Cathode and interdigitated air distributor geometry optimization

Materials Science and Engineering B 108 (2004) 241–252
Cathode and interdigitated air distributor geometry optimization
in polymer electrolyte membrane (PEM) fuel cells
M. Grujicic∗ , C.L. Zhao, K.M. Chittajallu, J.M. Ochterbeck
Department of Mechanical Engineering, Clemson University, Clemson, SC 29634-0921, USA
Received 31 May 2003; accepted 5 January 2004
Abstract
A steady-state single-phase three-dimensional electro-chemical model is combined with a nonlinear constrained optimization procedure
to maximize the performance of the cathode and the interdigitated air distributor in a polymer electrolyte membrane (PEM) fuel cell. The
cathode and the interdigitated air distributor design parameters considered include: the cathode thickness, the thickness of the interdigitated air
distributor channels and the width of the interdigitated air distributor channels. A statistical sensitivity analysis is used to determine robustness
of the optimal PEM fuel cell design. The results of the optimization analysis show that higher current densities at the membrane/cathode
interface are obtained in the PEM cathode and the interdigitated air distributor geometries that promote convective oxygen transport to the
membrane/cathode interface and reduce the thickness of the boundary diffusion layer at the same interface. The statistical sensitivity analysis
results show that, while the predicted average current density at the membrane/cathode interface is affected by uncertainties in a number of
model parameters, the optimal designs of the PEM cathode and the interdigitated air distributor are quite robust.
© 2004 Elsevier B.V. All rights reserved.
Keywords: Polymer electrolyte membrane (PEM) fuel cells; Design; Optimization; Robustness
1. Introduction
Due to their potential for reducing the environmental impact and the dependence on fossil fuels, fuel cells have
emerged as an attractive alternative to the internal combustion engines. 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. As
the hydrogen gas flows into the fuel cell on the anode side,
a platinum catalyst facilitates fuel oxidation which produces
protons (hydrogen ions) and electrons, Fig. 1. Protons diffuse through a membrane (the center of the fuel cell which
separates the anode and the cathode) and, with the help of
a platinum catalyst, combine with oxygen and electrons on
the cathode side, producing water. The electrons produced at
the anode side cannot pass through the membrane and flow
from the anode to the cathode through an external circuit
containing an electrical motor. The resulting voltage from
one single fuel cell is typically around 1.0 V. This voltage
∗ Corresponding author. Tel.: +1-864-656-5639;
fax: +1-864-656-4435.
E-mail address: mica.grujicic@ces.clemson.edu (M. Grujicic).
0921-5107/$ – see front matter © 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.mseb.2004.01.005
is generally 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 typically classified according to the type of
membrane (polymer electrolyte membrane fuel cells, solid
oxide fuel cells, molten carbonate fuel cells, etc.) they use.
One of the most promising fuel cells are the so-called polymer electrolytic membrane or proton exchange membrane
fuel cells (PEMFCs). 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, which consists of Teflon-like chains with
a fluorocarbon backbone and sulfonic acid ions, SO3 − , permanently attached to the side chains. When the membrane is
hydrated by absorbing water, protons attached to the SO3 −
ions combine with water molecules to form hydronium ions.
Hydronium ions are quite mobile and hop from one SO3 −
site to another within the membrane making the hydrated
solid electrolytes like NafionTM excellent conductors of the
hydrogen ions.
The anode and the cathode (the electrodes) in a PEM fuel
cell are made of an electrically conductive porous mate-
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M. Grujicic et al. / Materials Science and Engineering B 108 (2004) 241–252
Nomenclature
Ni
p
u
w
φ
ρ
molar flux of species (kg/m2 /s)
pressure (Pa)
gas velocity (m/s)
mass fraction
potential (V)
density (kg/m3 )
Subscripts
H2 O
water-related quantity
O2
oxygen-related quantity
Superscripts
eff
effective quantity
sym
symmetry
x
molar fraction
rial, typically 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 a
thick vertical line in Fig. 1. As also indicated in Fig. 1, oxidation and reduction fuel-cell half reactions take place in the
anode and the cathode active layers, respectively. 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 the minimum transport resistance
Fig. 1. A schematic of a polymer electrolyte membrane (PEM) fuel cell.
of the protons from the active sites in the anodic layer to the
active sites in the cathodic active layer.
