Fuel Cell Thesis

Water Management in Proton Exchange Fuel Cells: An Along-the-Channel Model with Dynamic Response for

the Solid Layer Temperature.

School of Engineering, University of Aberdeen

This thesis was submitted as part of the requirement for the MEng. Degree in Engineering

Project supervisor: Dr. M. Aleyaasin

Student: N. Guilló Vicente

Date: 13-01-2012

Water Management in PEM fuel cells – An Along the Channel Model

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Content

Content .............................................................................................................. 2

List of symbols and notation ........................................................................... 4

1. Aim of the thesis and introduction ............................................................. 7

1.1. Aim of the thesis ................................................................................................ 7

1.2. Introduction ........................................................................................................ 7

2. Basic principles of hydrogen fuel cells ...................................................... 9

2.1. Introduction ........................................................................................................ 9

2.2. Schematic description of a PEM fuel cell ........................................................ 9

2.3. Main fuel cell limitations ................................................................................. 10

2.4. Fuel cell design ................................................................................................ 11

3. Basic concepts for water management in PEM fuel cells ....................... 17

3.1. Overview of the problem ................................................................................. 17

3.2. Water sources and drains and relative humidity .......................................... 18

3.2.1. Water sources and drains in a PEM fuel cell .............................................. 18

3.2.2. Relative humidity and saturation pressure of the water .............................. 19

3.2.3. Studying water content in a fuel cell to take decisions ................................ 21

3.2.4. Methods for external humidification ............................................................ 24

3.3. PEM fuel cell cooling ....................................................................................... 25

3.3.1. Cooling using the anode air supply ............................................................. 25

3.3.2. Cooling using air or water separated from the reactant gases ................... 26

4. Description of an along the channel model for water management in a PEM fuel cell. .................................................................................................. 28

4.1. Introduction to the model ............................................................................... 28

4.2. Yi and Nguyen model and Gilbert and Lewin arrangements ....................... 28

4.3. Model applied to a specified PEM fuel cell geometry .................................. 29

4.4. Equations for the model .................................................................................. 30

4.4.1. Governing equations ................................................................................... 30

4.4.2. Supporting equations .................................................................................. 38


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5. Finite differences applied to the model .................................................... 44

5.1. Description of the applied finite differences ................................................. 44

5.2. Model discretization. Volume controls along the channel........................... 44

5.3. Other consideration about discretization ...................................................... 47

5.4. Finite differences applied to the equations of the model ............................ 47

5.4.1. Discretization of the governing equations ................................................... 48

5.4.2. Discretization of the supporting equations .................................................. 52

6. Conclusions ................................................................................................ 55

7. References .................................................................................................. 57


4

List of symbols and notation

Symbols for the basic concepts for water management in PEM fuel cells (section 3)

am Mass flow of air, g/s

wm Mass flow of water, g/s

inP Pressure in the fuel cell inlet, Pa

outP Pressure in the fuel cell outlet, Pa

satP Saturation pressure of the water, Pa

wP Partial pressure of the water vapour, Pa

inwP , Partial pressure of the water vapour in the inlet, Pa

outwP , Partial pressure of the water vapour in the outlet, Pa

inT Stream temperature in the inlet, ºC

outT Stream temperature in the outlet, ºC λ Stoichiometric coefficient, ºCφ Relative humidity of the air

Ψ Coefficient used to determine the external relative humidity

Symbols for the water management model (section 4)

a Heat transfer per unit length, cm

ka Activity of water in stream k

bA Heat transfer length between solid layer to bulk per unit area, cm

sA Conductive eat-transfer area of solid layer per unit area, cm2 d Channel height, cm oD Intra-diffusion coefficient of water in membrane, cm2/s

wD Diffusion coefficient of water, cm2/s

ipC , Heat capacity of gas i, J/mol/ºC

spC , Heat capacity of solid phase, J/g/ºC lwpC , Heat capacity of liquid water, J/mol/ºC

vkwpC ,, Heat capacity of water vapour ins stream k, J/mol/ºC


5

kwc , Concentration of water at k interface of the membrane, mol/cm3

F Faraday constant, 96487 C/equivalent h Channel width, cm

vkwH , Enthalpy of water vapour in k stream, J/mol

lkwH , Enthalpy of liquid water in k stream, J/mol

I Local current density, A/cm2

avgI Cell average current density, A/cm2 oI Exchange current density for the oxygen reaction, A/cm2

k Thermal conductivity of solid layer, J/s cm ºC

ck Evaporation and condensation rate constant, s-1

L Channel length, cm

M Molar flow rate of heat exchanger fluid, mol/s

iM Molar flow rate of species i, mol/s

drymM , Equivalent weight of a dry membrane, g/mol lkwM , Molar flow rate of liquid water in k channel, mol/s

vkwM , Molar flow rate of water vapour in k channel, mol/s

cn Number of channel for each side of membrane

dn Electro-osmotic drag coefficient

P Cell total pressure, atm

iP Partial pressure of species i, atm satkwP , Vapour pressure of water in k channel, atm

R Gas constant, 82.06 cm3 atm/mol/K or 8.314 J/mol/K

mt Membrane thickness, cm

aT Temperature of anode stream, ºC

cT Temperature of cathode stream, ºC

coolT Temperature of the coolant stream, ºC

infT Temperature of the surroundings, ºC

sT Temperature of the solid phase, ºC U Overall heat-transfer coefficient, J/s/cm2/ºC

bU Overall heat-transfer coefficient between channel and solid, J/s/cm2/ºC

infU Overall heat-transfer coefficient within the surroundings, J/s/cm2/ºC

ocV Cell open-circuit voltage, V

cellV Cell voltage, V α Net water flux per proton flux


6

HΔ Enthalpy change rate for the water formation reaction, J/mol η Over potential for the oxygen reaction, V

drym,ρ Density of a dry membrane, g/cm3

sρ Density of solid phase, g/cm3

mσ Membrane conductivity, 1/Ω/c

Specific symbols for the model discretization (section 5)

i Node number i n Number of nodes

wT Temperature in the exterior wall of the fuel cell,

iX Value of the variable X in the node i

inX Value of the variable X in the inlet

2/1+iX Mean value of the variable X between the node i and the node i+1

vapHΔ Enthalpy change for water vaporisation, J/mol xΔ Spatial step in the discretization, cm


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1. Aim of the thesis and introduction

1.1. Aim of the thesis

The aim of this thesis is to study the management of the water in a PEM fuel

cell and describe the equations for a control model for this issue based in the

model presented by Yi and Nguyen [1].

1.2. Introduction

Our energetic model, based in fossil fuels, is not sustainable in a long term.

Fossil fuels are a scarce resource that will soon be exhausted. In addition,

global warming requires the use of technologies resulting in a reduction in

emissions of carbon dioxide.

In the future, an energetic model based in hydrogen as an energetic vector may

be possible. Hydrogen could be generated by electrolysis using renewal ener-

gies, as huge arrays of solar panels or wind turbines, and be used when the

energy is needed. Of course, narrow technical circumstances, as renewal ener-

gies efficiency and hydrogen storage, have to be overcome.

Before that time in which the energy model change will be necessary, it is a

good invest to develop and get mature technologies to produce energy from

hydrogen fuel. PEM fuel cells (PEM – “polymer electrolyte membrane” or “pro-

ton exchange membrane”) are suitable devices to generate electrical energy

from hydrogen.

PEM fuel cells are based in the water formation reaction to get energy. There

are other kinds of fuel cells. In this thesis, the issue is focused in PEM fuel cells.

In the chart 1.1 the different types of fuel cells and the main characteristics of

each one are shown.

PEM fuel cells, as they are based in a chemical reaction, they are no under the

limitation of Carnot efficiency. Nevertheless, different losses in the process

make the efficient to be about 50%. But, the main advantage of PEM fuel cells

is that the only waste generated during their operation is just water.


8

Nevertheless, fuel cells are technologic devices with some design and opera-

tional complications, which will be described in this thesis. One of the main is-

sues in fuel cells is water management.

Fuel cell type Mobile ion

Operating tem-perature Applications

Alkaline (AFC) OH- 50-200ºC Used in space vehicles, e.g. Apollo, Shuttle.

Proton exchange membrane (PEMFC) H+ 30-100ºC

Vehicles and mobile applications, and for lower power CHP systems

Direct methanol (DMFC) H+ 20-90ºC

Suitable for portable electronic systems of low power, running for long times

Phosphoric acid (PAFC) H+ 220ºC

Large numbers of 200-kW CHP sys-tems in use.

Molten carbonate (MCFC) CO3

2- 650ºC Suitable for medium to large-scale CHP systems, up to MW capacity.

Solid oxide (SOFC) O2- 500-1000ºC Suitable for all sizes of CHP systems, 2 kW to multi-MW.

Chart 1.1. Different types of fuel cells, characteristics and applications [4].

Water management is dramatically important in the correct operation of a PEM

fuel cell, especially when the system is not stationary. In devices as motor vehi-

cles, for instance, the power demand is constantly changing. Also the external

conditions, such relative humidity of the air, change. This makes more compli-

cated the management of the water in this devices.

In this thesis a model to study the water and heat management in a fuel cell is

described. For run this model as a control model, all the equations are modified

and discretizated to make them suitable to numerical computational methods.


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2. Basic principles of hydrogen fuel cells

2.1. Introduction

A fuel cell is a device that transforms the chemical energy from a fuel into elec-

tric energy and heat through a chemical reaction with an oxidizing agent.

