Structural Analysis of Load Distribution within Single Cell
Fuel Cell
by
Eric J. O’Brien
An Engineering Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF ENGINEERING IN MECHANICAL ENGINEERING
Approved:
_________________________________________
Ernesto Gutierrez-Miravete, Adviser
Rensselaer Polytechnic Institute
Hartford, Connecticut
December, 2011
ii
CONTENTS
LIST OF SYMBOLS ........................................................................................................ iv
LIST OF TABLES ............................................................................................................. v
LIST OF FIGURES .......................................................................................................... vi
LIST OF KEY WORDS .................................................................................................. vii
ABSTRACT ................................................................................................................... viii
1. INTRODUCTION/BACKGROUND ....................................................................... 1
2. THEORY/METHODOLOGY .................................................................................. 4
2.1. Solid Mechanics of the PEM fuel cell................................................................ 4
2.2. Design of Experiment Factorial Methodology................................................... 5
2.3. Geometry ............................................................................................................ 7
2.4. Materials ............................................................................................................. 9
3. RESULTS ................................................................................................................. 15
3.1. Results Overview ............................................................................................. 15
3.2. Selecting a Surface for Evaluation ................................................................... 17
3.3. Cell Pressure Results – Z component of Stress ................................................ 21
3.4. Factorial Analysis Results ................................................................................ 26
4. DISCUSSION ........................................................................................................... 30
5. CONCLUSIONS ...................................................................................................... 34
6. BIBLIOGRAPHY...................................................................................................... 35
7. APPENDICES .......................................................................................................... 37
7.1. Comsol Files: ................................................................................................... 37
7.2. Minitab Files .................................................................................................... 37
iii
LIST OF SYMBOLS
σ = Normal Stress (Pa)
τ = Shear Stress (Pa)
λ = Lamé’s Constant (Pa)
ε = True Strain (-)
δ = Elongation (m)
G = Shear Modulus (Pa)
e = Linear Strain
E = Modulus of Elasticity (GPa)
ν = Poisson’s Ratio (-)
ρ = Density (kg/m^3)
F = External Force/Load (N)
A = Area (m2)
p = Pressure (Pa)
L = Length (m)
W = Width (m)
D = Diameter (m)
iv
LIST OF TABLES
Table 1: DOE factorial variables for analysis .................................................................... 7
Table 2: Material Properties .............................................................................................. 9
Table 3: Compression results by case .............................................................................. 22
v
LIST OF FIGURES
Figure 1 - How a PEM fuel cell works [1] ........................................................................ 1
Figure 2: Fuel Cell Stack Diagram .................................................................................... 3
Figure 3: Visual representation of a Full Factorial vs. Half Factorial ............................... 6
Figure 4: Geometry of the single cell ................................................................................ 8
Figure 5: Highlighted surfaces represent symmetry boundary conditions ...................... 10
Figure 6 – Applied pressure location on the pressure plate ............................................. 11
Figure 7 - Side view of cell mesh .................................................................................... 13
Figure 8: Mesh containing triangular prisms. .................................................................. 14
Figure 9 – Von Mises stress plot of the pressure plate .................................................... 16
Figure 10: Cross Section Planes within separator plates. ................................................ 18
Figure 11 - Plot of Z direction of stress in a cross section of the separator plate. ........... 19
Figure 12: Comparison Plots of locations within the separator plates (a) 1.5 mm from
center anode side (b) 1mm from center anode side (c) 1mm from center cathode side. . 20
Figure 13: 3D Plot of Z component stress in baseline separator plate ............................ 21
Figure 14: Plots of the Z component stress tensor (a) Case 1 (b) Case 2 (c) Case 3 (d)
Case 4............................................................................................................................... 23
Figure 15: Plots of the Z component stress tensor (a) Case 5 (b) Case 6 (c) Case 7 (d)
Case 8............................................................................................................................... 25
Figure 16: Main Effects plot for Average Values ........................................................... 26
Figure 17: Effects Pareto for Average ............................................................................. 27
Figure 18: Effects Pareto for Range ................................................................................ 28
Figure 19: Effects Pareto for Max Z Stress ..................................................................... 28
Figure 20: Pareto Chart of effects for Min Z stress ......................................................... 29
Figure 21: Optimization of Factorial Analysis ................................................................ 31
Figure 22: Low aspect ratio solution ............................................................................... 32
vi
LIST OF KEY WORDS
Fuel Cell
PEM – Polymer Electrolyte Membrane
Pressure Plate
Separator Plate
UEA – Unitized Electrode Assembly
GDL – Gas Diffusion Layer
DOE – Design of Experiments
Factorial Analysis
vii
ABSTRACT
A PEM fuel cell consists of a membrane within an electrode assembly, flow field
plates to deliver the reactant gases, and pressure plates to load the system for sealing and
conduction. Distribution of pressure on a fuel cell is important for maximum
performance and durability of the membrane. This study evaluates the factors which
affect the pressure within a single cell fuel cell by performing a structural analysis using
a Finite Element model to evaluate eight different cases. The cases were generated using
design of experiments factorial methods with four factors and two levels to vary the
pressure plate thickness, flow field plate thickness, aspect ratio, and tie rod load in each
case. The results show that there are areas of the cell which are not under compressive
load. Pressure plate thickness and applied load are the largest factors which affect the
load distribution of the cell.