As shown in Fig. 1, 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 temperature
(333–353 K) operation, a pollution-free character, and a relatively simple design, 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 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 decades a number of fuel-cell
models have been developed. 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 side 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 parameters (such as the cathodic and anodic
thicknesses, the geometrical parameters of the interdigitated
fuel/air distributors, etc.). However, a comprehensive optimization analysis of the PEM fuel cell design is still lacking.
Hence, the objective of the present work is to modify the
steady-state single-phase three-dimensional PEM fuel-cell
cathode model associated with a U-shaped air distribution
system [4] to include the effect of the interdigitated air distributor and to combine it with an optimization procedure
and a statistical sensitivity analysis in order to identify the
optimum geometry of the PEMFC cathode and the interdigitated air distributor.
The organization of the paper is as following: in Section 2,
the modification of the steady-state single-phase threedimensional model for a PEM fuel-cell cathode presented
in [4] and a solution method for the resulting set of partial
differential equations are briefly discussed. 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 conclusions resulting from the present work are
summarized in Section 5.
M. Grujicic et al. / Materials Science and Engineering B 108 (2004) 241–252
243
2. The model
As indicated in Fig. 1, the PEM fuel cell works on the
principle of separation of the oxidation of hydrogen (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. 1. Electrons
liberated in the anodic active layer via the oxidation halfreaction 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 and the cathode until they reach the
cathodic active layer. Simultaneously, protons (H+ ) 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.
Due to a relatively slow rate of the oxygen reduction reaction, the design of the fuel-cell cathode and the associated
interdigitated air distributor are considered as critical for
achieving a high performance of the PEMFCs. Therefore,
the work presented here focused on the cathode side of the
fuel cell.
2.1. Assumptions and simplifications
In this section, a simple steady-state single-phase threedimensional model for a PEM fuel-cell cathode developed
in [4] is modified to include the effect of an interdigitated
air distributor. The model is developed under the following
simplifications and assumptions:
• The computational domain, denoted with dashed lines in
Fig. 1, consists of a porous cathode which is in contact with an interdigitated air distributor. A three-dimensional view of the computational domain is shown in
Fig. 2.
• The membrane/cathode interface is considered as a zerothickness cathodic active layer.
• The portions of the cathode surface which are not in contact with the interdigitated air distributor are designated
as the current collector surface.
• Humidified air of fixed composition and pressure is supplied at the cathode inlet.
• Liquid water entering the cathode from the membrane side
and the one generated during the oxygen reduction reaction in the cathodic active layer are assumed to be of a
negligible volume (i.e., no two-phase flow in the cathode
is considered). It should be noted that this simplification
could be critical since liquid water residing in pores of the
cathode can significantly reduces its effective permeability. In fact, when the liquid water saturation level reaches
100% (the phenomenon known as “cathode flooding”),
the cathode becomes impervious to the gases.
• The inlet/outlet pressure difference specified in Table 1
has been selected in such a way that a compromise is
achieved between an effective removal of water droplets
Fig. 2. Schematic of the cathode and interdigitated air distributor in a
PEM fuel cell.
