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Proceedings of the 6th Annual ISC Graduate Research Symposium ISC-GRS 2012

April 13, 2012, Rolla, Missouri

OPTIMIZATION MODEL FOR BIO-INSPIRED DESIGN OF BIPOLAR PLATE FLOW FIELDS FOR POLYMER

ELECTROLYTE MEMBRANE FUEL CELLS

ABSTRACT The flow field of a bipolar plate distributes hydrogen and

oxygen for polymer electrolyte membrane (PEM) fuel cells and

removes the produced water from the fuel cells. It greatly

influences the performance of fuel cells, especially regarding

reduction of mass transport loss. Flow fields with good reactant

distribution capability reduce the mass transport loss, thus

allowing higher power density. The research described in the

present paper is inspired by natural structures such as veins in

tree leaves and blood vessels in lungs, which efficiently feed

nutrition from one central source to large areas and are capable

of removing undesirable by-products. A mathematical model is

developed to optimize the flow field with minimal pressure

drop, lowest energy dissipation, and uniform mass distribution.

The model can be used to generate optimum flow field designs,

leading to better fuel cell performance for different sizes and

shapes of bipolar plates. Finite element modeling (FEM) based

simulation of PEM fuel cells is conducted to compare the flow

field designs obtained using the developed optimization model

and the conventional designs. The numerical results show that

the fuel cell with the optimized design has a higher power

density than that of the conventional designs, with a lower

pressure drop and more uniform transport of the reactants.

1. INTRODUCTION As an energy conversion device the polymer electrolyte

membrane (PEM) fuel cell has great advantages such as low-

temperature operation, high-power density, fast start-up, system

robustness, and low emissions. It is a very promising alternative

power source for automotive applications [1]. The flow field of

a bipolar plate supplies fuel and oxygen to reaction sites and

removes reaction products (i.e., liquid water) out of the fuel

cell, significantly affecting the performance of PEM fuel cells.

Previous studies have demonstrated that the power density of a

fuel cell can greatly increase with a proper flow field design

[2,3].

Numerous flow field designs have been proposed and

investigated. These designs can be classified into four

categories of layouts: pin-type, parallel, serpentine and

interdigitated [4,5]. Among them parallel and serpentine

designs are the most widely known and utilized.

Besides conventional flow field layouts, researchers have

begun to investigate flow field designs with inspirations from

nature [8-13]. These bio-inspired flow field designs have

demonstrated the advantages in distributing reactants and

enhancing performance. Kloess et al. [8] combined serpentine

and interdigitated patterns with a leaf/lung layout to design the

bio-inspired flow layouts. They reported that these new designs

reduced pressure loss and increased power density as compared

to pure serpentine and interdigitated designs. Chapman et al. [9]

claimed that their bio-mimetic designs, which comprised main

and sub-feed channels that are tapered and non-linear, could

enhance the performance of PEM fuel cells by 16%. The above

studies have shown the potential benefits of incorporating

natural structures into flow field designs. However, the bio-

inspired designs that have been investigated thus far are only

bio-mimetic, i.e. they mimic some particular structures in

nature’s biological systems. There is a lack of comprehensive

and systematic studies on how to create an optimal bio-inspired

design of flow fields for bipolar plates.

With inspiration from the idea that nature structures (e.g., leaf

veins) provide excellent mass transport with uniform pressure

distribution and low energy dissipation, the present study

creates a mathematical model aimed to generate flow field

designs with these properties. The generated designs include

both flow field layouts and channel geometries. The model

starts with small grids on the flow fields and generates

optimum designs with structures similar to nature. This is

intended to lead to minimum pressure drop, uniform pressure

distribution, and better fuel cell performance. The developed

mathematical model provides a flexible approach to generate

optimal designs for different shapes of bipolar plates and

different locations of inlets and outlets, and flow field designs

with different desired properties. Given the bipolar plate’s

shape and the locations of the inlet and outlet, an example of

the generated optimal designs was investigated with FEM

simulation, and the FEM data of the optimal design was

compared with those of the conventional parallel and serpentine

designs. The results show that the generated design provides

better fuel cell performance, with more uniform pressure

distribution and less pressure drop over the flow field.