In PEM fuel cells (polymer electrolyte membrane fuel cells, PEMFC) hydrogen

is used as a fuel, and oxygen is used as an oxidizing agent. So, the hydrogen is

being combusted in the simple reaction

OHOH 222 22 →+

Therefore, hydrogen and pure oxygen or air are used to get the reaction, ob-

taining water as the only waste. To better understand how the system works,

first we should describe a PEMFC schematically.

2.2. Schematic description of a PEM fuel cell

In a PEMFC the core of the reaction is the set anode, electrolyte membrane and

cathode. A full fuel cell system involves many other elements, but the basics are

these ones. The basic operation of an acid electrolyte fuel cell is described in

the figure below.

Figure 2.1. Schematic description of a PEM fuel cell.


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• Anode: At the anode, hydrogen as a fuel is ionised, releasing electrons

and creating H+ ions (protons) as described in the figure. This reaction

release energy.

• Polymer electrolyte membrane: This membrane, also known as a pro-

ton exchange membrane, acts as a filter, allowing protons to pass

through it and avoiding electrons to pass.

• Cathode: At the cathode, oxygen reacts with incoming protons from the

membrane and incoming electrons from the external circuit, to form water.

• External circuit: The external circuit carries electrons from the anode to

the cathode and feeds the load.

2.3. Main fuel cell limitations

Before starting to explain the design of a PEMFC it is necessary to have some

considerations.

• Current limitation: Hydrogen reacts at the anode, releasing energy.

However, it does not mean that the reaction proceed at an unlimited rate.

Activation energy must be supplied to overcome the energy hill shown at

the figure below.

Figure 2.2. Activation energy of the reaction.

If the probability of a molecule having enough energy is low, then the re-

action will proceed slowly.


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There are three different ways to improve this: the use of catalysts, rais-

ing the temperature and increasing the electrode area.

The reaction involves hydrogen or oxygen, membrane and electrodes

contact. All the reaction proceeds on the solid phase surface (electrodes

and electrolyte membrane). Hence, contact surface optimisation will be a

dramatic factor in the performance of a fuel cell. Contact surface will de-

fine the current density through the fuel cell (current per cm2). Porous

electrodes are the best way to ensure a high rate contact surface-volume.

• Low voltage: Although ensuring a capable current density, the voltage

generated by a simple fuel cell is low, about 0.7 V. This means that to

get a useful voltage many fuel cells have to be connected in series. Such

a collection of fuel cells is known as stack.

At the same time, this means that every single voltage loss will be impor-

tant. Hence, they have to be avoided as much as possible.

2.4. Fuel cell design

Once the main operative limitations of fuel cells have been explained, we can

describe the standard and more common configuration to overcome these limi-

tations: Stack of fuel cells connected in series and fed by bipolar plates. This

system is described below.

In fuel cell stacks, electrodes (anode and cathode) are made flat and thin. The

main target is to get the largest possible contact surface with the gas. This is

the reason because highly porous materials are used. This kind of materials

raises the surface-volume rate. Usually, advanced carbon materials are used.

Small amounts of platinum are added on the surface of electrodes, to act as a

catalyst.

Also, properties as high temperatures working capacity and corrosion resis-

tance are important. To avoid electric losses, electrodes should be good con-

ductors.


12

The polymer electrolyte membrane is made with an ionomer with perfluori-

nated backbone like Teflon. This polymer is permeable to protons when it is

saturated with water, but it does not conduct electrons. The characteristic pa-

rameter of the membrane is the proton conductivity.

To connect several cells in series, bipolar plates are used. This makes con-

nections all over the surface of one cathode and the anode of the next cell.

Therefore, voltage losses are dramatically reduced. At the same time, the bipo-

lar plate serves as a means of feeding hydrogen to the anode and oxygen to

the cathode through channels made on its surface. The plates are made of a

good conductor, such as graphite or stainless Steel.

A scheme of the different parts and the assembly of a single fuel cell are shown

in the figure below.

Figure 2.3. Fuel cell basic assembly.

Bipolar plate

Anode

Membrane

Cathode

Single fuel cell

Hydrogen supply channels

Air supply channels


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The bipolar plate is one of the most important parts in a fuel cell. Its geometry

will define the fuel cell configuration, as it will restrict the series connexion in a

fuel cell stack.

Mainly, the bipolar plate geometry defines the way to supply hydrogen, air and

an optional cooling fluid as water. A typical bipolar plate has horizontal channels

for the anode face (hydrogen supply) and vertical channels for the cathode face

(air supply). This gives a cross configuration as it is shown in the figure below.

Figure 2.4. Bipolar plate with crossed configuration.

This kind of bipolar plate is used in external manifolding fuel cells stacks, which

will be described later.

Also, other kinds of bipolar plates let using a cooling fluid through the fuel cells.

They have internal tubes to spread the cooling fluid through the fuel cell. An

example of this kind of bipolar plates is shown in the figure below.

Figure 2.5. Bipolar plate with cooling channels.

Assembly


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Since the membrane and the electrodes are porous, it is necessary to seal their

edges to avoid gas leakages. To overcome this problem, an edge seal will be

used as is shown in the figure below.

Figure 2.6. Fuel cell assembly with edge seal.

As has been explained above, several cells are stacked in series connexion to

get a useful voltage. Usually, due to the low voltage achieved by each cell,

hundreds of them have to be stacked. The figure below shows an example of a

fuel cell’s stack with crossed channels bipolar plates.

Figure 2.7. Fuel cell stack.

Edge seal

Assembly


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The type of stack shown in the figure above is used in external manifolding fuel cells. External manifolds are fitted to the fuel cell stack to provide hydrogen

and air as it is shown in the figure below.

Figure 2.8. External manifolding fuel cell.

The main advantage of external manifolding fuel cells is its simplicity. Neverthe-

less, it has two major disadvantages.

The first one is that it is difficult to cool the system. During their operation, fuel

cells release high quantities of heat. So, to avoid membrane damages or dehy-

dration, cooling the system is necessary. In this kind of fuel cell stacks, any ex-

ternal cooling fluid cannot be used. The only way to cool the system is using the

same air used to supply the cathode. This means that the air has to be supplied

at a higher rate than it is demanded by the cell chemistry, which increases

pressures drops and, therefore, it increases losses.

The second disadvantage is that the edge seal shown in the figure 2.6 is not

pressed firmly onto the electrode at the point where a channel is. This increases

the probability of a leakage of the reactant gases.

Also it is possible to use bipolar plates for internal manifolding as is shown in

figure 2.5. With this type of bipolar plates the system can be cooled using cool-

Anode hydrogen supply

Cathode air or oxygen supply


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ing fluids as water. This is thanks to narrow channels for cooling fluid made over

the bipolar plate surface. These systems are more complicated and expensive.

It should now be clear that the bipolar plate is a complex part of a fuel cell stack.

Also, the question of its material is very important as the bipolar plate makes a

major contribution to the cost of a fuel cell. Graphite is often used, but it is diffi-

cult to work. Stainless steel can also be used, but it is easy to corrode.

At the same time, it is important to consider that leaks are a major problem in

fuel cells. A good design has to overcome this problem, and the joins between

bipolar plate, edge seal, electrodes and membrane are a very important issue in

this aspect.


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3. Basic concepts for water management in PEM fuel cells

3.1. Overview of the problem

Water management is a dramatically important issue in PEM fuel cells, which is

directly related to the temperature of the fuel cell solid phase and, by extension,

to the fuel cell cooling.

The electrolyte membrane must be hydrated to work properly, as the proton

conductivity is directly proportional to the water content. If it is partially not hy-

drated, the performance will be significantly deteriorated. Also, if this happens,

the membrane could be physically damaged. However, there must not be so

much water that the electrodes bonded to the membrane flood, blocking the

pores in the electrodes or the gas diffusion layer. Hence, a balance is needed,

which take care to achieve.

If an ideal PEM fuel cell is supposed, which would be almost perfectly design

and working at optimum conditions, the membrane could keep at the correct

level of hydration just within the water formed at the cathode trough the water

formation reaction. Air would be blown over the cathode, and apart of supplying

the necessary oxygen it would dry out any excess water. At the same time, wa-

ter would diffuse trough the membrane from the cathode to the anode, keeping

the whole membrane in a correct hydration level. Sometimes, this situation can

be achieved, but PEM fuel cell presents some operative problems, which are

difficult to deal with.

Basically, at high current densities, a phenomenon related with water molecules

being dragged by protons is especially strong. This phenomenon can be greater

than the water back diffusion and, consequently, dry the membrane.

Also, at high temperatures the drying effect of the air at the anode would ex-

ceed the rate of water created by the reaction. This effect is directly related with

the temperature, so keeping the temperature at the design value is strongly im-

portant.


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If it is taken into account that above 50% of the energy of the chemical reaction

is released as thermal energy, it is obvious that, for big fuel cells, a good cooling

system will be necessary to keep the temperature at the design values.

3.2. Water sources and drains and relative humidity

3.2.1. Water sources and drains in a PEM fuel cell

As it has been previously described, keeping a suitable hydration level of the

electrolyte membrane is very important.

In the chart below, the water sources and drains for anode and cathode are de-

scribed. (+) Means a source and (-) means a drain.

Anode Cathode

• (+) Water may back diffuse from the

cathode to the anode. It is due to the

water concentration differences be-

tween both sides and due to the

membrane thickness.

• (-) Water will be dragged from the an-

ode to the cathode sides by protons

moving through the electrolyte.

• (+) Water may be supplied by exter-

nally humidifying the hydrogen supply.