Table 1
General parameters and reference-case cathode and interdigitated air
distributors’ parameters used for 3D modeling the PEM fuel cell
Parameter
Symbol
SI units
Value
Faraday’s constant
Universal gas constant
Temperature
Atmospheric (reference)
pressure
Gas viscosity
Pressure differential
F
R
T
p0
A s/mol
J/mol K
K
Pa
96,487
8.314
353
1.013 × 105
µ
p
kg/m s
Pa
Activation overpotential
Molecular weight of oxygen
Molecular weight of water
Molecular weight of
nitrogen
Molar diffusion volume of
oxygen
Molar diffusion volume of
water
Molar diffusion volume of
nitrogen
Porosity of cathode
Inlet mass fraction of oxygen
Inlet mass fraction of oxygen
Inlet mass fraction of oxygen
Empirical value
η
MO2
MH 2 O
MN 2
V
kg/mol
kg/mol
kg/mol
2.0 × 10−5
p0 × 2.5 ×
10−1
0.5
32 × 10−3
18 × 10−3
28 × 10−3
vO 2
m3 /mol
16.6 × 10−6
vH 2 O
m3 /mol
12.7 × 10−6
vN 2
m3 /mol
17.9 × 10−6
ε
wO2 ,0
wH2 O,0
wN2 ,0
kMS
0.5
0.1447
0.3789
0.4764
99.9 × 10−8
Specific surface area
Thickness of the active
layer
Number of electrons
Exchange current density
Cathode conductivity
Sa
δ
No units
No units
No units
No units
Nm2 kg1/2 /
K1.75 mol7/6
m2 /m3
m
n
i0
k
No units
A/m2
S/m
4.0
4.0 × 10−3
1.0 × 105
1.0 × 108
1.0 × 10−5
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M. Grujicic et al. / Materials Science and Engineering B 108 (2004) 241–252
Fig. 3. Dependent variables, governing equations and boundary conditions in the interdigitated air distributor.
from the cathode and the retention of structural integrity
of the cathode.
• The behavior of the O2 +N2 +H2 O gas mixture in the cathode is considered to be governed by the ideal gas law while
their diffusivities are described by the Maxwell–Stefan
diffusion model.
• The cathode is considered as a homogeneous medium in
which the porosity and permeability are distributed uniformly.
• The kinetics of the electro-chemical oxygen reduction
half-reactions taking place in the cathode active layer is
assumed to be governed by the Tafel equation.
2.3. The governing equations
The governing equations for the interdigitated air distributor and the cathode are given in Figs. 3 and 4, respectively.
They include:
• For the interdigitated air distributor: (a) a momentum
conservation equation; (b) a continuity equation; (c) an
oxygen mass-balance equation; and (d) a water-vapor
mass-balance equation.
• For the cathode: (a) a momentum conservation equation;
(b) a continuity equation; (c) an oxygen mass-balance
equation; (d) a water-vapor mass-balance equation; and
(e) an electric charge conservation equation.
2.2. The dependent variables
As indicated in Fig. 2, there are two sub-domains in the
present model: (a) the interdigitated air distributor and (b) the
porous cathode. As indicated in Figs. 3 and 4, the following
dependent variables are used in the present model:
• In the interdigitated air distributor: (a) the gas-phase velocity, u; (b) the gas-phase pressure, p; (c) the oxygen
mass fraction in the gas-phase, wO2 ; and (d) the watervapor mass fraction in the gas phase, wH2 O ;
• In the cathode: (a) the gas-phase velocity, u; (b) the gasphase pressure, p; (c) the oxygen mass fraction in the gasphase, wO2 ; (d) the water-vapor mass fraction in the gas
phase, wH2 O ; and (e) the electronic potential, φ.
2.4. The boundary conditions
The boundary conditions used are also listed in Figs. 3
and 4. The boundary conditions can be briefly summarized
as following:
2.4.1. For the interdigitated air distributor
(a) The composition of the air at the inlet to the interdigitated air distributor is set equal to that of steam-saturated
air at 353 K. The air pressure head is set to a fixed value;
(b) At the interdigitated air distributor outlet, a straight-out
flow is assumed, the pressure is fixed and the mass transfer is taken to be dominated by convection;
M. Grujicic et al. / Materials Science and Engineering B 108 (2004) 241–252
245
Fig. 4. Dependent variables, governing equations and boundary conditions in the porous PEMFC cathode.
(c) At the remaining boundaries, no-slip impervious boundary conditions are assumed.