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2. MATHEMATICAL MODEL 2.1 Flow field

A flow field involves its layout and the channel geometries,

which include cross-section shapes and dimensions. Usually for

a flow field design, researchers start from a particular flow field

layout (e.g., serpentine [14] or interdigitated [15]) and then

optimize the cross-section shapes or dimensions of the channels

to achieve desired designs. Our proposed approach starts from

the flow field grids (see Fig. 1), where the flow field is meshed

by 𝑁×𝑁 nodes and (2×𝑁×(𝑁-1)) channels. After optimizing

the channel dimensions, the corresponding flow field layout can

be determined because the flow field layout essentially is the

configuration of all the flow channels.

2.2 Optimization model

The mathematical model is formulated into a constrained

optimization problem. The steps for solving the problem are as

follows: (1) Mesh the flow field into small grids, which can be

triangle, rectangular, hexagonal, etc.; (2) Establish flow

relationships using these grids, and define objective function

and constraints; (3) Optimize the objective function using the

Lagrangian method; and (4) Generate the optimum flow field

design. The flow chart of the solution approach is given in Fig.

1.

Fig. 1 Flow chart of the solution approach.

2.2.1 Flow relationship

In the proposed mathematical model, several assumptions are

made. Firstly, the flow through the channels is laminar flow.

This assumption is fairly true in reality, since, under most

conditions, gas flow in fuel cell channels is laminar; only at

extremely high flow rates can gas flow become turbulent [16].

Secondly, only the pressure drop along each channel is

considered in the model, while the pressure losses at the

junctions of channels are ignored. Finally, the gases in the flow

channel obey ideal gas laws.

For laminar flow, the pressure drop in a channel for a given

flow rate can be determined by

∆𝑝 = 𝑓𝐿

𝐷

𝜌𝑣

2 (1)

where 𝐿 is channel length, 𝜌 is fluid density, 𝑣 is average flow

velocity in the channel, 𝐷ℎ is hydraulic diameter, and 𝑓 is

friction factor. For a channel with a rectangular cross-section

(width: 𝑤, depth: 𝑑) [16],

𝐷ℎ =4𝐴

𝑃=

2𝑤𝑑

𝑤+𝑑 (2)

where 𝐴𝑐 and 𝑃 are the channel’s cross-section area and

perimeter, respectively. For laminar flow,

𝑓 =64

𝑅𝑒 (3)

where 𝑅𝑒 is Reynolds Number of the flow,

𝑅𝑒 =𝜌𝑣𝐷

𝜇 (4)

where 𝜇 is fluid viscosity. The volumetric flow rate has the

relationship

𝑞 = 𝑣 ∗ 𝐴𝑐 =𝜋𝐷

𝑣

4 (5)

By substituting equations (3)-(5) into equation (1), the pressure

drop in a channel can be expressed in terms of flow rate and the

channel’s dimensions as

∆𝑝 =128𝜇𝐿𝑞

𝜋𝐷 (6)

Consider a flow field composed of 𝑁×𝑁 nodes that are

connected by channels (Fig. 2). Channel (𝑘, 𝑙) is designated as

the channel connecting two neighboring nodes designated by 𝑘

and 𝑙. Therefore, once the flow rate and the channel dimensions

are known, the pressure drop in each channel (𝑘, 𝑙) can be

determined as follows:

∆𝑝𝑘𝑙 =128𝜇𝐿 𝑞

𝜋𝐷 (7)

Fig. 2 Mesh of the flow field with 𝑁×𝑁 nodes (one inlet, one

outlet), which are connected by (2×𝑁×(𝑁-1)) channels.

There are varying volumetric flow rates through the channels.

For a given node 𝑘, the in-flow rate should equal the out-flow

rate. The relationship can be expressed as a zero sum of all the

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flow rates through a node if the in-flow rate is defined as

positive and the out-flow rate is negative, i.e.