• (-) Water may back diffuse from the

cathode to the anode. It is due to

the water concentration differences

between both sides and the mem-

brane thickness.

• (+) Water will be dragged from the

anode to the cathode sides by pro-

tons moving through the electrolyte.

• (+) Water will be produced as water

formation reaction proceeds.

• (-) Water will be removed by evapo-

ration into the air circulating over the

cathode.

• (+) Water may be supplied by exter-

nally humidifying the air supply.

Chart 3.1. Sources and drains of water in a PEM fuel cell.


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The key to solve the problem of water management in a PEM fuel cell is to find

a balance between the inputs and outputs of water at the anode and cathode

sides. When the air is blown over the cathode at the stoichiometric rate, it is

easy to get that necessary balance. Nevertheless, this will produce concentra-

tion losses and the oxygen reaction will not proceed at a good rate. In practice,

stoichiometry (λ ) will be at least 2, even more. Problems arise because the

drying effect of air is non-linear in its relationship to temperature. To understand

this, it is necessary to consider the meaning and effects of relative humidity and

saturated vapour pressure.

3.2.2. Relative humidity and saturation pressure of the water

First of all, relative humidity is defined.

sat

w

PP

=φ

Equation 3.1. Relative humidity.

Where wP is the partial pressure of the water in the air, and satP is the saturation

pressure of the water at the air temperature. Typical values of relative humidity

are 0.3 (30%) in the dry conditions of the Sahara desert and 0.7 (70%) in New

York on an average day.

The drying effect of the air, or the rate of evaporation of the water, is directly

proportional to the difference between satP and wP . When watsat PP = , air will not

be able to hold any more water, and that drying effect will disappear. If wP is

greater than satP water will condense.

The cause of the complication for PEM fuel cells is that the saturation pressure

of the water varies with temperature in a highly non-linear way, which means

that satP increases more and more rapidly at higher temperatures. Basically,

warm air with a high content in water can have really higher drying effect than

cold air with the same content in water (the same partial pressure of water va-

pour), as its relative humidity is much more low. This effect can be checked in

the chart below.


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Temperature (ºC) Water saturation pressure (kPa)

20 2.338

30 4.246

40 7.383

50 12.35

60 19.94

70 31.19

80 47.39

90 70.13

Chart 3.2. Temperatures and water saturation pressures.

To give a quick example, air at 20ºC with a water partial pressure of 1.64 kPa,

has a relative humidity of 70%. At the temperature of 60ºC, this air has a rela-

tive humidity of 8%. The relative humidity goes dramatically down with the tem-

perature.

Usually, air used to feed fuel cells cathode is directly taken from the atmosphere

(an average of 20ºC and 70% of relative humidity). As the PEM fuel cell opera-

tion temperature is up to 100ºC, this air has a strong drying effect. Therefore,

the auto hydration mechanism for the fuel cell membrane described before

could not be enough.

In general, it is experimentally demonstrated that in fuel cells operating at tem-

peratures over 60ºC, the generated water at the cathode reaction and the water

back diffusion are not enough to keep the electrolyte membrane hydrated

(Büchi and Srinivasan, 1997). The blown air over the cathode (with a high dray-

ing effect at 60ºC or more) removes the water at a higher rate than it is pro-

duced. This means that external humidification is necessary in these cases.

This external humidification is usually added in the air inlet. Nevertheless, it

could be added in the hydrogen fuel inlet as well. In this section, just air humidi-

fication is considered.


21

3.2.3. Studying water content in a fuel cell to take decisions

Now, some formulas to study the humidity in a fuel cell are going to be de-

scribed. Basically, the air humidity in the outlet should be above 80% to prevent

excess drying, but must be below 100% to prevent electrodes to be flood.

Therefore, be able to predict the relative humidity of the out coming air is very

important. It is supposed that the incoming air removes all the water produced

by the reaction. The water produced is related to the fuel consumption and, by

extension, to the stoichiometric coefficient λ( ). Also, some water is considered

in the incoming air. Within these assumptions, the next equation is given to cal-

culate the water vapour pressure in the outlet.

[ ][ ] 210.01420.0

,++

+=

λψψλ out

outwPP

Equation 3.2.

Where outP is the pressure of the air at the outlet and ψ is calculated in the next

equation.

inwin

inw

PPP

,

,

−=ψ

Equation 3.3.

Where inP is the total inlet pressure, a bit larger than outP , and inwP , is the water

vapour partial pressure at the inlet.

The saturation pressure of the water satP can be calculated, if the temperature

of the air in the outlet is known, just using the equation 3.1. The main target is

to keep ψ in a suitable value to keep the ratio satw PP / (relative humidity) be-

tween 0.8 and 1. These two different scenarios are possible.

• The relative humidity is under 0.8 in the outlet. In this case, an external

humidification system is necessary to improve the water content of the

air in the inlet.


22

• The relative humidity is over 1 in the outlet. In this case, electrodes can

be flood. The first action in this case is suppressing any external humidi-

fication. If the relative humidity is still over 1, the stoichiometric coefficient

can be improved blowing a higher rate of air. This will remove more wa-

ter and, at the same time, the temperature will increase and, therefore,

the saturation pressure of the water will increase too, making lower the

relative humidity.

If external humidification is needed, the mass of water to be added to the air is

given by the next equation.

[ ] aextwinwin

extwinww m

PPPPPm

,,

,,622.0−−

−=

Equation 3.4. Water mass flow rate related to the air mass flow rate.

Where inwP , is the desired water vapour pressure in the inlet and extwP , is the

water vapour pressure content by the atmospheric air.

Now, a numerical example is given. A fuel cell operating at 80ºC and 2 bar is

assumed. The pressure drop along the channel is neglected. Therefore,

barPP outin 2== . The atmospheric air is at 20ºC and the external relative humid-

ity is 70% ( )7.0=Φext . For an air mass rate of sgma /11=& and 2=λ , the de-

sired suitable relative humidity in the outlet is 100% ( )1=Φout . Is it necessary to

add water? And, if it is necessary, how many water must be added?

First of all it is necessary to find the water saturation pressure for the inlet and

outlet temperatures. They are shown at the chart 3.2.

PaPCTPaPCT

satout

satin

47390º802338º20=→==→=

Knowing the saturation pressure and the relative humidity, it is easy to find the

water vapour pressure at the inlet and at the outlet using equation 3.1. First, the

water pressure in the external air is calculated.

( ) PaCPP satextextw 6.163623387.0º20, =×=Φ=


23

And the desired water vapour pressure in the outlet, if the desired relative hu-

midity is 100%.

( ) PaCPP satoutoutw 47390473901º80, =×=Φ=

With all this information it is possible to calculate ψ applying the equation 3.2.

[ ][ ]

[ ][ ] 06637.0

210.0211022420.047390

210.01420.0 5

, =→++

×+=→

+++

= ψψ

ψλψψλ out

outwPP

So ψ is a positive value. This means that it is necessary to add water in the

inlet to keep the membrane hydrated. Now the desired water pressure in the

inlet for the ψ is calculated using the equation 3.3.

PasPP

PPP

Pinw

inw

inw

inwin

inw 83.12447102

06637.0 ,,

5,

,

, =→−×

=→−

=ψ

This pressure is greater than the water saturation pressure in the inlet. This

means that liquid water must be added. Now, the mass rate of water to be

added is calculated using the equation 3.4.

[ ] [ ] sgmPPP

PPm aextwinwin

extwinww /39.011

6.163683.124471026.163683.12447622.0622.0 5

,,

,, =×−−×

−=

−−−

=

This way to determine the necessary amount of water to be added is an easy

and quick approach. It is working well in small stationary fuel cells, with not too

high current densities, stoichiometric ratios and dynamical responses.

This system does not take into account the temperature change along the

channel, and the water vaporisation ratios for the liquid water added in the inlet

neither. This model does not consider the dynamical response of the fuel cell if

it is not working in stationary conditions, as the solid phase temperature change,

which is very important to study the membrane dehydration.

All this makes this just an approximation. This is the reason because an along

the channel model is described in this paper. With a differential model which

studies all the variables changes along the channel, depending of the tempera-


24

ture response of the stream and the solid phase, it is rather effective to avoid

local dehydrations of the membrane.

3.2.4. Methods for external humidification

As it has been explained above, adding water in the air inlet will be probably

necessary in PEM fuel cells. Sometimes, it will be advisable to add water in the

hydrogen inlet as well. The humidification systems described in this section are

valid for both cases.

The method used to humidify the reactant gases of a PEM fuel cell is one of the

features of the PEM fuel cell where no standard has yet emerged. As fuel cells

are not the only systems where gas streams have to be humidified, used else-

where technologies can be adapted. Useful areas are air conditioning and hu-

midification at the entering air in the four-stroke engine with gasoline. The

methods used in these areas, together with others created especially for fuel

cells, are used. Some humidification methods are described below.

• In laboratory test systems the reactant gases of fuel cells are humidified

by bubbling them though water, whose temperature is controlled. This is

good for experimental work, but this will rarely be used in the application

fields.

• One of the easiest and more extended systems to control the humidity is

the direct injection of water as a spray. At the same time, this system lets

to cool the inlet gas. This is especially useful when the gas is com-

pressed after be injected in the fuel cell. This method involves the use of

pumps and a solenoid valve to open and close the injector. This adds an

extra cost to the whole equipment, but is a mature and effective tech-

nique.

• Another method is to directly inject liquid water into the fuel cell. To avoid

electrodes flooding, a special bipolar plate and flow field is design that

forces the reactant gases to blow the water through the cell and over the

entire electrode. The possibility of cooling the system is lost when this

method is used.