2.4.2. For the cathode
(a) At the portions of the top surface of the cathode, which
are not in contact with the interdigitated air distributor,
no-slip, impervious, and zero electric potential boundary
conditions are assumed;
(b) At the portions of the top surface of the cathode which
are in contact with the interdigitated air distributor which
is a boundary only with respect to the electric charge
conservation equation, a zero electric current boundary
condition is assumed;
(c) At the membrane/cathode interface, no-slip boundary
conditions for the gas-phase velocity are assumed while
the electric current and the oxygen and the water vapor
fluxes are given by the Tafel equation;
(d) At the remaining surfaces, no-slip and zero-flux boundary conditions for the electric potential, oxygen and water vapor mass fractions are prescribed.
2.5. Closure relationships
To express the governing equations in terms of the dependent variables, the following closure relations are used:
The molar mass of the O2 + N2 + H2 O gas mixture is
defined as:
M = MO2 wO2 + MH2 O wH2 O + MN2 wN2
(1)
The molar density of the gas mixture is defined using the
ideal gas law as:
p
(2)
ρ=
RT
The symmetric binary diffusivities appearing in the mass
balance equations are dependent on the gas-phase composition and are given as:
sym
Dii
=
((wj + wk )2 /xi Djk ) + (w2j /xj Dik ) + (w2k /xk Dij )
(xi /Dij Dik ) + (xj /Dij Djk ) + (xk /Dik Djk )
(i, j = O2 , H2 O, N2 )
sym
Dij
(3)
(wi (wj + wk )/xi Djk ) + (wj (wi + wk )/xj Dik )
− (w2k /xk Dij )
=
(xi /Dij Dik ) + (xj /Dij Djk ) + (xk /Dik Djk )
(i, j = O2 , H2 O, N2 )
(4)
The Maxwell–Stefan diffusivities used in Eqs. (3) and (4)
are defined as:
T 1.75
1
1 12
Dij = kMS
+
1/3
1/3
Mj
p(vi + vj )2 Mi
(i, j = O2 , H2 O, N2 )
(5)
The effective binary diffusivities within the porous cathode are given as:
Dijeff = Dij ε1.5
(i, j = O2 , H2 O, N2 )
(6)
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M. Grujicic et al. / Materials Science and Engineering B 108 (2004) 241–252
The Tafel coefficient which relates the oxygen and the
water vapor fluxes with the local current density at the membrane/cathode interface is defined as:
Sa δ i0
0.5Fη
kTafel =
exp
(i, j = O2 , H2 O, N2 ) (7)
nF
RT
2.6. Computational method
The steady-state, nonlinear three-dimensional system of
governing partial differential equations (discussed in Section
2.3 and in Figs. 3 and 4) subjected to the boundary conditions (discussed in Section 2.4 and in Figs. 3 and 4) are implemented in the commercial mathematical package FEMLAB [5] and solved (for the dependent variables discussed
in Section 2.2 and in Figs. 3 and 4) using the finite element
method. The FEMLAB provides a powerful interactive environment for modeling various scientific and engineering
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 visualization package [6]. As a result, the models
developed in the FEMLAB can be saved as MATLAB programs for parametric studies or iterative design optimization.
erence) design of the PEMFC cathode and the interdigitated
air distributor. A schematic of the cathode is given in Fig. 5
to explain the three design parameters (denoted as x(i), i =
1–3) defined above. The width and the length of the cathode
are kept constant at 0.0009 and 0.0013 m, respectively.
The objective function f[x(1), x(2), x(3)] is next defined
as the average electric current density at a typical value of
the cell voltage of 0.7 V. The average electric current density
can be considered as a measure of the degree of utilization of
the expensive Pt-based catalyst in the cathode active layer.
