the oxygen mole fraction in the reference

case and in the optimal design of the PEM fuel cell is

shown in Fig. 8a–d. �0.01 and 0.01 m/s contours are

used in Fig. 8a and b to denote the region in cathode in

which the gas velocity normal to the cathode/mem-

brane interface is essentially zero and in which, hence,

the oxygen transport is controlled by diffusion. By

comparing the results shown in Fig. 8a and b, it is clear

that the thickness and the length of such regions are

significantly reduced in the optimal fuel-cell design.

The results displayed in Fig. 8c and d, on the other

hand, show that the oxygen concentration at the

cathode/membrane interface is consistently higher

in the case of the optimal design than in the case of

the reference design. Thus, the results displayed in

Fig. 8a–d provide a direct evidence for the enhanced

species transport to the cathode/membrane interface

and for an enhanced rate of the reduction half-reaction

in PEM fuel cells with the optimal design.

4.3. Statistical sensitivity analysis

The six cathode-based model parameters (the fac-

tors) whose values are associated with the largest

uncertainty along with their three levels (one of which

corresponds to the reference value) are listed in

Table 5. The non-reference levels are arbitrarily

selected to be 10% below and 10% above their corre-

sponding reference values.

The L18 (36) orthogonal matrix [10] whose rows

define the 18 finite element computational analyses

which are carried out as a part of the statistical

sensitivity analysis is given in Table 6. The values

1, 2 or 3 in this table correspond, respectively, to the

three levels of the corresponding factor as defined in

Table 6. It should be noted that the reference case

corresponds to the analysis 2 in Table 6. The values of

the objective function (the averaged current density in

A/m2) obtained in the 18 analysis are given in the last

column in Table 5.

The results of the statistical sensitivity analysis are

displayed in Table 7. The results presented in this table

are obtained in the following way. In the first column,

the factors are listed in the decreasing order of the

variance ratio, F (given in the last column of the same

table). The difference from the mean of the objective

function associated factor B and its level 1 is obtained

by first calculating the mean of the objective function

obtained in the analyses 1, 4, 9, 11, 15 and 17 in which

the level 1 of factor B is used. The overall mean of the

objective function (3550.4 A/m2, the bottom row, the

Fig. 8. x-component of the gas velocity in m/s (a) and (b), and

oxygen mole fraction (c) and (d), contour plots for the reference

case (a) and (c), and the optimal design (b) and (d), of the PEM fuel

cell.


rightmost column in Table 6) is next subtracted from

the mean associated with the level 1 of factor B to yield

the value �2375.73 A/m2. The same procedure is next

used to obtain the remaining values in columns 2–4,

Table 7.

The sums of squares associated with each factor are

listed in column 5, Table 7. They are obtained by first

squaring the difference from the mean for each level of

each factor (columns 2–4, Table 7) and then by multi-

plying these with 6 (the number of times each level

appears in each column of the orthogonal matrix).

The values obtained for different levels of the same

factor are next summed to obtain the sum of squares

associated with that factor.

The number of degrees of freedom associated with

each factor (column 7, Table 7) is one less than the

number of levels of that factor. The number of degrees

of freedom associated with the error is obtained using

Eq. (13) to yield (18 � 1 � 6ð2Þ ¼ 5).

Since the number of degrees of freedom associated

with the error is nonzero, the sum of squares asso-

ciated with the error is calculated using Eq. (12),

Table 5

PEM fuel cell factors and levels used in the statistical sensitivity analysis

Factor Symbol Units Designation Levels

1 2 3

Reference oxygen concentration crefO2

mol/m3 A 0.6 0.665 0.73

Cathode/membrane equilibrium potential difference Dfeq;c V B 0.9 1 1.1

Active layer thickness dl m C 0.9 � 10�5 1.0 � 10�5 1.1 � 10�5

Gas diffusive coefficient inside the agglomerate Dagg m2/s D 4.49 � 10�12 4.98 � 10�12 5.487 � 10�12