viii
1. INTRODUCTION/BACKGROUND
Fuel Cells utilize an electrochemical reaction between a fuel, usually hydrogen, and
oxygen, typically from air to generate electricity. There are multiple types of fuel cells
which have different benefits for different applications. These include Direct Methanol,
Alkaline, Phosphoric Acid, Molten Carbonate, Solid Oxide and Polymer Electrolyte
Membrane fuel cells. The fuel cell type which will be evaluated in this paper is a
Polymer Electrolyte Membrane (PEM) fuel cell. A single fuel cell does not generate a
large amount of electricity; therefore, multiple cells are usually stacked together into a
cell stack assembly. The area of each cell and the number of cells in the stack can be
varied to meet specific operating conditions.
Figure 1 - How a PEM fuel cell works [1]
As seen in Figure 1, for a PEM fuel cell to function it needs to have a hydrogen flow
field plate on the anode side, an oxygen flow field plate on the cathode side, and the
polymer electrolyte membrane (PEM) in the center. The flow field plates usually are
made from carbon graphite material. This material allows for electrical conductivity
1
through the plates, as well as channels to be machined into them for fuel and air flow.
These flow channels allow for fuel and air to pass by the electrode assembly which
contains the membrane, catalyst layers as well as gas diffusion layers (GDLs) which are
located on either side of the membrane. The GDLs help the flow diffuse equally into the
membrane, including areas between the actual flow channels. Often the electrode
assembly is often bonded into a single assembly [2], or unitized, and referred to as a
unitized electrode assembly or UEA. It is referred to as unitized because it is made into a
single assembly which is then sandwiched between the flow field plates, also known as
separator plates
Electricity is generated in the fuel cell when a fuel, pure hydrogen in the case of
PEM fuel cells, flows over the anode side of the electrode assembly and air flows over
the cathode side of the electrode assembly. When this occurs, hydrogen reacts with a
catalyst in the electrode which causes positive ions to pass through the membrane, while
the negative ions create an electrical current. The positive ions then react with the
oxygen in the air on the cathode side of cell to produce water. Below are the following
equations for this reaction [3]:
Anode Reaction: 2H 2  4H   4e 
Cathode Reaction: O2  4H  4e  2H 2 O
Overall Cell Reaction: 2H 2  O2  2H 2O
Although there are many factors which affect the performance of a fuel cell, the load
on the UEA and more specifically on the GDL can significantly affect the performance.
[4]. Proper load distribution of the fuel cell is important for both performance and
durability. Low loads on a cell increases resistance and therefore reduces performance.
High loads can create excessive stress on the cell membrane decreasing lifetime of the
cell. Fuel cell stacks with dozens of cells end up with relatively even pressure
distribution because the ratio of the length of the stack vs. the distance from the tie rod
loads to the center of the cell is large. When this number is small the distribution of load
at the UEA can be poor, such as in single cell stacks or stacks with only a few cells.
2
The pressure within a given single cell has been proven to vary as much as 4 times
from the lowest pressure to the highest pressure even with different loads [5]. The
question to be answered in this study is what variable of the design affect this ratio from
the highest load to the lowest load.
To apply load to the stack, there are stainless steel pressure plates on the either end
of the assembly with threaded tie rods connecting the two pressure plates as seen in
Figure 2 below [6]. The pressure plates are large plates of stainless steel represented by
numbers 120 and 140 in the diagram below. The threaded tie rods, represented by
number 102 in the figure below, extend through each of the pressure plates and nuts on
either side pull the pressure plates together, compressing the cell stack assembly. None
of the other features in the diagram represent the fuel cell discussed in this study. For
instance, there are coolant ports in the pressure plate below, however the pressure plate
used in this study does not have any ports in it.
Figure 2: Fuel Cell Stack Diagram
3
2. THEORY/METHODOLOGY
To ensure proper loading, optimization of the pressure plate and bipolar plate design
is needed. The objective of this paper is to evaluate the change in load distribution when
changing the configuration of the cell, including the thickness of the pressure plate,
thickness of the separator plate, load on the cell and aspect ratio of the fuel cell. Using a
constant pressure plate configuration, the pressure on the electrode assembly (UEA) will
be evaluated.
A finite element model was developed using the program Comsol. The geometry
was originally modeled in Pro/Engineer and was then refined within Comsol such that
the parametric features could be used within Comsol. Some simplifications were made in
order to keep the mesh of reasonable size, such as the number of channels in the
separator plates as well as having the channels run parallel instead of perpendicular.