Thus, the fuel-cell design optimization problem can be
defined as: minimize 1/f[x(1), x(2), x(3)] with respect to x(1),
x(2), and x(3) subject to:
0.00002 m ≤ x(1) ≤ 0.0005 m
0.00002 m ≤ x(2) ≤ 0.005m, and
0.000025 m ≤ x(3) ≤ 0.000175 m
The upper limits for the three design parameters are chosen based on considerations of the size constraints for a
PEMFC stack. The lower limits for the three design parameters, on the other hand, are selected based on the consideration of minimal feature sizes which can be attained using the
current PEMFC cathode and the interdigitated air-distributor
manufacturing processes.
3. Design optimization and robustness
3.1. Optimization
There are many PEMFC cathode and the interdigitated
gas distributor parameters which affect the performance of a
PEM fuel cell. Some of these parameters such as the cathode
permeability and porosity are controlled by the microstructure of the cathode porous material. 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 PEMFC cathode optimization procedure described below. Instead, geometrical parameters (e.g. the cathode thickness, the width
of the interdigitated air distributor channels, etc.) will be
considered. As explained earlier, the kinetics of reduction
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
design of its cathode side. This approach is adopted in the
present work and, consequently, the following three cathode and interdigitated air distributor design parameters have
been identified as the most important:
The optimization problem formulated above is solved using the MATLAB optimization toolbox [6] which contains
an extensive library of computational algorithms for solving different optimization problems such as: unconstrained
and constrained nonlinear minimization, quadratic and linear programming and the constrained linear least-squares
method. The problem under consideration in the present
work belongs to the class of multidimensional constrained
nonlinear minimization problems which can be solved using the MATLAB fmincon() optimization function of the
following syntax:
• the cathode thickness (0.0001 m);
• the thickness of the interdigitated air distributor channels
(0.0001 m); and
• the width of the interdigitated air distributor channels
(0.0001 m).
The numbers given within the parentheses correspond to
the values of the three design parameters in the initial (ref-
fmincon(fun, x0 , A, B, Aeq , Beq , LB, UB, confun, options);
(8)
where fun denotes the scalar objective function of a multidimensional design vector x, while confun contains nonlinear
non-equality (c(x) ≤ 0) and equality (ceq (x) = 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 constraining equations of the type Aeq x = 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 [7] within
which the original problem is approximated with a quadratic
programming subproblem which is then solved successively
M. Grujicic et al. / Materials Science and Engineering B 108 (2004) 241–252
247
Fig. 5. Definition of the three cathode and the interdigitated air distributor-based design parameters used in the PEMFC optimization procedure.
until convergence is achieved for the original problem. The
method has the advantage of finding the optimum design
from an arbitrary initial design point and typically requires
fewer function and gradient evaluations compared to other
methods 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 continuous and, for a
given initial (reference) design of the PEMFC cathode and
the interdigitated air distributor, the method can find only a
local minimum in the vicinity of the initial design.
3.2. Statistical sensitivity analysis
Once the optimum combination of the design parameter is
found using the optimization procedure described above, it
is important to determine how sensitive is the optimal fuelcell design to the variation in the model parameters whose
magnitudes are associated 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 statistical sensitivity analysis
[8] 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 formulation 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 performed 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 a 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 fac-
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M. Grujicic et al. / Materials Science and Engineering B 108 (2004) 241–252
tors appear and each combination appear an equal number
of times. A limited number of standard orthogonal matrices [9] 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
η (the objective function in the present case) is given by:
h = overall mean +
(factor effect) + error
(9)
quantity in Eq. (10) 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 = DOFgrand − 1 −
(DOFfactor )
(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 the error, SSerror , can be calculated using the following relationship:
where MEANfactor is a mean value of the objective function
for a given factor. The varaince ratio, F, 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
significant.
SSerror = SSgrand − SSmean − SSfactors
(10)
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
For Eq. (10) 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 the 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:
SSerror
VARerror =
, and
(12)
DOFerror
F=
(MEANfactor )2
VARerror
(13)
4. Results and discussion
4.1. The reference case
Contour plots of the current density and the oxygen mole
fraction at the membrane/cathode interface in the case of the
reference design of the PEMFC cathode and the interdigitated air distributor are shown in Fig. 6a and b, respectively.