25

All the systems described above require liquid water to operate. To obtain this

water, the best way is to treat the exit air to avoid the necessity of add an exter-

nal supplying system for water. This means that the water content in the air out-

let must be condensed, stored and pumped to the humidification system. All this

means extra equipment.

Therefore, engineers at the Paul Scherer Institute of Switzerland have devel-

oped a method to avoid all that extra equipment. This system uses the exit wa-

ter to humidify without a separate condenser, pump and tank. The method is

shown in the figure 3.1.

Figure 3.1. Humidification of reactant air using exit air [4].

The warm, damp air leaving the cell passes over one side of a membrane,

where it is cooled. Some of the water condenses on the membrane. The liquid

water goes through the membrane and is evaporated by the dryer gas going

into the cell on the other side. The membrane used is similar or the same that

the membrane used as an electrolyte in the fuel cell.

3.3. PEM fuel cell cooling

3.3.1. Cooling using the anode air supply

PEM fuel cells are note 100% efficient. The efficiency average is 50%. This

means that an a half of the energy released during the reaction is released as

thermal energy. All this heath must be removed to keep the fuel cell in a suit-

able operating temperature. If the temperature rises too much, the membrane

will be dehydrated and performance will be deteriorated. In the worst case, the


26

membrane would be damaged. An approximation to the rate of released heat in

a PEM fuel cell is given by the expression below.

wattsV

PrateHeatingc

e⎭⎬⎫

⎩⎨⎧ −= 125.1

Equation 3.5. Heating rate released in a PEM fuel cell.

The way to remove this heat depends on the size of the fuel cell. With fuel cells

up to 100 W it is possible to use just the cathode air supplied and convected air

to cool the cell. This is done with open cell construction with cell spacing be-

tween 5 and 10 mm per cell. For more compact fuel cell small fans can be used

to blow the reactant and cooling air through the cell. In this case, a large propor-

tion of the heat is still removed by natural convection and radiation.

However, if the power of the fuel cell goes beyond 100 W, a lower portion of

heat is removed by convection and radiation from and around the external sur-

faces.

3.3.2. Cooling using air or water separated from the reactant gases

The usual way of cooling cells in the range from about 100 to 1000 W is to

make extra channels in the bipolar plates through which cooling air can be

blown. This is shown in the figure below.

Figure 3.2. Three cells from a stack with a modified bipolar plate for external cooling.


27

A commercial PEM fuel cell which uses this cooling system is shown in the pic-

ture 3.3. The blower for the cooling air can be seen at the bottom left of the unit.

The reactant air passes through the humidifier on the front of the unit.

Figure 3.3. Ballar Nexa PEM fuel cell that uses air cooling [4].

The issue is when to change from air cooling to water cooling. Air cooling is

simpler than water cooling. Air is just taken from the atmosphere and blown

trough the cooling channels. If water is used, it must be cooled using and exter-

nal system, such a heat exchanger.

Nevertheless, air cooling becomes harder to ensure that the fuel cell is well

cooled when the power goes beyond 1000 W. Also, the cooling channels

needed are bigger, making the fuel stack bigger too. To remove high rates of

heat, a high blowing energy is also required. When all this disadvantages be-

come important, water must be used.

The system used too cool a fuel cell using water is the same as the system

used to the air. Jus the same channels are used or maybe smaller channels, to

inject water trough them using a pump.


28

4. Description of an along the channel model for water management in a PEM fuel cell.

4.1. Introduction to the model

In this section a model for the water and heat management in a PEM fuel cell is

described. This model is based in Yi and Nguyen papers [1] and Golbert and

Lewin papers [3].

As it has been explained before, water management is a very important and

complicated issue in a PEM fuel cell. If this PEM fuel cell is used for non-

stationary purposes, like a motor vehicle, it becomes even more complicated

and important.

The temperature of gas streams in a fuel cell is very important because it is re-

lated to the saturation pressure of the water, and by extension to the drying ef-

fect of the gas. Also, the solid phase temperature of the fuel cell is important, as

this directly affects to the gas streams temperature and the membrane dehydra-

tion.

All this happens in a scenario where the temperature changes along the chan-

nel and this affects the solid phase temperature and cooling system. Water is

generated by the reaction, can evaporate or condense, moves through the

membrane and can be added in the inlet. To manage all this and avoid local

membrane dehydrations, a differential model which studies all the x -dependent

variables is necessary.

As this model will be rather complicated to solve analytically, a finite difference

version of this model is presented as well. This model can be easily solved us-

ing numerical methods in computing software such MATLAB.

4.2. Yi and Nguyen model and Gilbert and Lewin arrangements

Yi and Nguyen described an along the channel model for PEM fuel cells, where

the temperature of the solid phase is supposed to be at quasi-steady-state.


29

Nevertheless, Gilbert and Lewin introduced some arrangements, modelling the

temperature of the solid phase dynamically.

Each model is formulated using different dimension terms. Yi and Nguyen use

energy per length for the energy balances, while Gilbert and Lewin use energy

per volume. Also, coefficients used for heat transfer by convection and coeffi-

cients for the geometric relationships are different. In the section 4.4 both mod-

els are joined together, covering the lacks of each other and adapting coeffi-

cient and units to the Yi and Nguyen model.

The energy balance on the solid is modeled dynamically whereas all the other

equations are assumed to be at quasi-steady-state. Therefore, there is a partial

differential equation (equation 4.10) to calculate the solid phase temperature

and nine differential ordinary equations (equation 4.1 to 4.9) to calculate the

rest of the variables. There are, as well, fourteen supporting equation (equation

4.10 to 4.24) to calculate different parameters which appears at the main gov-

erning equations.

As it has been said before, the fuel cell is modeled along the channel, which

means that the variables change as the gas stream move forward in the x di-

rection into the channel. The model accounts for heat transfer between the solid

and the two gas channels (anode and cathode), and between the solid and

cooling water. Also, the liquid water and the water vapour content are modeled

for each stream.

4.3. Model applied to a specified PEM fuel cell geometry

The geometry of the fuel cell, almost defined by the type of bipolar plate used,

is dramatically important to a correct application of equations. This model as-

sumes that anode and cathode streams flow trough parallel channels in hori-

zontal direction. This geometry has been chosen because is one of the most

common. Also, the equations are easy to adapt to other different geometries.

The fuel cell geometry and the coordinate axes for this model are shown in the

figure below. L is the channels length, h is the channel width and d the chan-


30

nel height. A differential of x , dx( ) is also shown in the image. In this case, the

bipolar plate for the anode stream is the same that the one for the cathode

stream. Hence, the geometry is the same.

Figure 1.2 Fuel cell bipolar plate geometry.

It must be noticed that to check the bipolar plate geometry is very important.

Other kind of bipolar plates are used. For instance, sometimes anode stream is

in horizontal direction and cathode stream is in vertical direction. If the geometry

changes, all the equation must be reviewed and arranged. Specially the spatial

parameters as dx .

4.4. Equations for the model

4.4.1. Governing equations

The water and heat management model for a PEM fuel cell has 10 governing

equations for the 10 unknowns below.

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) )(,,,,,,,,, ,,,,22 xTxTxTxTxMxMxMxMxMxM scoolcalcw

vcw

law

vawOH

The model also has 14 supporting equations for the parameters use in the gov-

erning equations like heat capacity or activity of different species, for instance.

dx

x

y


31

The governing equations are referred to the cathode stream, the anode stream

or the solid layer temperature. Some equations for anode and cathode will be

explained together, as they described the same or very similar phenomena.

The first equation is about hydrogen fuel consumption along the anode channel.

( ) ( )xIFh

dxxdMH

22 −=

Equation 4.1. Change of the hydrogen molar flow rate along anode channels.

This ordinary differential equation shows how hydrogen is being consumed as

the stream go along the anode channel in the x direction. The change in the

number of moles of the single-phase species (like hydrogen or oxygen) along

the channel is due to the normal flux in the y direction into or out the mem-

brane. In this case, this normal flux is about protons going through the mem-

brane, from de anode to the cathode, as the reaction proceeds.

It is obvious that hydrogen consumption is proportional to local current density

(A/cm2), which varies along the channel as the membrane conductivity and the

electrodes overpotential change. h is the channel width.

F is the Faraday constant (96485 C) and it is related to the electrical charge of

a mole of electrons. As each mole of oxidised hydrogen releases two electrons,

two multiplies the Faraday constant.

The second equation is about oxygen consumption along the cathode channel.

( ) ( )xIFh

dxxdMO

42 −=

Equation 4.2. Change of the oxygen molar flow rate along cathode channels.

This equation is rather similar to the equation 4.1, as the oxygen consumption is

stoichiometrically related to the hydrogen consumption. If the chemical reaction

for the water formation is remembered:

⎪⎩

⎪⎨⎧

→++

+→→+

+−

−+

)(244

)(442022

22

2222 cathodeOHHeO

anodeeHHHOH


32

It is straightforward to see that for each mole of oxygen molecules, two moles of

molecules of hydrogen are consumed. Hence, four moles of electrons are re-

leased and consumed during the reduction of one mole of oxygen molecules.

This is the reason because in equation 4.2 the Faradays constant is multiplied

by four.

The next equation is about the change in the flow rate of liquid water along an-

ode channels.