Meshing the perpendicular flow field would have exponentially increased the amount of
elements needed. Minitab was used to develop a set of analyses with two levels for each
variable using Design of Experiment (DOE) factorial methods. The results were
evaluated for each of the cases required for the DOE analysis. The thru plane stress was
the main output compared for each case. Because good load distribution on the cell is
important for its performance, as stated above in the introduction, reducing the pressure
distribution at the surface of the UEA was the goal of the analysis.
2.1. Solid Mechanics of the PEM fuel cell
It is assumed that there are no initial stresses on the materials from machining or
material processing and all materials are assumed to be isotropic. Due to the relatively
low temperature of a PEM fuel cell, which is usually less than 100C, the material
properties are assumed to be the same as room temperature. There is also assumed to be
no thermal stress in the system.
4
The stress within the material is therefore represented with the following equations [7]:
 x  xy  xz
   ij   xy  y  yz
 zx  zy  z
[1]
Since the system is in equilibrium the following equation applies:
div  0
[2]
The stress-stain relationship is Hooke’s Law:
 ij   kk  ij  2Geij
[3]
Where:
G
E
2(1  )

2G
1  2
[4a]
[4b]
2.2. Design of Experiment Factorial Methodology
Design of Experiments is a method of experimentation to test or establish a
hypothesis to see which inputs have the greatest effect on the output. Generally used for
physical experiments to ensure variation is of the experiment does not affect the
outcome, it can also be used for a set of analyses used to determine which parameters in
the analysis have the greatest effect on the output. One type of analysis method is
factorial analysis. The factorial analysis is a statistical method to describe the effect of
variables on a system with a reduced number of factors. In this case there are four factors
5
being analyzed with two levels for each factor. To produce a full factorial of tests, 16
analyses would have to be run because the factorial equation is the number of levels
raised to the power of the number of factors as represented in equation. However, a full
factorial compares all interactions between all of the factors. For this case the
assumption is that the higher level interactions are not large factors and a half factorial
can be used. This allows for the most significant data to be collected with half of the
cases, which would now be 8.
The half factorial set selects 8 of the cases in a logical manner which allows for
the least error. A graphical representation of full factorial vs. ½ factorial can be seen in
Figure 3. This figure represents a three factor experiment where the number of cases
reduces from 8 to 4; however the same logic applies to four factors.
Figure 3: Visual representation of a Full Factorial vs. Half Factorial
The experiment variables were selected to be the pressure plate thickness, tie rod
load, aspect ratio of the cell, and the separator thickness. Minitab’ half factorial logic
was used to calculate the set of cases and these are shown below in Table 1. In general
the table is just a set of zeros or ones representing the two levels of each factor. To
generate the cases specific to this analysis, the following four parameters were entered
into Minitab: pressure plate thickness, load, aspect ratio, tie rod location. These cases
were entered as variables within a finite analysis model as parameter such that the model
could be run with any set of variables. The outputs from the analysis were then entered
back into Minitab for post processing. The details of the post processing are within the
Results and Discussion sections below.
6
Table 1: DOE factorial variables for analysis
Run #
1
2
3
4
5
6
7
8
Pressure
Plate
Thicknes
s (mm)
16
20
16
20
16
20
16
20
Load
(kPa)
200
200
350
350
200
200
350
350
Separator
Aspect
Thickness
Ratio
(mm)
1.5:1
1.5:1
1.5:1
1.5:1
2:01
2:01
2:01
2:01
2
4
4
2
4
2
2
4
The table above contains the 8 cases which were analyzed. The factors were selected
as bounds of the design space for each of the variables. The pressure plate thickness is
the through thickness of the plate which was varied from 16mm to 20mm which is
scaled for size from similar documented analyses [5]. The load is the pressure on the
active area of the cell ranging from 200 kPa to 350 kPa which are standard loads on a
fuel cell. The aspect ratio of the cell varied from 1.5:1 to 2:1 which is within a standard
range of aspect ratios where fuel cells are not perfectly square as found in various studies
[8]. The separator thickness was set to 2mm because standard fuel cell channels are
around 1mm deep or less [9] . To add significant thickness the high level of the factor
was set to 4mm.
2.3. Geometry
The geometry is a simplified representation of a single cell fuel cell containing
pressure plates, a fuel flow field plate, an air flow field plate and an electrode assembly
(UEA). To reduce the number of elements needed, the model was made as ¼ of the
actual assembly by taking advantage of symmetry. This is possible because the cell is
symmetrical in both directions. This means instead of four tie rod loading points in the
model, there is only one corner which is loaded. The gaskets within the flow field plates
were ignored as they are only on the very edge of the plate and this study is focused on
7
the lack of pressure within the center of the assembly. The cell stack is assumed to have
external manifolds for the reactants and coolant flows. Therefore, there are no features
on the pressure plates other than the flanges for the tie rods.
The resultant model including all of the modifications above is shown in Figure 4.