Based on the results displayed in Fig. 6a and b, the following main observations can be made:
• The variation of the current density over the membrane/cathode interface is significant (∼50%) and the
largest values of the current density are found in the region adjacent to the inlet of the interdigitated air distributor. The average current density at the membrane/cathode
interface is ∼10,600 A/m2 .
M. Grujicic et al. / Materials Science and Engineering B 108 (2004) 241–252
249
Fig. 6. Variations of: (a) the current density and (b) the oxygen mole fraction over the membrane/cathode interface in the reference design of the PEMFC
cathode and the interdigitated air distributor.
• The variation of the oxygen mole fraction over the membrane/cathode interface is also significant (∼50%) and the
largest values of the oxygen mole fraction are also found
in the region adjacent to the inlet of the interdigitated air
distributor. However, the lowest levels of the oxygen mole
fraction are not found in the region adjacent to the outlet
of the interdigitated air distributor but further away from
it. A careful examination of the diffusion flux fields revealed oxygen back-diffusion at the outlet of the interdigitated air distributor which results in higher oxygen mole
fractions in this region.
• There is a clear correlation between the current density and the local oxygen mole fraction at the membrane/cathode interface.
The variations of the oxygen mole fraction over three
planes normal to the membrane/cathode interface and
parallel with the fuel-cell width are shown in Fig. 7a-c.
In the first plane which bisects the inlet channel of the
interdigitated air distributor, Fig. 7a, the values of the
oxygen mole fraction are the highest and the peak val-
ues are found in the vicinity of the air entrance. In the
third plane which bisects the outlet channel of the interdigitated air distributor, Fig. 7c, the oxygen mole fractions are the lowest. However, due to the effect of oxygen
back-diffusion, the oxygen mole fractions are slightly increased in the region near the air exit. In the second plane
which is located halfway between the other two planes,
Fig. 7b, intermediate values of the oxygen mole fraction
are found.
4.2. Fuel-cell design optimization
The optimization procedure described in Section 3.1
yielded the following optimal PEMFC cathode and the
interdigitated air distributor geometric parameters:
• the cathode thickness: 0.000037 m;
• the channel thickness of the interdigitated air distributor:
0.0005 m (the upper bound); and
• the channel width of the interdigitated air distributor:
0.000175 m (the upper bound).
Fig. 7. Variations of the oxygen mole fraction throughout the PEMFC cathode thickness at three different locations relative to the fuel cell front. Please
see text for details.
250
M. Grujicic et al. / Materials Science and Engineering B 108 (2004) 241–252
The results of the optimization analysis presented above
show that the optimal design of the cathode side of a PEM
fuel cell is associated with the upper bounds of the interdigitated air distributor channel thickness and the channel
width. The optimal value of the cathode thickness, on the
other hand, is quite low but larger than its lower bound. A
brief explanation of these findings is given below.
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/current-collector interface.
This causes the thickness of the diffusion boundary layer
adjacent to the membrane/cathode interface to increase and,
in turn, gives rise to a decrease in the rate of oxygen transport to the cathodic active layer. This causes a decrease in
the local current density at the membrane/cathode interface.
It is hence justified that the optimal PEMFC cathode design
is associated with a low value of the cathode thickness.
However, 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 further decrease in the cathode thickness. Hence, the intermediate optimal value of the
cathode thickness (∼0.000037 m) found in the present work,
is justified. A similar finding was obtained in our recent
two-dimensional analysis of the effect of cathode thickness
on the average current density on the membrane/cathode
interface [2].