( )( )[ ]

( )( ) ( ) ( )

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−+⎭

⎬⎫

⎩⎨⎧

+= xPP

xMxMxM

xTRdhk

dxxdM sat

awH

vaw

vaw

a

claw

,,

,,

2273

Equation 4.3. Change in the liquid water molar rate along anode channels.

Where ck is the homogeneous rate constant for the condensation and evapora-

tion of water reaction and h and d are the width and height of the channel, re-

spectively.

The change in the liquid water molar rate is proportional to the difference be-

tween the partial pressure of the water vapour and the saturated vapour pres-

sure at the temperature of the anode stream. The saturation pressure varies

with x as the temperature of the stream also does. The partial pressure of the

water is calculated using the molar rates of water vapour and hydrogen, as they

are the only present gases in the anode. P is the operation pressure of the fuel

cell, considered the same at anode and cathode.

The first term on the right side of the equation is always positive. Therefore, if

the partial pressure of the water vapour is greater than the saturation pressure

of the water, water will condense, as the derivative of the molar rate of liquid

water will be positive. In the opposite case, the water will evaporate.

Now, the equation for the change in the flow rate of liquid water along cathode

is shown.


33

( )( )[ ]

( )( ) ( ) ( )

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−++⎭

⎬⎫

⎩⎨⎧

+= xPP

MxMxMxM

xTRdhk

dxxdM sat

cwNO

vcw

vcw

c

ccw,

,

,1,

22273

Equation 4.4. Change of the liquid water molar rate along cathode channels.

This equation is almost the same as the equation 4.3 for the anode. The only

difference is that in the cathode the gaseous species are different. They are

water vapour, oxygen and nitrogen. Hence, the partial pressure is calculated

using these species.

Now, the change of the water vapour molar rate along the anode and cathode

channels is going to be dealt. To understand the equations that drive this phe-

nomenon it is important to remember that the water vapour in the flow channels

is affected by some different mechanisms. These mechanisms are:

• Water vapour generated at the cathode by the reaction of oxygen with

the protons and electrons from the anode.

• Water vapour diffusing through the membrane because the different con-

centration between anode and cathode.

• Water vapour dragging through the membrane by protons migrating from

the anode to the cathode.

• Finally, liquid water condensing or evaporating.

The next equation is for the change in the flow rate water vapour along anode

channels.

( ) ( ) ( )xIFxh

dxxdM

dxxdM l

awvaw )(,, α

−−=

Equation 4.5. Change of the water vapour molar rate along anode channels.

The first term on the right side of the equation is referring to the liquid water

condensation or evaporation at the anode, and is calculated at equation 4.3. If

water evaporates, this derivative will be negative (as liquid water decrease).

Hence, the first term on the right side will be positive. The term will be negative

if the water condenses, as it will reduce the amount of water vapour.


34

The second term on the right is the net migration of water across the membrane,

due the combined effect of diffusion by concentration and pressure gradients

between anode and cathode, and also water molecules being dragged across

the membrane by protons associated to the current density. The α coefficient,

called net water flux per proton flux, does include both of the effects, diffusion

and dragging, and the equation to calculate it will be described at the supporting

equations section.

In general, if the partial pressure of the water vapour is lower than the water

saturation pressure, what means that the presence of liquid water is no ex-

pected, α should be negative, as this means that the water diffusion from cath-

ode to anode is greater than the water dragging form anode to cathode.

Otherwise, the anode side of the membrane could get dehydrated, and the fuel

cell performance would be dramatically deteriorated.

The equation for the change in the flow rate water vapour along cathode chan-

nels is described below.

( ) ( ) ( ) ( ) ( )xIFxhxI

Fh

dxxdM

dxxdM l

cwvcw α

++−=2

,,

Equation 4.6. Change of the water vapour molar rate along cathode channels.

The first term on the right side of the equation works exactly in the same way as

in equation 4.5 does.

The second term is related to the water vapour formation due the reaction of the

oxygen with the electrons and protons. As for each mole of oxygen molecules

consumed, two moles of water vapour are formed. Therefore, this term is the

same as it is described at the equation 4.2 for the oxygen consumption, but

multiplied by two.

The last term on the right hand of the equation is, as in equation 4.5 was, about

the net migration of water across the membrane. But, in this case, the sign is

opposite. If the net flux of water through the membrane is positive, water mole-

cules are living from the anode to the cathode and vice versa.


35

Now, the temperature of the gases streams at the anode and at the cathode is

going to be discussed. First, for the anode, the equation that described the

temperature of the stream is shown below.

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) xTxTaUdxxdM

xHxHdxxdTxCxMxCxM

as

lawl

awvaw

avawp

vawHpH

−+

−=+ ,,,,,,, 22

Equation 4.7. Change of the stream temperature along the anode channel.

The temperature of the stream at the anode varies with x . Hence; all tempera-

ture-dependent properties vary along the channel too.

In general, the local temperatures in the anode, cathode and coolant channels

are affected by heat transfer between the mass surface and the fluid.

At equation 4.7 an energy balance is proposed to determine the change at the

anode stream temperature. On the left side of the equation it is applied the well-

known formula TCmQ pΔ= , to describe the amount of energy that stream tem-

perature increases or decrease involve. As the stream is almost in gas state,

just gas-phase species are considered. At the anode, they are hydrogen, water

vapour and oxygen. The molar heat capacity of each species is temperature-

dependent, and expressions to calculate them are given at the supporting equa-

tions section.

On the right side of the equation, the first term is related to liquid water vapori-

sation or condensation and the energy that this phenomenon involves. The

change of the liquid water rate described at equation 4.4 is multiplied by the

enthalpy variation between water vapour and liquid water. If liquid water is con-

densing, heat is being released to the stream. Otherwise, if liquid water is being

vaporised, heat is being taking from the stream. The enthalpy variation changes

with the temperature, therefore is x -dependent. An expression to calculate it is

given at the supporting equations section.

The second term on the right hand of the equation is related to the heat transfer

between the stream and the solid phase of the fuel cell (bipolar plate, elec-

trodes and membrane). It is just the Newton’s law. Obviously, if the stream tem-


36

perature is greater than the solid temperature, heat from the stream to the solid

phase will be transferred. This heat transfer is proportional to the overall heat-

transfer coefficient U and the heat transfer per unit length coefficient A .

It is important to notice that the temperature of the solid phase sT is x -

dependent but it does not vary in the y direction. This is due to the thinness of

each fuel cell and the high thermal conductivity of the materials. These reasons

make reliable to consider sT constant in the y direction. This will be further ex-

plained at the equation 4.10, which is to calculate the solid phase temperature.

Equation 4.8 describes the change of the stream temperature along the cath-

ode.

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) xTxTaUdxxdM

xHxHdxxdTxCxMxCMxCxM

cs

lcwl

cwvcw

cvcwp

vcwNpNOpO

−+

−=++ ,,,,,,,, 2222

Equation 4.8. Change of the stream temperature along the cathode channel.

This equation is rather similar to equation 4.7. In this case, the present gas-

phase species at the cathode are oxygen, nitrogen and water vapour. Other

possible species at the atmospheric air are neglected.

The equation below is to calculate the coolant temperature change through the

heat exchanger.

( ) ( ) ( ) xTxTAUdxxdTCM coolsbb

coollwp −=,

Equation 4.9. Change of the coolant flow temperature along the heat exchanger.

M is the molar flow rate of heat exchanger fluid. In this case, the coolant fluid is

supposed to be water. bA is the heat transfer length between solid layer to bulk

of coolant fluids and bU is the overall heat-transfer coefficient between solid

layer and heat exchanger fluids. If the solid temperature is greater than the

coolant temperature, heat will be transferred from the solid layer to the coolant.

Therefore the coolant temperature will increase.


37

Equation 4.10 is an energy balance for the solid phase of the fuel cell. This bal-

ance has been modelled dynamically. This gives the most complicated equation

of the model, as it is a partial differential equation.

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )xIVFHhn

dxxdM

xHxHndxxdM

xHxHn

xTxTUAxTxTxTUandxxTdkA

dtxdTCA

cellccw

cwvcwc

awaw

vawc

scoolbbscacs

ss

spss

⎭⎬⎫

⎩⎨⎧ +Δ

−−+−+

−+−++=

2

2

1,1

,,

1,1

,,

2

2

,ρ

Equation 4.10. Change of the solid layer temperature.

The firs term of the left side of the equation is the dynamic thermal response of

the solid. In the right side of the equation all the heat transfer mechanism for the

solid phase has been described. It is important to fully understand each one of

them.

• The first term of the right side of the equation is related to heat transfer

by conduction in the solid phase along the x direction. As mentioned

above, temperature along y direction in the solid phase is considered to

be constant. Therefore, heat conduction is just in the x direction.

• The second and third terms are related to heat transfer by convection be-

tween the solid phase and the streams flow and coolant channels. a is

the heat-transfer length between channel to solid layer per unit area. U

is the overall heat-transfer coefficient between channel and solid layer.

bA is the heat transfer length between solid layer to bulk of coolant fluids

and bU is the overall heat-transfer coefficient between solid layer and

heat exchanger fluids.

• The fourth and fifth terms are about heat related to condensation or

evaporation of water in the flow channels. It is important to notice that

when liquid water exists in the flow channel, it tends to be on the surface

of the flow plate (bipolar plate). Since heat transfer between the liquid

water droplets and the flow plate is always faster than between the liquid

water droplets and the gas phase, it is assumed that the liquid and the


38

solid phases exist at the same temperature. Therefore, when water con-

denses or vaporizes in the channels, it is assumed that the phase

change takes place on the surface of the flow plate with heat transferred

to or from the solid phase.