The highlighted parts are the pressure plates while the remainders of pieces in the middle
are the separator plates, which have channels in them, as well as the UEA. In the figure
below, the pressure plates are 20mm thick, the separator plates are 2mm thick and the
UEA is 0.5mm thick. The area of the cell for both cases of active area was held at a
constant 200cm2. The 1.5:1 aspect ratio had a length of 175mm and width of 115mm.
The 2:1 aspect ratio has a length of 200mm and width of 100mm. The actual lengths in
the model are half of that due to the use of symmetry.
Figure 4: Geometry of the single cell
The hole for the tie rods was also removed to create a simpler mesh and not generate
unrealistic stresses on the hole due to the coarse mesh in that area. Since that area was
not of concern for this analysis the simplification was considered appropriate.
8
2.4. Materials
Three different materials are included in the fuel cell. The pressure plates are made
of Type 316 stainless steel. The flow field plate is made from graphite. Material
properties from a standard fuel cell plate supplier, Graftech, were used for the separator
plate materials. The UEA made up of multiple materials; however the thickest is the
GDL, which is made from a fiberous graphite. Therefore the standard GDL material
properties were used for the UEA. The material properties are listed in table 2.
Table 2: Material Properties
Material
Density
(kg/m3)
Young’s
Modulus
(GPa)
Poisson's
Ratio
Source
316 SS
7.92
193
.28
[10]
1.5
10
.27
[11]
.9
10
.27
[11]
Graftech
Graphite Plate
UEA
2.5. Boundary Conditions
As mentioned above, in order to simulate the entire cell while still maintaining a
manageable model, the symmetry boundary condition was used on two sides of the cell.
Both the sides in highlighted in Figure 5 have symmetry boundary conditions applied.
Therefore the actual cell size has four times the area analyzed in the model.
9
Figure 5: Highlighted surfaces represent symmetry boundary conditions
The load on the fuel cell is created by four tie rods that run between the pressure
plates similar to Figure 2. The load from the tie rod is distributed onto the pressure plate
through a washer. The load from this washer is represented by an applied pressure on the
circular surfaces on the pressure plate flanges. The area highlighted in Figure 6 is the
area where load was applied to the model.
10
Figure 6 – Applied pressure location on the pressure plate
The opposing side of the other pressure plate has the equal but opposite applied
pressure to the same size area. This pressure was scaled such that the correct pressure
was applied to the cell. The applied pressure is calculated from the force from the tie rod
and the area of the washer which is in contact with the pressure plate and is represented
by the following equations:
pT 
FT
AT
11
[5]
Where the Force on the tie rod is ¼ of the force on the cell because there are four tie
rods:
FT 
Fc
4
[5a]
And the force on the cell is calculated from the pressure and area of the cell separated
into length and width within this equation:
Fc  LcWc p c
[5b]
The diameter of the washer which is loaded by the tie rod is 12mm, and the area which is
loaded in the model is:
AT 
D 2
4
[5c]
The resultant equation is for the applied pressure is:
pT 
LcWc pc
D 2
[6]
This pressure is a parameter in Comsol calculated for each cell pressure case. For
200Kpa the pressure is 8.897MPa. For a 300KPa pressure on the cell, the pressure is
13.35MPa.
2.6. Mesh
The mesh is made up of triangular prism elements. In order to mesh the separator
plate with channels, a surface mesh of triangles was first created on the side of the plate.
The remainder of the pressure plate and the UEA were also surface meshed with
triangles as seen in Figure 7.
12
Figure 7 - Side view of cell mesh
The tabs at the top of the pressure plate where the tie rods are connected were
surfaced meshed with triangles on the planes perpendicular to the rest of the model. This
enabled the circle to be meshed appropriately while still having the mesh align with the
triangular prisms below. The surfaces of triangular mesh were then swept across the
volumes to create triangular prisms. The flanges at the top of the pressure plates were
swept perpendicular to the remainder of the model. The entire mesh of the pressure plate
is shown in Figure 8 below.
13
Figure 8: Mesh containing triangular prisms.
14
3. RESULTS
3.1. Results Overview
The results below show how the pressure within the cell for each case of the DOE
table. The main focus of results is the pressure distribution across the separator plate. In
the finite element analysis this load is represented by the Z component of the stress
tensor. Most of the plots below display a cross section within the separator plate of each
of the results. The maximums, minimums and averages of the Z component of stress are
then tabulated in the DOE table.
The model detailed in the above section was run in Comsol. To get a general idea of
the resultant stresses in the model, a plot of the von Mises’ stresses is displayed below in
Figure 9. The von Mises’ stress is a criterion that summarizes the complete distribution
of the state of stress on the system with a single number and is defined as [7]:
0 
1
2
[( 1   2 ) 2  ( 2   3 ) 2  ( 3   1 ) 2 ]1 / 2
15
[7]
Figure 9 – Von Mises stress plot of the pressure plate
Evaluating the von Mises’ stress is good for understanding the general stresses in
the cell and also makes sure that the materials are not being stressed beyond their yield
stress. The yield stress for the pressure plate material, which is 304 SS is 276MPa [12]
and the results in Figure 9 show the max stress at 192 MPa which means that the
16
analysis is not producing unrealistically high stresses to the point where the material
would yield.