∑ 𝑞𝑘𝑙𝑘 = 0 (8)

2.2.2 Constraints

The wider the channel is, the lower the pressure drop that

occurs in the channel. However, the channel cannot be

arbitrarily wide. The total surface area of the channel is

𝑆𝑐 = ∑ 𝐿𝑘𝑙𝑤𝑘𝑙(𝑘,𝑙) (9)

This area must be less than the area of the flow field. Surface

area is an important parameter in the design of a bipolar plate

flow field because it directly influences the mass transport

efficiency and the electrical conductivity of bipolar plates.

Because the total active area equals the sum of the surface area

of channels and the land area, which is the contact area between

the bipolar plate and the gas diffusion layer (GDL), a larger

surface area of channels would be good for mass transport but

would reduce the conduction of electricity. Therefore, a trade-

off between mass transport and electrical conductivity is needed

in order to decide a proper channel surface area. For the

purpose of providing an illustrated example, assuming that the

surface area is 60% of the total active area,

𝑆𝑐 = 0.6𝐴 (10)

where 𝐴 is the active area of the fuel cell.

2.2.3 Objective function

For PEM fuel cells, a uniform pressure distribution over the

flow field means a uniform reactant concentration distribution,

which will reduce the local over-potential caused by the large

concentration gradient. Therefore, in this study lowest pressure

difference over the flow field is used as the objective function:

𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝐹 = ∑ ∆𝑝𝑘𝑙(𝑘,𝑙) = ∑128𝜇𝐿 𝑞

𝜋𝐷 (𝑘,𝑙) (13)

2.2.4 Optimization algorithm

The Lagrangian method [17-23] is used to solve the constrained

optimization problem. The Lagrangian function is defined as

ϕ({qkl}, {Dklh }) = ∑

128μL q

πD (k,l) − λ∑ LklDkl

h(k,l) (17)

where λ is the Lagrange multiplier. The necessary conditions

for a minimum F subjected to constraint (10) are ∂ϕ

∂q = 0,

∂ϕ

∂D = 0 (18)

Consider the derivative of Φwith respect to Dklh

Dklh = (−

512μq

λπ)

(19)

Then, with the constraint (10), in the minimum configuration an

implicit scaling relationship between hydraulic diameter and

flow rate is q

D − K

1

(2d−D ) = 0 (20)

where K =d

C∑

q L (2d−D )

D (m,n) is a constant and C = 0.6A.

To start the calculation, a set of values (e.g., 1.5 mm for all

channels) is used to initialize the hydraulic diameter {Dklh }. The

iteration process continues; at each iteration, the hydraulic

diameter varies by solving the implicit scaling relationship

given in equation (20), until the minimum pressure drop is

achieved, thereby resulting in the optimum flow field design. A

Matlab program has been developed to implement this

optimization algorithm.

2.3 Optimized flow field designs

Several designs for different situations (i.e., positions of inlets

and outlets) were obtained using the proposed optimization

algorithm, for a 5×5 cm2 flow field with the total surface area

of channels being 60% of the total active area. The results for

several scenarios are shown in Fig. 3. These design examples

show the great potentials and capabilities of this optimization

algorithm, which can generate the desired flow field design

with the desired properties from well-defined objective

functions and constraints. This algorithm can be used for flow

fields of any shape and with any locations of inlets and outlets.

CAD models were constructed based on the channel width data

generated from the optimization model for FEM analysis and

fabrication of bipolar plates for experimental study. Although

the optimized designs have complex geometries with tapered

channels, they can be easily fabricated with the Selective Laser

Sintering (SLS) process [24-26]. Fig. 4 (a) and (b) show the

corresponding CAD models from Fig. 3(c) and (d).

Fig. 3 Optimum flow field designs obtained from the

optimization model when (a) N = 11, one inlet with one sink;

(b) N = 11, one inlet with uniform sink; (c) flow field without

loops, N = 11, one inlet with uniform sinks; (d) flow field

without loops, N = 16, one inlet with uniform sinks.