• The sixth and last term is related to the heat generation by the reaction.

HΔ is the total enthalpy change of the reaction of water formation.

For the equation 4.10, the following boundary conditions are given.

( ) ( )

( ) ( ) infinf

infinf0

TxTaUndxxdTk

TxTaUndxxdTk

scLx

s

scx

s

−−=

−=

=

=

Equation 4.11. Boundary conditions for equation 4.10.

These boundary conditions reflect heat losses to the surroundings from the

edges. It is assumed that there is not heat loss along the channel, since the

channel is part of a large, symmetrical proton exchange membrane stack. infU

is the overall heat transfer coefficient between de channel edges and the sur-

roundings. infT is the surroundings temperature. In this case, heat transfer by

convection in the very closed to the wall area is equalled to the heat transfer by

convection between the wall and the surroundings. Heat transfer by radiation is

neglected.

4.4.2. Supporting equations

It is important to notice equation 4.12, as it is used in different numerical solu-

tion techniques.

( )∫=L

avg dxxIL

I0

1

Equation 4.12. Average of current intensity.

This equation will let to determine ( )xI values along the channel, starting in a

specified average current. L is the channel length.


39

Equation 4.13 is to calculate net water flux per proton flux.

( ) ( ) ( ) ( ) ( ) ( )m

awcwwd t

xcxcxD

xIFxnx ,, −

−=α

Equation 4.13. Net water transfer coefficient per proton.

As it was explained previously, this parameter does combine electro-osmotic

dragging and water diffusion. First term of the right side of the equation ( dn ,

electro-osmotic coefficient) is related to water dragged by protons through the

membrane from the anode to the cathode.

In the second term, wD is the effective diffusion coefficient of water in mem-

brane. These water molecules go in backwards direction (from the cathode to

the anode). The number of molecules is proportional to the water concentration

difference between the cathode membrane face and the anode membrane face.

At the same time, it is inversely proportionally to local current, as the current is

dragging from the anode to the cathode. Also it is inversely proportionally to the

membrane thick.

If α is positive, in balance water is going from anode to cathode and vice versa.

To calculate the electro-osmotic coefficient, which appears at equation 4.13, the

expression below is used.

( ) ( ) ( )[ ] ( )[ ] ( )( )109.453.4024.20049.0 32≤+−+= xaforxaxaxaxn aaaad

( ) ( ) ( )( )11159.059.1 >−+= xaforxaxn aad

Equation 4.14. Electro-osmotic coefficient.

This coefficient is water activity-dependent. Just the water activity at the anode

is used because at high current densities, the water transport rate by electro-

osmosis from the anode to cathode exceeds the back diffusion rate of water

from the cathode to the anode. This result in a net water transport rate from the

anode to the cathode, and as a result, partial dehydration along the anode and

saturation of the cathode. Based on this observation, it is reasonable to assume

that the water content in the membrane is lower on the anode side. So this is


40

the reason because just the anode water activity is used to calculate the elec-

tro-osmotic coefficient across the whole membrane.

Based in the same assumption, the diffusion coefficient of water is given by the

following expression.

( ) ( ) ( ) ⎭⎬⎫

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧

+−=

xTDxnxD

s

odw 273

130312416exp

Equation 4.15. Diffusion coefficient of water.

The temperature taken into account is the solid temperature, as the membrane

is supposed to be at that temperature.

The concentration of water at k interface of the membrane, were k means an-

ode or cathode, is given by the next expression.

( ) ( ) ( )[ ] ( )[ ] ( ) 10.3685.398.17043.0 32

,

,, ≤+−+= xaforxaxaxa

Mxc kkkk

drym

drymkw

ρ

( ) ( )[ ] ( ) 114.114,

,, >−+= xaforxa

Mxc kk

drym

drymkw

ρ

Equation 4.16. Concentration of water at k interface of the membrane.

In this case, the water activity taken into account is the water activity of the side

concentration (anode or cathode) that is being calculated. drym,ρ is the density of

the dry membrane and drymM , is de equivalent weight of a dry membrane

(g/mol).

Equation 4.17 is the expression to calculate the activity of water in the anode

stream.

( ) ( )( ) ( ) ( )xP

PxMxM

xMxa sat

awHvaw

vaw

a,,

,

2⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+=

Equation 4.17. Activity of water in the anode stream.


41

The activity of the water is proportional to the partial pressure of the water va-

pour divided by the saturation pressure. Only the gas-phase of the species is

taken into account. In the anode, they are water vapour and hydrogen.

For the cathode the expression is rather similar. Just it is necessary to add the

nitrogen to calculate the partial pressure of the water vapour. The nitrogen mo-

lar rate remains constant, as the hydrogen does not take part of the reaction.

The equation for the cathode is shown below.

( ) ( )( ) ( ) ( )xP

PMxMxM

xMxa sat

cwNOvcw

vaw

c,,

,

22⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

++=

Equation 4.18. Activity of water in the cathode stream.

Equation 4.19 is to calculate the saturation pressure of the water. It just de-

pends of the temperature of the stream. Again, k means anode or cathode, as

they are at different temperatures.

( )[ ] ( ) ( )[ ] ( )[ ] 18.21044.11018.91095.2log 37252,10 −⋅+⋅−⋅= −−− xTxTxTxP kkksatkw

Equation 4.19. Water saturation pressure.

It is important to notice that the saturation pressure of the water is not lineal

temperature-dependent at all. As the temperature increases, saturation pres-

sure grows faster and faster. This is the reason because at high temperatures

the draying effect of the gas stream is very high.

The expression to calculate the enthalpy increase between vapour and liquid

water is given below.

( ) ( )[ ] ( ) ( )[ ] ( )[ ]( )[ ]410

3623,,

1098.8

1054.21044.394.41070.45

xT

xTxTxTxHxH

k

kkklkw

vkw

−

−−

⋅−

⋅+⋅+−=−

Equation 4.20. Enthalpy increase between water vapour and liquid water.

The enthalpy change depends on the temperature and it is used many times at

the governing equations to evaluate heat transfer due water phase changes.


42

The next group of equations is to calculate the heat capacity of the different

species, which appear at the model. They are liquid water, water vapour, hy-

drogen, oxygen and nitrogen. The heat capacities are just functions of the tem-

perature.

CmolJCl wp /º/38.75, =

( ) ( ) ( )[ ] ( )[ ] CmolJxTxTxTxC kkkv

kwp /º/1059.31060.71088.646.33 39263,,

−−− ⋅−⋅+⋅+=

( ) ( ) ( )[ ] ( )[ ] CmolJxTxTxTxC kkkHp /º/1070.81029.31065.784.28 310265, 2

−−− ⋅−⋅+⋅+=

( ) ( ) ( )[ ] ( )[ ] CmolJxTxTxTxC kkkOp /º/1031.11008.61016.110.29 39263, 2

−−− ⋅+⋅−⋅+=

( ) ( ) ( )[ ] ( )[ ] CmolJxTxTxTxC kkkNp /º/1087.21072.51020.200.29 39263, 2

−−− ⋅−⋅+⋅+=

Equation 1. Group of expression to calculate the heat capacity of different species.

Equation 4.22 is to calculate the voltage of the fuel cell.

( ) ( )( )xtxIxVV

m

moccell σ

η −−=

Equation 4.22. Voltage of the fuel cell.

The first term on the right side of the equation is the open circuit voltage of the

fuel cell. Some expressions to calculate it theoretically do exist. But they are not

too approached to the reality, as they do not consider different losses. The best

way to obtain the open voltage circuit is experimentally.

The second term represents the over potential for the oxygen reaction, which is

a voltage drop. This phenomenon is a chemical reaction given at the cathode

interface.

The third term represents the ohmic losses due to the resistance of the mem-

brane to conduct electricity current. This is proportional to the current intensity

and the membrane thick, and inversely proportional to the membrane conductiv-

ity.


43

The equation shown below is to calculate the over potential for the oxygen reac-

tion.

( ) ( )[ ] ( )( ) ⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧+

=xPI

xIFxTRx

Oo

s

2

ln5.0

273η

Equation 4.23. Over potential for the oxygen reaction.

Where 0I is the exchange current density at one atmospheric pressure, and

( )xPO2 is the partial pressure of oxygen in the cathode stream. The exchange

current is temperature-dependent. Nevertheless, to make the model simpler,

the dependence is neglected and an experimental value is given.

The last supporting equation is to calculate the membrane conductivity. This

value depends of the kind of membrane. This is an experimental expression for

a ionomer with perfluorinated backbone membrane.

( ) ( ) ( ) ⎭⎬⎫

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧

+−⋅

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−=xT

xcM

xs

awdrym

drymm 273

130311268exp00326.000514.0 ,

,

,

ρσ

Equation 4.24. Membrane conductivity.


44

5. Finite differences applied to the model

5.1. Description of the applied finite differences

The model does contain complex differential equations, which are complicate or

even impossible to resolve analytically. As it has been explained before, the

model has a partial differential equation to calculate the solid phase tempera-

ture and nine differential ordinary equations to calculate the rest of variables.

Applying finite differences to the model lets solve it using numerical methods in

computational software such as MATLAB and predict the dynamic response of

the system. Also, it lets to create different scenarios just changing the parame-

ters and study how the system responds.

The fuel cell support has been spatial discretized to study the evolution of dif-

ferent control volume and the relationship between them. This will turn the spa-

tial dependent ordinary differential equations into algebraic equations.