3.2. Selecting a Surface for Evaluation
However, von Mises’ is not what is needed for the evaluation of stress on the UEA.
For that the Z component of the stress tensor will be used as that component is normal to
the surface under evaluation. For the baseline case there were multiple surfaces
evaluated. Due the thinness of the UEA layer itself a cross section through that material
was not used. Instead cross sections of the separator plate adjacent to the UEA were
evaluated for stresses. Below in Figure 10, are the 3 surfaces that were evaluated in the
baseline case. The first was on the anode separator plate ¾ thickness of the separator
away from the UEA. The second was on the anode separator plate through the ½
thickness of the separator. This location intersected with the channels and therefore the
results have gaps of material missing. The third location was on the cathode plate ½
through the thickness. This section also included the channels in the cross section. The
purpose of the cathode results was to ensure the results were symmetrical since the
model is symmetrical.
17
Figure 10: Cross Section Planes within separator plates.
Figure 11 shows the actual location of the cross section through the pressure plate
with the Z stress plotted. The Z direction of stress is through the thickness of the fuel
cell. In this figure the cross section is through the ¾ anode thickness. Because the anode
thickness increases for 4mm during some of the cases, the surface was held at a set value
of 2mm from the center of the cell. Although there are no gaps in the results shown, the
horizontal lines represent the uneven stress due to the close proximity to the channels in
the plate. In general this plot shows the stresses highest closer to corner of the pressure
plate with the tie rods as would be expected. The following plots will be 2D plots of the
cross section shown in Figure 11.
18
Figure 11 - Plot of Z direction of stress in a cross section of the separator plate.
To choose which plane is best for the comparison, the three locations were
plotted and are shown in Figure 12. The tie rod load is on the top right hand corner of the
plots which is why the highest negative stress is in these locations. All three of the plots
display very similar results, however the first one shown, which is ¾ through the
thickness of the separator plate will be used for comparison because it is easier to view
without the lines from the channels actually shown, however, it does have the stresses
from the channels on the plot. The second and third plots are the same location on both
the anode and cathode plate. There is a slight difference likely due to some difference in
mesh and the line constraints which were applied to the cathode side of the model to
hold it in space.
19
Figure 12: Comparison Plots of locations within the separator plates (a) 1.5 mm from center anode
side (b) 1mm from center anode side (c) 1mm from center cathode side.
The plots in figure 10 have a scale which goes from 0.1 MPa to -0.1 MPa.
Therefore the light yellow represents values close to zero while the blue represents
compression on the cell and the red is the area the cell is not being compressed.
However, the maximum on the compression side, -9.1 MPa, is significantly higher than
the value of the load in the uncompressed area, 0.53 MPa. Figure 13 shows this stress
with the compression stress peaking in the corner nearest the tie rod load.
20
Figure 13: 3D Plot of Z component stress in baseline separator plate
To get more resolution on the upper ends of the scale since that is where the significant
loads are the scale was changed from +/- 0.1 MPa to +/- 0.5MPa. It is still symmetric
around zero such that the edge of the compression area can easily be seen.
3.3. Cell Pressure Results – Z component of Stress
Below are the results from the study of the 8 cases which were evaluated. There
are four factors which will be used to compare the compression on the UEA. The max
compression value is the largest negative of the Z direction stress. Alternately the
minimum compression is the maximum positive value on the surface evaluated. The
average compression (negative Z direction stress) is a calculated average of the stresses
21
on the surface under evaluation and the range column is simply the difference between
the max and min compression. The range is obviously heavily driven by the max
compression because it is 1 to 2 orders of magnitude higher than the min. The details of
these results are represented by two methods. One is an observation of the stress plots
output from Comsol. The other is the output of the factorial analysis of the data itself
within Minitab.
Table 3: Compression results by case
Run #
1
2
3
4
5
6
7
8
Pressure
Plate
Thickness
(mm)
16
20
16
20
16
20
16
20
Load
(kPa)
Aspect
Ratio
200
200
350
350
200
200
350
350
1.5:1
1.5:1
1.5:1
1.5:1
2:01
2:01
2:01
2:01
Separator
Max
Min
Average
Thickness Compression Compression Compressi
(mm)
(MPa)
(MPa)
on (MPa)
-6.95
-4.50
-10.04
-9.62
-9.15
-5.31
-12.05
-7.87
2
4
4
2
4
2
2
4
0.34
0.20
0.40
0.44
0.53
0.31
0.68
0.38
-0.26
-0.20
-0.35
-0.35
-0.35
-0.20
-0.35
-0.35
Range
(MPa)
7.29
4.70
10.43
10.06
9.68
5.62
12.73
8.25
The Plots for all 8 cases are below in Figures 14 & 15. All of the plots have the same
scale, however, cases 1-4 have a different aspect ratio as cases 5-8 and this can be seen
in the shape of the images. The images are all images of the cross section plane within
the anode separator plate. The vertical lines representing spikes in stress are due to the
cross section surface being in close proximity to the channels in the plate. The applied
stress from the tie rod is not shown, but is on the top right corner of the cross section.