3. FEM SIMULATION The operation of PEM fuel cells involves complicated

processes including electrochemical reaction, flow mechanics,

and thermal mechanics. Most of these processes are difficult or

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even impossible to investigate with experimental study only.

Finite element modeling (FEM) can be used to provide

predictive models of PEM fuel cell operation [27-32]. In our

present work, the fuel cell module from the commercial

software ANSYS Fluent was used to investigate the

performance of a fuel cell, including polarization curves,

pressure distribution, and reactants distribution within a fuel

cell. Since the performance of a fuel cell is determined largely

by the oxygen reduction reaction at the cathode side, this study

mainly focuses on the distributions of pressure and reactants at

the cathode side. Fig. 5 shows the pressure distribution in the

flow channels and also at the interface of the GDL and catalyst

layer, when the operation voltage is 0.55 V. The corresponding

oxygen distribution at the interface of the GDL and catalyst

layer is shown in Fig. 6. To give a detailed and quantitative

comparison of pressure and mass distribution for different

designs, the pressure and oxygen mass fraction values on three

cross-sections of the interface of GDL and catalyst layer are

reported in Figs. 7 and 8, respectively. The positions of these

cross-sections are indicated in Figs. 5 and 6, and the distances

from inlet are: 10 mm for section 1, 25 mm for section 2, and

40 mm for section 3. Fig. 9 shows the polarization curves and

power density curves for the three different designs. It can be

seen that the optimized design provides the highest

performance among the three flow field designs.

4. DISCUSSION 4.1 Pressure and mass distribution

It can be seen in Fig. 5 that the optimized design obtained from

the proposed optimization model has the lowest pressure drop

(259 Pa) over the flow field (from inlet to outlet) among the

three designs. It is four times lower than that of the multiple-

serpentine design (1191 Pa). A lower pressure drop is better for

the operation of PEM fuel cells because it could reduce the

need for an additional power supply, especially for a fuel cell

with a large active area or a stack with a large number of fuel

cell units. Moreover, the optimized design does not have

stagnant areas in flow channels like the parallel design. This is

because the pressure drop in each channel is required to be

greater than zero in the optimization model in order to satisfy

the required flow rate. This point can also be seen from the

mass distribution in the optimized design in Fig. 8(c), where all

the mass fractions of oxygen over the catalyst layer are

substantially greater than zero.

The optimized design also provides a more uniform pressure

distribution over the whole flow field, as shown in Fig. 5 and

Fig. 7 (which provides some more details than Fig. 5). From

Fig. 7, it can be seen that the optimized design has the lowest

pressure variation, followed by the parallel design, and the

serpentine design has the largest pressure variation within the

flow field. For the serpentine design, pressure drops down

sharply across each land area and then stays almost the same

within the multiple channels. The pressure of the parallel design

decreases greatly initially, and then remains almost the same in

the middle channels, where the large stagnant areas form. For

the optimized design, the overall trend is dropping down from

left to right and from bottom to top (i.e., from section 1 to

section 3), although the pressure difference is small among

adjacent areas. This is why the optimized design could

eliminate the stagnant areas even with small pressure drop. The

distributions of oxygen at the interface of GDL and catalyst for

different designs are compared in Fig. 6 and Fig. 8. These

figures show that in the optimized design the catalyst surface

has the largest amount of oxygen and the most uniform

distribution of oxygen among the three designs, thus providing

the best oxygen mass transport.

4.2 Fuel cell performance

The polarization and power density curves obtained from the

FEM simulation as shown in Fig. 9 indicate that the optimized

design has the highest performance among the three designs.

The serpentine design initially performs similarly to the

optimized design, but after the fuel cells work above 0.4 A/cm2,

the optimized design provides a better performance. Also, the

optimized design produces a higher maximum power density

when the fuel cell operates around 1.0 A/cm2. The parallel

design has the lowest performance among the three designs, as

expected, because the large stagnant areas (Fig. 6(a)) formed in

the parallel channels result in poor mass transport capabilities

[4].