As the thermal dynamic response of the fuel cell has been also modeled, there

is a partial differential equation which is spatial and temporal dependent (equa-

tion 4.10). This means that discretization could have been also applied in the

time dimension to study the system along different time steps, using a Crank

Nicolson scheme, for instance. Nevertheless, this has not been done as MAT-

LAB is able to apply finite difference to study the temporal evolution of the

model by itself. So in this equation (equation 4.10) just the spatial derivative is

expressed as a finite differences ant the temporal derivative remains. This turns

this partial differential equation into an ordinary differential equation.

5.2. Model discretization. Volume controls along the channel.

As the studied model describes the variables evolution along the channel, in-

cluding streams and solid phase, the physic domain has been divided in n

nodes in the x direction. The anode and cathode interfaces have been discre-

tized, these ones relates to the gas streams along the channel. Also the solid


45

phase (including membrane, electrode supports and bipolar plates) has been

discretized. The cooling channels have been discretized as well.

In the picture (figure 5.1) a discretized channel is shown on the left side. On the

right side, the whole fuel cell is shown with a sector of parallel nodes indicated.

Figure 5.1. Model discretization.

xΔ Is the length of each node. It is the result of dividing the channel length by

the number of nodes.

nLx =Δ

In the grid below, a scheme of the discretization is shown. As it was explained

before, anode stream, cathode stream, solid phase and cooling channels have

been discretized. So there are four different elements discretized. And these

four elements interact between them.

xΔ 1=i

ni = ∆x

x

y


46

1=i 2=i ... ←→ ... 1−= ni ni = Cooling channel

↑↓

1=i 2=i ... ←→ ... 1−= ni ni = Anode interface

↑↓

1=i 2=i ... ←→ ... 1−= ni ni = Solid phase

↑↓

1=i 2=i ... ←→ ... 1−= ni ni = Cathode interface

↑↓

1=i 2=i ... ←→ ... 1−= ni ni = Cooling channel

Figure 5.2. Nodes structure.

In the same element there is n nodes in the x direction. Each node is related

to the previous one and the next one. For instance, for the anode channel the

molar rate of hydrogen 1,2 +iHM in the node 1+i is related to iHM ,2 in the previ-

ous node i and to the hydrogen consumption in that node.

In the case of the solid layer, heat transfer from node 1+i to i is related to the

solid temperatures 1, +isT and isT , .

But there is not just a series relationship between the nodes of the same ele-

ment; there is also a parallel relationship between the nodes of each element.

For instance, there is a convection heat transfer between the anode stream

nodes and the solid phase nodes. This heat transfer from the i node of the an-

ode stream and the i node of the solid phase depend on the temperature of

node i in the anode iaT , and the temperature of the node i in the solid phase

isT , . It is the same between the solid phase nodes and the cooling channel

nodes.

This is the reason because the geometry of the solid phase is that important.

The solid have to be carefully studied before being discretized in order to

choose the most functional nodes grid.

xΔ


47

The discretization chosen in this case of study is the most suitable for an along

the channel model, where the nodes of the same element interact just in the x

direction and the nodes of the different elements interact between them in a

parallel relationship.

5.3. Other consideration about discretization. Mean value for physi-cal properties.

When finite differences are applied to a derivative expression it turns in an ex-

pression like it is shown below.

( )xYY

dxxdY ii

Δ−

≈ +1

Where the difference ii YY −+1 is the change of the variable ( )xY between 1+i

and i . Now the expression shown below should be studied as an example.

( ) ( ) lKCxYYlxKC

dxxdY

iii +=

Δ−

→+= ++

2/11

This means that the change in the variable ( )xY depends on the value of the x -

dependent variable ( )xC , which will have different values 1+iC and iC for each

node. As the change ii YY −+1 depends of both values of C the most correct way

to solve the problem is to use the mean value of 1+iC and iC as it is shown in

the expression below.

21

2/1+

++

= iii

CCC

5.4. Finite differences applied to the equations of the model

In this section all the equations of the section 4 are discretized. For the differen-

tial equations, which express the change of a variable between two nodes,

there is a general form of the equation for the node i and a specific form for the

node 1=i , as in the first node the boundary conditions must be applied.


48

5.4.1. Discretization of the governing equations

Hydrogen molar rate is a variable in the anode discretization. The discretized

equations for the nodes i and 1=i are shown below. This equation is based in

the differential equation 4.1.

2/1,1,

222

++ −=Δ−

iiHiH I

Fh

xMM

Equation 5.1. Discretized equation for the hydrogen molar rate.

iinHH I

Fh

xMM

22,1, 22 −=

Δ−

Equation 5.2. Discretized equation for the hydrogen molar rate, first node.

It must be noticed that iI has the same value in the i node of each element

discretized (anode, solid phase, cathode), as the current flows in the y direc-

tion.

Oxygen molar rate is a variable in the cathode discretization. The discretized

equations for the nodes i and 1=i are shown below. This equation is based in

the differential equation 4.2.

2/1,1,

422

++ −=Δ−

iiOiO I

Fh

xMM

Equation 5.3. Discretized equation for the oxygen molar rate.

iinOO I

Fh

xMM

42,1, 22 −=

Δ

−

Equation 5.4. Discretized equation for the oxygen molar rate, first node.


49

For the molar flow rate of the liquid water in the anode the following discretized

equations are given. They are based in the differential equation 4.3.

[ ] ⎭⎬⎫

⎩⎨⎧

−+⎭

⎬⎫

⎩⎨⎧

+=

Δ−

+++

+

+

+ satiaw

iHv

iaw

viaw

ia

cl

iawl

iaw PPMM

MTR

dhkxMM

2/1,,2/1,2/1,,

2/1,,

2/1,

,,1,,

2273

Equation 5.5. Discretized equation for the liquid water molar rate in the anode, first node.

[ ] ⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−+⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+=

Δ− sat

awH

vaw

vaw

a

cl

inawlaw PP

MMM

TRdhk

xMM

1,,1,,

1,,

1,

,,1,,

1,22732

Equation 5.6. Discretized equation for the liquid water molar rate in the anode, first node.

For the molar flow rate of the liquid water in the cathode the following discre-

tized equations are given. They are based in the differential equation 4.4.

[ ] ⎭⎬⎫

⎩⎨⎧

−++⎭

⎬⎫

⎩⎨⎧

+=

Δ−

+++

+

+

+ saticw

NiOv

icw

vicw

ic

cicwicw PPMMM

MTR

dhkxMM

2/1,,2/1,2/1,,

2/1,,

2/1,

1,,

11,,

22273

Equation 5.7. Discretized equation for the liquid water molar rate in the cathode.

[ ] ⎭⎬⎫

⎩⎨⎧

−++⎭

⎬⎫

⎩⎨⎧

+=

Δ− sat

cwNO

vcw

vcw

c

cincwcw PPMMM

MTR

dhkxMM

1,,1,1,,

1,,

1,

1,,

11,,

222732

Equation 5.8. Discretized equation for the liquid water molar rate in the cathode, first node.

The discretized versions of the differential equation 4.5 for water vapour molar

rate in the anode is given below.

2/12/1,,1,,,,1,,

++++ −

Δ−

−=Δ−

ii

liaw

liaw

viaw

viaw I

Fh

xMM

xMM α

Equation 5.9. Discretized equation for the water vapour molar rate in the anode.


50

11,,1,,,,1,,

22I

Fh

xMM

xMM l

inawlaw

vinaw

vaw α

−Δ−

−=Δ−

Equation 5.9. Discretized equation for the water vapour molar rate in the anode, firs node.

And now for the cathode, the equation 4.6 for the cathode water vapour molar

rate is discretized as follows.

2/12/1

2/1,,1,,,,1,,

2 ++

+++ ++Δ−

−=Δ−

ii

i

licw

licw

vicw

vicw I

FhI

Fh

xMM

xMM α

Equation 5.10. Discretized equation for the water vapour molar rate in the cathode.

11

1,,1,,,,1,,

222I

FhI

Fh

xMM

xMM l

incwlcw

vincw

vcw α

++Δ−

−=Δ−

Equation 5.11. Discretized equation for the water vapour molar rate in the cathode, firs node.

The next equations are the discretized versions of the equation 4.7 for the an-

ode stream temperature.

2/1,2/1,

,,1,,2/1,,

,1,2/1,,,2/1,,2/1,,2/1, 22

++

++

+++++

−+Δ−

Δ=Δ−

+

iais

liaw

liaw

iavapiaiav

iawpv

iawiHpiH

TTaUxMMH

xTTCMCM

Equation 5.12. Discretized equation for the stream temperature in the anode.

1,1,

,,1,,1,,

,1,1,,,1,,1,,1, 2222

as

linaw

law

avapinaav

awpvawHpH

TTaUxMM

HxTT

CMCM

−+Δ−

Δ=Δ−

+

Equation 5.13. Discretized equation for the stream temperature in the anode, first node.


51

And for the cathode, the discretized versions of the equation 4.8 is as follows.

2/1,2/1,

,,1,,2/1,,

,1,2/1,,,2/1,,2/1,,2/1,,2/1, 2222

++

++

++++++

−+Δ−

Δ=Δ−

++

icis

licw

licw

icvapicicv

icwpv

icwiNpNiOpiO

TTaUxMMH

xTTCMCMCM

Equation 5.14. Discretized equation for the stream temperature in the cathode.

1,1,

,,1,,1,,

,1,1,,,1,,1,,1,,1, 222222

cs

lincw

lcw

cvapinccv

cwpvcwNpNOpO

TTaUxMM

HxTT

CMCMCM

−+Δ−

Δ=Δ−

++

Equation 5.15. Discretized equation for the stream temperature in the cathode, firs node.