The effect of the interdigitated air distributor channel
thickness can also be explained using the analogy with gas
flow through a pipe. The pressure drop between the interdigitated air distributor inlet and the outlet is fixed and equal
to p. As the channel cross section area increases with an
increase in the channel thickness, the pressure drop inside
the channel decreases. Consequently, a larger pressure difference exists between the surfaces at which the air enters
the cathode and the surfaces at which the air leaves the cathode. The resulting higher air velocity promotes convective
oxygen transport to the membrane/cathode interface. However, the effect of the interdigitated air distributor channel
width is found to be quite small and if the channel width is
changed from the optimal (upper bound) value to the lower
bound value the magnitude of the average current density is
decreased by only 0.8%.
The effect of the interdigitated air distributor channel
width can be explained using the same argument as the one
used to explain the effect of the interdigitated air distributor channel thickness. However, as discussed above, this effect is found to be relatively small. A careful examination
of the oxygen mole fraction and the air velocity fields revealed that as the interdigitated air distributor channel width
increases, a larger fraction of the membrane/cathode interface is benefiting from the convective oxygen transport and
this is believed to represent the major contribution that the
air distributor channel width has on the average current density. This is consistent with the fact that the effect of the air
distributor channel width is substantially larger than that associated with the air distributor channel thickness. In fact,
changing the air distributor channel width from its upper
bound to its lower bound value causes the average current
density to decrease by 4.7%.
4.3. Statistical sensitivity analysis
Six model parameters (the factors) 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 2. The non-reference levels are arbitrarily selected to be 10% below and 10% above their corresponding
reference values.
The L18 (36 ) orthogonal matrix [9] 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 3. The values 1, 2 or 3 in this table correspond
respectively to the three levels of the corresponding factor
as defined in Table 2. It should be noted that the reference
case corresponds to the analysis 2 in Table 3. The values
of the objective function (the averaged current density at
the membrane/cathode interface in A/m2 ) obtained in the 18
analyses are given in the last column in Table 3.
The results of the statistical sensitivity analysis are displayed in Table 4. 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
Table 2
PEM fuel cell factors and levels used in the statistical sensitivity analysis
Factor
Symbol
Units
kMS
δ
vO 2
vH 2 O
vN 2
ε
Nm2
Designation
Levels
1
Empirical value
Thickness of the active layer
Molar diffusion volume of oxygen
Molar diffusion volume of water
Molar diffusion volume of nitrogen
Porosity of cathode
kg1/2 /K1.75
m
m3 /mol
m3 /mol
m3 /mol
No units
mol7/6
A
B
C
D
E
F
89.9 ×
0.9 ×
14.9 ×
11.4 ×
16.1 ×
0.45
2
10−8
10−5
10−6
10−6
10−6
99.9 ×
1.0 ×
16.6 ×
12.7 ×
17.9 ×
0.50
3
10−8
10−5
10−6
10−6
10−6
109.9 ×
1.1 ×
18.3 ×
14.0 ×
19.7 ×
0.55
10−8
10−5
10−6
10−6
10−6
M. Grujicic et al. / Materials Science and Engineering B 108 (2004) 241–252
Table 3
L18 (36 ) Orthogonal matrix used in the statistical sensitivity analysis
Analysis
number
Factors
A
B
C
D
E
F
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
2
3
1
3
1
2
2
3
1
3
1
2
1
2
3
2
3
1
1
2
3
3
1
2
3
1
2
2
3
1
1
2
3
2
3
1
3
1
2
2
3
1
1
2
3
3
1
2
1
2
3
3
1
2
2
3
1
2
3
1
3
1
2
1
2
3
1
2
3
3
1
2
3
1
2
1
2
3
2
3
1
2
3
1
Overall mean of the average current density (A/m2 )
Average current
density (A/m2 )
11,093
11,880
13,084
11,185
11,933
12,777
11,935
12,811
11,135
13,123
11,087
11,897
11,961
13,043
11,068
13,211
11,189
11,664
12,004
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 (12,004 A/m2 ,
the bottom row, the rightmost column in Table 3) is next
subtracted from the mean associated with the level 1 of factor B to yield the value −0.8781 A/m2 . The same procedure
is next used to obtain the remaining values in columns 2–4,
Table 4.