22
(a)
(b)
(c)
(d)
Figure 14: Plots of the Z component stress tensor (a) Case 1 (b) Case 2 (c) Case 3 (d) Case 4
Cases 1-4 all have varying levels of pressure plate thickness, separator thickness, and
load. The largest difference is when comparing cases 1 and 2 to cases 3 and 4, is that 1
23
and 2 have a load of 200 kPa while 3 and 4 have a load of 350 kPa. The results of the
change in load are both the area and magnitude of the compression stress (blue) is larger
in cases 3 and 4. Likewise the red area representing high tensile stresses is also larger for
cases 3 and 4. Cases 1 and 3 have a pressure plate thickness of 16mm while 2 and 4 have
a thickness of 20 mm and cases 1 and 4 have a separator thickness of 2mm while cases 2
and 3 have a thickness of 4mm. While it is difficult to see any major differences in the
plots, there appears to be a small effect of the width of the local stresses due to the
change in thickness of the separator plate.
Comparing cases 1-4 with cases 5-8, the aspect ratio does not appear to have a
significant effect on the areas of significant compression or lack thereof. The excess
length in cases 5-8 appears to be areas of low stress. The area of low stress appears to be
a large percentage of the cell which could mean high electrical resistivity in the cell. Due
to the short width direction in these cases, the area of low stress between the large
compression and large tensile stress is extremely narrow.
Cases 5 and 7 have a pressure plate thickness of 16mm while 6 and 8 have a
thickness of 20 mm. Also cases 6 and 7 have a separator thickness of 2mm while cases 5
and 8 have a thickness of 4mm. Neither of these appears to have significant effects on
the trends in the plots. Similarly to cases 1-4, the load difference between cases with low
loaded (5 & 6) and those with high load (7 & 8) are significant.
24
(a)
(b)
(c)
(d)
Figure 15: Plots of the Z component stress tensor (a) Case 5 (b) Case 6 (c) Case 7 (d) Case 8
25
3.4. Factorial Analysis Results
The second part of the results is the DOE factorial analysis. All of the plots below
were generated using Minitab’s factorial analysis feature. It takes into account the data
from all 8 cases which were run. This allows for the comparison of each of the variables
by evaluating the results in table 3. Below in Figure 16 is the plot of the main affects
which cause a change in the average Z direction of stress on the cell. Within this plot
there are 4 individual graphs, one for each variable changed, which has a point for each
of the values used in the analysis. The steeper the slope of the curve, the more impact
that variable had on the results. For this analysis the load on the pressure plate has the
largest impact on the average, which the separator thickness had the least. The pressure
plate thickness was also significant, while the aspect ratio was not very significant.
Main Effects Plot for Average
Data Means
-250000
Pressure Plate Thickness
Load
-275000
-300000
Mean
-325000
-350000
16
20
200
2.0
2
Aspect Ratio
-250000
350
Separator Thickness
-275000
-300000
-325000
-350000
1.5
4
Figure 16: Main Effects plot for Average Values
To further understand if any of these factors together are affecting the pressure on
the UEA, a Pareto analysis of the effects was produced and the results are shown in
Figure 17. The Pareto chart again shows that load is most crucial to the average pressure.
Also significant is the thickness of the pressure plate and the interaction between the
load and thickness which is represented by the “AB” term on the Pareto chart. Neither
26
aspect ratio nor separator thickness significantly affects the average Z direction stress in
the UEA.
Pareto Chart of the Effects
(response is Average, Alpha = 0.05)
126672
F actor
A
B
C
D
B
A
N ame
P ressure P late Thickness
Load
A spect Ratio
S eparator Thickness
Term
AB
C
AD
AC
D
0
20000
40000
60000
80000
Effect
100000 120000 140000
Lenth's PSE = 33652.5
Figure 17: Effects Pareto for Average
Average stress being driven by the load makes sense since the average does not take
into account the actual distribution of the load. The items which are affected by this are
the max and min stresses on the UEA. The combination of these two is represented by
the range value in the results table. The Pareto chart for the range of stress in the Z
direction, shown in Figure 18 also shows that the load is the largest factor; however, the
pressure plate thickness is almost as significant. The third most effective variable is the
combination of the aspect ratio and the load.