Agglomerate particle radius Ragg m E 0.9 � 10�7 1.0 � 10�7 1.1 � 10�7

Cathodic exchange current density i0,c A/m2 F 0.9 1 1.1

Table 6

L18 (36) orthogonal matrix used in the statistical sensitivity analysis

Analysis number Factors Average current

density (A/m2)A B C D E F

1 1 1 1 1 1 1 1085.6

2 2 2 2 2 2 2 3214.3

3 3 3 3 3 3 3 6472.7

4 1 1 2 2 3 3 1333.4

5 2 2 3 3 1 1 3357.8

6 3 3 1 1 2 2 5641.0

7 1 2 1 3 2 3 3362.8

8 2 3 2 1 3 1 5734.3

9 3 1 3 2 1 2 1197.7

10 1 3 3 2 2 1 6576.9

11 2 1 1 3 3 2 1069.1

12 3 2 2 1 1 3 3260.4

13 1 2 3 1 3 2 3432.5

14 2 3 1 2 1 3 6361.2

15 3 1 2 3 2 1 1016.3

16 1 3 2 3 1 2 6816.0

17 2 1 3 1 2 3 1345.8

18 3 2 1 2 3 1 2629.3

Overall mean of the average current density (A/m2) 3550.4


where the SSgrand term is obtained by summing the

squares of the values given in the last column of

Table 6, the SSmean term is computed as 18�(3550.4 A/m2)2 and the SSfactor term is obtained by

summing the sum of squares values associated with

the six factors (column 5, Table 7).

The percent contribution of each factor and the error

to the total sum of the sum of squares is next calculated

and listed in column 6, Table 7.

The mean sum of squares given in column 8, Table 7

is obtained by dividing the values given in column 5,

Table 7 with the number of degrees of freedom,

column 7, Table 7. Finally, the variance ratio, F, for

each factor (column 9, Table 7) is obtained by dividing

its mean sum of squares (column 8, Table 7) by the

mean sum of squares associated with the error (¼23,559.2 A2/m4, Table 7).

The results displayed in Table 7 show that an

uncertainty in the value of the equilibrium cathode/

membrane potential has by far the largest effect on the

predicted average current density at the cell voltage of

0.7 V. The effect of other factors is comparably smal-

ler, but since their values of the variance ratio, F, are

either above or near 4, the effect of these factors is also

significant.

The results displayed in Table 6 show that the levels

of the six factors associated with analysis 16 gives rise

to the largest deviation from the reference case, ana-

lysis 2. To test the robustness of the optimal design of

the fuel cell discussed in the previous section, the

optimization procedure is repeated but for the factor

levels corresponding to analysis 16. The optimization

results obtained show that the optimal values of the

four fuel-cell design parameters are identical to the

ones for the reference case. This finding suggest that

while uncertainties in the values of various model

parameters can have a major effect on the predicted

polarization and current density distribution curves,

the optimal design of the fuel-cell is not affected by

such uncertainties. Therefore, experimental polariza-

tion curves for a given design of the fuel cell, in

conjunction with a regression analysis, can be used

to assess various model parameters which, in turn,

despite some uncertainties in their values, can be used

for optimization of the fuel-cell design.

5. Conclusions

Based on the results obtained in the present work,

the following conclusions can be drawn:

(1) Optimization of the design of PEM fuel cells can

be carried out by combining a multi-physics

model consisting of electrical and electrochemi-

cal potential and species conservation equations

with a nonlinear constrained optimization algo-

rithm.

(2) The optimum PEM fuel-cell design is found to be

associated with the cathode geometrical and

operation parameters which reduce the thickness

of the boundary diffusion layer at the cathode/

membrane interface.

(3) The predicted electrical response of PEM fuel

cells is highly dependent on the magnitude of a

number of parameters associated with the oxygen

Table 7

Statistical sensitivity analysis of the optimal design of the PEM fuel cell

Factor Difference from mean (A/m2) Sum of squares

(A/m2)2

Percent of

sum of squares

Number of

DOF

Mean sum of

squares (A/m2)2

Variance

ratio FLevel 1 Level 2 Level 3

B �2375.73 �340.88 2716.61 78,841,504.8 97.94 2 39,420,752.4 1673.26

A 217.48 �36.66 �180.82 488,026.4 0.60 2 244,013.2 10.36

C �192.24 12.04 180.20 417,446.3 0.52 2 208,723.1 8.86

F 150.35 11.37 138.98 252,310.8 0.31 2 126,155.4 5.35

D �133.79 1.73 132.06 212,055.7 0.26 2 106,027.9 4.50

E 129.39 �24.21 �105.18 170,339.0 0.21 2 85,169.5 3.61

Error 117,796.2 0.15 5 23,559.2

Total 80,499,479.3 100.00


transport and the reduction half-reaction. How-

ever, the optimal design is essentially unaffected

by a �10% variation in the value of these

parameters.

References

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electrodes in proton exchange membrane fuel cells using the

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lington, MA 01803, 2003, pp. 2-279–2-294.

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[6] FEMLAB 2.3a, COMSOL Inc., Burlington, MA 01803, 2003,

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HTTP://WWW.COMSOL.COM

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