The equation to study the cooling stream temperature 4.9 is discretized as it is

shown below.

2/1,2/1,,1,

, +++ −=Δ−

icoolisbbicoolicooll

wp TTAUxTTCM

Equation 5.16. Discretized equation for the cooling stream temperature.

1,1,,1,

,2

coolsbbincoolcooll

wp TTAUxTTCM −=

Δ−

Equation 5.17. Discretized equation for the cooling stream temperature, first node.

The equation 4.10 for the solid phase temperature is discretized as it is shown

below. In this case, just the spatial partial derivative is discretized. The temporal

derivative remains as MATLAB is able to discretize the temporal dimension by

itself. This means that the partial differential equation turns into an ordinary dif-

ferential equation.

As the equation 4.10 contents a second order derivative for the spatial dimen-

sion; the central difference method has been used to discretize it.


52

( )

2/1

1,,

11,,

2/1,,

1,,

11,,

2/1,,2/1,2/1,

2/1,2/1,2/1,21,,1,

,

2

22

+

++

++++

+++−+

⎭⎬⎫

⎩⎨⎧ +Δ

−

Δ−

Δ+Δ−

Δ+−+

−++Δ

+−=

icellc

icwicwicvapc

iawiawiavapcisicoolbb

isiciacisisis

ss

spss

IVFHhn

xMM

HnxMM

HnTTUA

TTTUanx

TTTkA

dtxdTCA ρ

Equation 5.18. Discretized equation for the solid phase temperature.

In this case, it does not make sense to give a special equation for the first node.

What is done is to discretize the boundary conditions (equations 4.11) for equa-

tion 4.10. Nevertheless, some arrangements must be done.

The discretized boundary conditions are shown below. The boundary is defined

by the first node 1=i and the last node ni = , as they are the nodes in contact

with the surroundings, where heat is transferred to the exterior of the fuel cell.

infinf,

infinf,1,

2

2

TTaUnxTdTk

TTaUnxTTk

ncwsn

wcwss

−−=Δ−

−=Δ−

Equation 5.19. Discretized boundary conditions for the solid layer temperature equation.

In this case an energy balance is made. The heat transfer by conduction be-

tween the node 1=i and the wall is equalled to the heat transfer by convection

between the wall and the surroundings. The same is done for the last node

ni = . wT is the temperature of the exterior wall of the fuel cell, where heat is

transferred to the surroundings by convection. Radiation is neglected.

5.4.2. Discretization of the supporting equations

In this section, as governing equations are just algebraic equations, there is not

necessity of special boundary equations.

∑=

=n

iiavg I

LI

1

1

Equation 5.20. Discretized equation for the current average.


53

m

iawicwiw

iidi

tccD

IFn ,,,,

,,−

−=α

Equation 5.21. Discretizated equation for the net water flux per proton coefficient.

( )109.453.4024.20049.0 ,3,

2,,, ≤+−+= iaiaiaiaid aforaaan

( )11159.059.1 ,,, >−+= iaiaid aforan

Equation 5.22. Discretized equations for the electro-osmotic drag coefficient.

⎭⎬⎫

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧

+−=

is

oidiw

TDnD

,,,

2731

30312416exp

Equation 5.23. Discretized equation for the diffusion coefficient of water.

10.3685.398.17043.0 ,3,

2,,

,

,,, ≤+−+= ikikikik

drym

drymikw aforaaaM

c ρ

[ ] 114.114 ,,,

,,, >−+= ikik

drym

drymikw aforaM

c ρ

Equation 5.24. Discretized equations for the concentration of water in the membrane.

satiawiH

viaw

viaw

iaPP

MMMa

,,,,,

,,,

2 ⎭⎬⎫

⎩⎨⎧

+=

saticwNiO

vicw

viaw

icPP

MMMMa

,,,,,

,,,

22 ⎭⎬⎫

⎩⎨⎧

++=

Equation 5.25. Discretized equations for the water activity.


54

( ) 18.21044.11018.91095.2log 3,

72,

5,

2,,10 −⋅+⋅−⋅= −−−

ikikiksatikw TTTP

Equation 5.26. Discretized equation for the water saturation pressure.

CmolJTTTC ikikikv

ikwp /º/1059.31060.71088.646.33 3,

92,

6,

3,,,,

−−− ⋅−⋅+⋅+=

CmolJTTTC ikikikiHp /º/1070.81029.31065.784.28 3,

102,

6,

5,, 2

−−− ⋅−⋅+⋅+=

CmolJTTTC ikikikiOp /º/1031.11008.61016.110.29 3,

92,

6,

3,, 2

−−− ⋅+⋅−⋅+=

CmolJTTTC ikikikiNp /º/1087.21072.51020.200.29 3,

92,

6,

3,, 2

−−− ⋅−⋅+⋅+=

Equation 5.27. Discretized equations for heat capacity of species.

im

miioccell

tIVV,σ

η −−=

Equation 5.28. Discretized equation for the fuel cell voltage.

[ ]⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧+

=iO

oii

i PII

FTR

,2

ln5.0

273η

Equation 5.29. Discretized equation for the electro-osmotic drag coefficient.

⎭⎬⎫

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧

+−⋅

⎭⎬⎫

⎩⎨⎧

−=is

imdrym

drymim

TcM

,,

,

,,

2731

30311268exp00326.000514.0

ρσ

Equation 5.29. Discretized equation for the membrane conductivity.


55

6. Conclusions

Our energetic model, based in fossil fuels, is not sustainable in a long term.

Fossil fuels are a scarce resource that will soon be exhausted. In addition,

global warming requires the use of technologies resulting in a reduction in

emissions of carbon dioxide. PEM fuel cells are a promising technology in the

transition from fossil fuels to other energy sources. This type of fuel cells con-

sumes hydrogen as a fuel, producing water as the only residue. Hydrogen, the

best energetic vector, shall be produced in the future by sustainable and more

efficient technologies.

To enable widespread use in practical applications of fuel cells in a not too dis-

tant future, it is necessary to develop a mature technology nowadays. Water

management is one of the most important and complex issues in a PEM fuel

cell, which should be improved, especially in appliances with a dynamic re-

sponse. Heat engine vehicles are one of those appliances which are under con-

tinuous transient states, as well they are one of the devices in which most inter-

ested in applying the use of fuel cells, especially to reduce pollution in the cities.

All this together makes very important to study water management in fuel cells.

When the fuel cell has to work under transient conditions, local dehydration or

flooding of the electrolyte membrane and electrodes may happen. To get a

good control over all the local fuel cell parameters involved in the water man-

agement, it is necessary to build up an advanced differential model. This is the

reason why Yi and Nguyen [1] presented their model; and Golbert and Lewin [3]

adapted it to make it suitable to build a control model.

In this project, both models are described and adapted to make them suitable to

been run by a numerical computational software, such MATLAB. As the model

presented is no suitable to be analytically solved, it is arranged to be numerical-

ly solved. To make this possible finite differences are applied to the entire mod-

el. Basically, all the equations are discretized. This will let to use the model in a

control system to manage a device under transitory conditions, like in a car, for

instance. When a motor vehicle is working, power demand is continuously


56

changing, as well the external relative humidity of the air changes. This makes

water management specially complicated in this type of devices.

When equations are discretized, also the solid must be discretized or divided in

several nodes. In this stage, it is very important to carefully study the geometry

of the solid, chiefly the channel distribution of the reactants (hydrogen and air)

and the coolant fluid. All this is strongly influenced by the type of bipolar plate

used in the fuel cell.

In this case of study, a standard fuel cell has been assumed. The channels for

the anode and cathode streams are parallel, facing each other. These channels

are vertically distributed in the bipolar plate, and parallel to the horizontal direc-

tion. Also the cooling channels are horizontal and closed to the anode and ca-

thode stream channels. All the channels are made into a solid phase, which

includes anode, cathode, membrane and bipolar plates. The solid phase be-

haves as an only element.

Described this case of study, the best way found to discretized the fuel cell is to

divide it into four different elements. The nodes of the same element interact in

series in the x direction, and nodes of different elements interact in parallel in

the y direction.

In future works, the model should be numerically solved. Therefore, a flow chart

and introducing all the discretized equations in computational software will be

necessary.


57

7. References

[1] J.S. Yi, T.V. Nguyen (1998), “An along-the-channel model for proton ex-

change membrane fuel cells”, J. Electrochem. Soc. 145, 4 1149-1159.

[2] T.V. Nguyen and Ralph E. White (1993), “A water and heat management

model for proton-exchange-membrane Fuel Cells”, J. Electrochem. Soc. 140,

2718.

[3] J. Golbert and D.R. Lewin (2004), “Model-based control of fuel cells: (1)

Regulatory control”, PSE Research Group, Wolfson Department of Chemical

Engineering.

[4] J. Larminie and A. Dicks (2003), “Fuel Cell Systems Explained”, Wiley (ed).

[5] C. Spiegel (2008), “PEM Fuel Cell Modeling and Simulation Using Matlab”,

Elsevier-AP (ed).

[6] C. Spiegel (2007), “Designing and Building Fuel Cells”, Elsevier (ed).

[7] F. Barbir (2005), “PEM Fuel Cells: Theory and Practice (Sustainable World

Series)”, Elsevier (ed).

[8] B. Sorensen (2011), “Hydrogen and Fuel Cells, Second Edition: Emerging

Technologies and Applications (Sustainable World)”, AP (ed).

Fuel Cell Thesis

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