27
Pareto Chart of the Effects
(response is Range, Alpha = 0.05)
5371441
F actor
A
B
C
D
B
A
N ame
P ressure P late Thickness
Load
A spect Ratio
S eparator Thickness
Term
AC
C
AD
D
AB
0
1000000
2000000
3000000
Effect
4000000
5000000
6000000
Lenth's PSE = 1427010
Figure 18: Effects Pareto for Range
The Pareto chart for the maximum pressure, seen in Figure 19, is very similar
because the range factor is largely driven by the maximum pressure.
Pareto Chart of the Effects
(response is Max, Alpha = 0.05)
4247484
F actor
A
B
C
D
B
A
Term
AC
C
AD
D
AB
0
1000000
2000000
3000000
Effect
4000000
Lenth's PSE = 1128412
Figure 19: Effects Pareto for Max Z Stress
28
N ame
P ressure P late Thickness
Load
A spect Ratio
S eparator Thickness
The Pareto chart of effects for the minimum stress, see Figure 20, does not have a
dominant factor. All of the factors are roughly equal, although the pressure plate
thickness is the most significant.
Pareto Chart of the Effects
(response is Min, Alpha = 0.05)
609590
F actor
A
B
C
D
A
C
N ame
P ressure P late Thickness
Load
A spect Ratio
S eparator Thickness
Term
B
AC
D
AB
AD
0
100000 200000 300000 400000 500000 600000
Effect
Lenth's PSE = 161948
Figure 20: Pareto Chart of effects for Min Z stress
29
4. DISCUSSION
Each of the Z component stress plots has a very distinct shape even if their values
differ. There is an arc of zero loads where the pressure goes from positive to negative.
This point is the equilibrium point where the deflection of the pressure plate is equal to
the compression of the system. The tie rod loads put a moment on the edge of the
pressure plate which from basic beam theory will cause a deflection away from the
UEA. This causes the lack of load on the UEA. Changing the pressure plate design could
yield improved results.
The cell loaded by the pressure plate in this design would likely have a short lifetime
due to the extremely high loads which are 20-50 times higher than the average load
expected in the cell. These loads could likely damage the UEA especially during thermal
cycling.
On the opposing side, the areas with little or no pressure on the cell would definitely
suffer from conductivity issues. The electron flow through the cell is a basic function of
the fuel cell and an air gap between layer will be extremely resistive and performance
with not be as good as if there were load on those sections of the cell.
To investigate ways to improve the distribution of the cell an optimization
analysis was run in Minitab. The analysis is shown below in Figure 21. This
optimization analysis used the relationship between each of the factors and the outputs to
determine the optimized solution for a target output. The settings can then be modified
interactively to see how the settings affect the responses. The two outputs which were
evaluated in Figure 21 were the range of Z direction stress and the average of Z direction
stress.
This optimization analysis used the relationship between each of the factors and
the outputs to determine the optimized solution for a target output. The settings can then
be modified interactively to see how the settings affect the responses. The two outputs
which were evaluated in Figure 21 were the range of Z direction stress and the average
of Z direction stress and can be seen in the left side column. The range was set to a target
value of minimum which means the optimizer will try to get the value as close to zero as
possible. The average was set to 300 kPa which is in between the applied pressures of
30
200 kPa and 350 kPa. The row of plots aligned with each of the outputs show the
relationship between each of the factors, which are across the top are of the chart, and
that individual output. The steeper the curve is on the plot, the greater the impact on the
output. Although they are cut off the factors from left to right are: pressure plate
thickness, applied load, aspect ratio and separator thickness. The bracketed numbers are
the current values being input into the optimization while the numbers above and below
are the two levels which were used in the analysis.
Optimal
High
D
Cur
0.55169 Low
Pressure
20.0
[20.0]
16.0
Load
350.0
[300.0]
200.0
Aspect R
2.0
[2.0]
1.50
Separato
4.0
[4.0]
2.0
Composite
Desirability
0.55169
Range
Minimum
y = 6.918E+06
d = 0.30826
Average
Targ: -300000.0
y = -2.99E+05
d = 0.98735
Figure 21: Optimization of Factorial Analysis
The top row of plots displays the relationship between each factor and its ability
to satisfy the targeted solution. The higher the point on the plot the more desirable the
solution is. The plots for pressure plate thickness, aspect ratio, and separator thickness
all increase to the right on the desirability plots and therefore the maximum values for
those factors are best for the solution. The load however has a maximum point which
means that point is optimal for the solution. The composite desirably value on the left is
31
the geometric mean of each of the factor desirabilities, meaning how well the variables
satisfy the solution. The composite desirability scale is from 0 to 1 with 1 being an
optimal solution. In Figure 21 the composite desirability is 0.55 meaning that the factors
do not completely satisfy the output targets. However, that is the optimal solution for the
input range and output targets for this analysis. There are no other values which will
yield a higher desirability of this analysis.
The final optimized values are therefore a pressure plate thickness of 20mm, a
load of 300 kPa, an aspect ratio of 2:1 and a separator thickness of 4. Although these
exact values are not optimal they show which levels of each factor were in the direction
of improving distribution of the cell.