The sums of squares associated with each factor are listed
in column 5, Table 4. They are obtained by first squaring
the difference from the mean for each level of each factor
(columns 2–4, Table 4) and then by multiplying 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 4) is one less than the number of
levels of that factor. The number of degrees of freedom
251
associated with the error is obtained using Eq. (11) to yield:
18 − 1 − 6(2) = 5.
Since the number of degrees of freedom associated with
the error is nonzero, the sum of squares associated with the
error is calculated using Eq. (10), where the SSgrand term
is obtained by summing the squares of the values given in
the last column of Table 3, the SSmean term is computed as
18 × (12,004 A/m2 )2 and the SSfactor term is obtained by
summing the sum of squares values associated with the six
factors (column 5, Table 4).
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 4.
The mean sum of squares given in column 8, Table 4 is
obtained by dividing the values given in column 5, Table 4
with the number of degrees of freedom, column 7, Table 4.
Finally, the variance ratio, F, for each factor (column 9,
Table 4) is obtained by dividing its mean sum of squares
(column 8, Table 4) by the mean sum of squares associated
with the error (=0.0032 A2 /m4 , Table 4).
The results displayed in the last column in Table 3 show
that an uncertainty in the value of the thickness of the active
layer has by far the largest effect on the predicted average
current density at the membrane/cathode interface. The effect of other factors is comparably smaller, 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 4 show that the levels of
the six factors associated with analysis 16 gives rise to the
largest deviation from the reference case, analysis 2. To test
the robustness of the optimal design of the PEMFC cathode
and the interdigitated air distributor 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
three fuel-cell design parameters are almost 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 average current density
at the membrane/cathode interface, the optimal design of a
PEM fuel-cell is not significantly affected by such uncertainties (Fig. 8). Therefore, despite some uncertainties in the
Table 4
Statistical sensitivity analysis of the optimal design of the PEM fuel cell
Factor
A
B
C
D
E
F
Error
Total
Difference from mean (A/m2 )
Level 1
Level 2
Level 3
804
−8781
−711
−496
478
−556
−137
−1259
44
8
−89
43
−667
10,039
666
488
−389
513
Sum of squares
(A/m2 )2
0.0667
10.7684
0.0570
0.0290
0.0232
0.0344
0.0161
10.9948
Percent of sum
of squares
0.6064
97.9406
0.5187
0.2639
0.2114
0.3129
0.1461
100.00
Number
of DOF
Mean sum of
squares (A/m2 )2
Variance
ratio F
2
2
2
2
2
2
5
0.0333
0.5384
0.0285
0.0145
0.0116
0.0172
0.0032
10.37
167.55
8.87
4.51
3.62
5.35
252
M. Grujicic et al. / Materials Science and Engineering B 108 (2004) 241–252
Fig. 8. Variations of: (a) the current density and (b) the oxygen mole fraction over the membrane/cathode interface in the optimal design of the PEMFC
cathode and the interdigitated air distributor.
values of a number of model parameters, the optimal design
of the PEMFC cathode and the interdigitated air distributor
is quite robust.
5. Conclusions
Based on the results obtained in the present work, the
following conclusions can be drawn:
• Optimization of the PEMFC cathode and the interdigitated
air distributor designs can be carried out by combining
a multi-physics model consisting of the continuity, momentum, mass and electric charge conservation equations
with a nonlinear constrained optimization algorithm.
• The optimum design of the cathode side of a PEM fuel
cell is found to be associated with the cathode and the
interdigitated air distributor geometrical parameters which
promote the role of convective oxygen transport to the
membrane/cathode interface and reduce the thickness of
the boundary diffusion layer at the same interface.
• The predicted average current density at the membrane/cathode interface of PEM fuel cells is highly dependent on the magnitude of several model parameters
associated with the mass transport and the oxygen reduction half-reaction. However, the optimal design of the
PEMFC cathode and the interdigitated air distributor is
essentially unaffected by a ±10% variation in the values
of these parameters.
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