The polarization curves from the multiple-serpentine design

and the optimized design in Fig. 9 can be compared further for

different regions of the polarization curves. There are three

regions of a typical fuel cell polarization curve: activation

region, ohmic region, and mass transport region, with activation

loss dominant in the first region, ohmic loss dominant in the

second region, and mass transport loss dominant in the third

region [16] (see Fig. 9). The activation loss mainly depends on

the operation condition under which the electrochemical

reaction takes place. The ohmic loss is caused by the electrical

(proton) resistances of the membrane, catalyst layer, GDL, and

bipolar plates, as well as the contact resistances among them.

The mass transport loss is caused by the depletion of reactants.

In this study, since both of the fuel cells operate under the same

condition, the activation losses are essentially the same. Also,

all of the various resistances should be the same because the

same materials and the same land areas between the GDL and

the bipolar plate are used. Therefore, nearly the same ohmic

losses exist for the two designs. The main difference between

the two designs is the mass transport loss, which is the main

cause of different performance between the two designs. In Fig.

9, the first portions of the polarization curves are almost the

same for these two designs because of the same activation

losses. For the middle and the last portions of the polarization

curves, the performance of the multiple-serpentine design is

initially similar to the optimized design due to nearly the same

ohmic losses. However, as the current density increases, the

performance of the optimized design becomes higher and

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higher than that of the serpentine design, because the effect of

the mass transport loss gradually becomes more and more

dominant. This indicates that the optimized design has more

efficient mass transport capabilities.

The above comparisons among these three designs in terms of

performance, pressure distribution, and oxygen distribution

have led to the conclusion that the optimized design provides

higher performance capability, because of more uniform

pressure distribution and more efficient mass transport.

Therefore, our proposed optimization model has a great

potential for the design of flow fields for bipolar plates and the

improvement of PEM fuel cell performance. We are currently

performing experiments to validate the results of analysis and

simulation that lead to the conclusion on the higher

performance of the optimized design.

Fig. 5 Pressure (unit: Pa) distribution in the flow channels on

the cathode side at 0.55 V for (a) parallel design, (b) multiple-

serpentine design, and (c) optimized design.

Fig. 6 Distribution of oxygen (mass fraction) at the interface of

the GDL and catalyst layer on the cathode side at 0.55 V for (a)

parallel design, (b) multiple-serpentine design, and (c)

optimized design.

Fig. 7 Pressure distribution at the interface of GDL and catalyst

layer at cathode for (a) parallel design, (b) multiple-serpentine

design and (c) optimized design. The three symbols are for the

three different sections (y direction) shown in Fig. 5.

Fig. 8 Distribution of Oxygen at the interface of GDL and

catalyst layer at cathode for (a) parallel design, (b) multiple-

serpentine design and (c) optimized design. The three symbols

are for the three different sections (y direction) shown in Fig. 6.

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Fig. 9 Polarization curves (in solid line) and power density

curves (in dashed line) for the parallel design, the multiple-

serpentine design, and the optimized design.

5. CONCLUSIONS

A mathematical model has been proposed for optimizing the

design of flow fields in order to achieve lower pressure drop

and more uniform distribution of reactants in bipolar plates,

thereby increasing the PEM fuel cell performance. The

optimization model is capable of generating flow field designs

that are similar to the structures of leaf veins found in nature.

We have demonstrated this model’s ability to optimize designs

for any shapes of flow fields, any locations of inlets and outlets,

and any desired flow properties. The comparison of FEM

simulation results between the optimized design and the

conventional parallel and multiple-serpentine designs has

shown that the optimized design can provide higher power

density and less pressure drop, due to more uniform pressure

distribution and more efficient mass transport. Experiments are

being performed to validate the optimized designs from the

simulation results. Future work will include investigating the

effect of the model parameters including the number of nodes,

the inlets and outlets positions, and the objective functions in

the optimization model, in order to further understand and

improve the design of flow fields and to increase the

performance of PEM fuel cells.

8. ACKNOWLEDGMENTS This project is funded by the National Science Foundation

grant #CMMI-1131659 and by the Air Force Research

Laboratory contract #FA8650-04-C-5704.

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