Because there are other target outputs besides stress on the cell when designing a
fuel cell this analysis could be used to evaluate alternate near optimal solutions. For
example, if for flow reasons a 1.5:1 aspect ratio is desired, the value in analysis could be
set to 1.5 and it would yield the plots shown in Figure 22.
New
High
D
Cur
0.50785 Low
Pressure
20.0
[20.0]
16.0
Load
350.0
[300.0]
200.0
Composite
Desirability
0.50785
Range
Minimum
y = 7.362E+06
d = 0.26385
Average
Targ: -300000.0
y = -2.98E+05
d = 0.97750
Figure 22: Low aspect ratio solution
32
Aspect R
2.0
[1.50]
1.50
Separato
4.0
[4.0]
2.0
Surprisingly the aspect ratio was not a large factor in the analysis. The results
show that changing from an aspect ratio of 2:1 to 1.5:1 only changed the composite
desirability from .55 to .50. From the plots in the aspect ratio column it can also be
observed that the affect on average stress is almost zero since the line is nearly
horizontal. The actual value for average stress went from -299 kPa to -298 kPa. The
range of stress went from 6.92 MPa to 7.36 MPa. If these differences were insignificant
for a given cell design the aspect ratio could be changed to anything between 2:1 and
1.5:1
The fact that the aspect ratio was could be due to the range of the values used for
the analysis or the fact that the load distribution was so poor due that the aspect ratio did
not matter. The percentage of the cell which had near zero loads was much greater in the
higher aspect ratio cell. Depending on the sensitivity to loads of a given UEA this could
justify having a cell with a smaller aspect ratio.
Based on the current results it does not appear that the current design can ensure
a fully loaded cell assembly without a change in basic design. Moving the actual tie rod
locations closer to the center of the pressure plate would help the load distribution
because the load would no longer be concentrated in the corner of the cell.
The pressure plate itself could also change shape. Having a pressure plate which
applies higher load to the center of the cell due to its shape would avoid the cantilever
affect given by the current design.
33
5. CONCLUSIONS
Although the exact design of pressure plate did not evenly distribute the load, the
analysis pointed out the items which were most greatly affecting the distribution of load.
From the Pareto charts of the factorial analysis it is important to note that pressure plate
thickness was a large factor in the load distribution. The thicker pressure plate was better
due to the stiffness and therefore lack of deflection across the surface of the cell.
Alternate designs could also change the load such as designs which load from the middle
of the cell as seen in some existing industry designs [13].
Changing the load on the cell is something else which had significant impact to
the results. A further study should be done on any specific cell to ensure the loading is
correct for the membrane and GDLs being used as there could some which are more
tolerable to high pressure and others which may have bad conductivity without
significant load.
One of the weaknesses of the 2 level factorial analyses is that the system is
assumed linear within the range being examined. To further understand the loading of a
single cell an analysis with more levels would have to be evaluated. However, factorial
analysis identified the factors which have the greatest impact on the load on the cell so
they can be the factors which are analyzed in more detail and the insignificant ones can
be assumed to have minimal impact. The variables with the greatest impact could be
analyzed with more levels and curve fit relations could be developed. Those curves
could then be used to optimize a given design or pinpoint exactly how the design could
be changed to improve the fuel cell.
34
6. BIBLIOGRAPHY
1. Energy, U.S. Department of. How Fuel Cells Work. FuelEconomy.gov. [Online] 11
2011. http://www.fueleconomy.gov/feg/fcv_pem.shtml.
2. Skiba, Tommy, Chi-Hum, Paik and Jarvi, Thomas D. Ultrasonically Welded Fuel
Cell Unitized Electrode Assembly. US200901699446A1 United States of America, July
2, 2009.
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cell. Woo-kum Lee, chien-Hsien Ho, J.W. Van Zee, Mahesh Murthy. 1999, Journal
of Power Sources 84, pp. 45-51.
5. Analyses of the fuel cell stack assembly pressure. Lee, Shuo-Jen, Hsu, Chen-De and
Huang, Ching-Han. 145, TaoYuan : Journal of Power Sources, 2005.
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Bejan, A. 19-20, Tallahassee : International Journal of Heat and Mass Transfer, 2004,
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10.
Stainless
Steel
AISI
Type
316.
efunda.com.
[Online]
http://www.efunda.com/Materials/alloys/stainless_steels/show_stainless.cfm?ID=AISI_
Type_316&prop=all&Page_Title=AISI%20Type%20316.
11.
Grafcell.
[Online]
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http://www.graftechaet.com/CMSPages/GetFile.aspx?guid=30e6203d-ad27-4936-948130d75f37ac36.
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12. Shigley, Joseph, Mischke, Charles and Budynas, Richard. Mechanical
Engineering Design. New York, NY : McGraw Hill, 2004.
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Structure. 5009968 United States of America, April 23, 1991.
36
7. APPENDICES
7.1. Comsol Files:
7.2. Minitab Files
37