MODELING OF A FUEL CELL

by

PRAVEEN MAROJU, B.Tech.

A THESIS

IN

ELECTRICAL ENGINEERING

Submitted to the Graduate Faculty

of Texas Tech University in Partial Fulfillment of the Requirements for

the Degree of

MASTER OF SCIENCE

IN

ELECTRICAL ENGINEERING

Approved

August, 2002

ACKNOWLEDGEMENTS

I am very grateful for the advice and support my advisor. Dr. Micheal E Parten,

for his feedback, for keeping me focused all the time, and for providing valuable

resources. I would like to thank Dr. Sunanda Mitra and Dr. Michael G Giesselmann for

their comments and for being part of my thesis committee. Discussions with Wallace D L

Turner and folks working on HEV team have greatly improved and clarified this work. I

owe a lot to a number of friends. Our conversations and work together have greatly

influenced my thesis.

11

CONTENTS

ACKNOWLEDGEMENTS ii

ABSTRACT vi

UST OF TABLES vii

UST OF nOURE viii

L INTRODUCTION 1

LI Hybrid Electric Vehicles 1

1.2 Fuel Cell 6

1.3 Course of the Report 7

2. FUEL CELL 9

2.1 Course of the Report 9

2.1.1 Alkaline Fuel Cells 9

2.1.2 Phosphoric Acid Fuel Cells 9

2.1.3 Molten Carbonate Fuel Cells 10

2.1.4 SoUd Oxide Fuel Cells 10

2.1.5 Polymer Electirolyte Fuel Cell 10

2.2 Typical Electrochemical Reactions in Fuel Cells 11

2.3 Characteristics of fuel cells. 12

2.4 Hydrogen Fuel Cell 12

2.5 Proton Exchange Membrane Fuel Cell 13

2.5.1 Electrolyte 15

111

2.5.2 Electrodes 16

2.5.3 Operation of PEM fuel cell 17

3. MATHEMATICAL MODEL OF A PEM FUEL CELL 21

3.1 Electrochemical Reaction in the Fuel cell 21

3.2 Voltage produced by the Fuel Cell 22

3.2.1 Gibbs free energy of formation 22

3.2.2 Nemst Equation 24

3.3 Causes of the Voltage drop 26

3.3.1 Activation Losses and Tafel's Equation ..28

3.3.2 Fuel Crossover and Internal Currents 30

3.3.3 Ohmic losses 31

3.3.4 Mass Transportation or Concentration Losses 33

3.4 Efficiency of the Fuel Cell 37

3.5 Power Lost 38

3.6 Heat Produced 38

4. SIMULINK MODEL OF THE FUEL CELL 40

4.1 Top Level Block 40

4.2 Fuel Cell System 42

4.3 Electrochemical Current 42

4.4 Load Current 45

4.5 Determination of Actual Voltage of the Fuel Cell 46

4.6 Intemal Resistance 48

IV

4.7 Cell Temperature 49

4.8 Efficiency of the Fuel Cell 52

4.9 Power lost by Fuel Cell 53

5. ANALYSIS OF SIMULINK MODEL 55

5.1 Load limited mode 55

5.2 Comparison with practical Fuel Cell data 62

5.3 Reactant hmited mode 67

6. CONCLUSION 69

REFERENCES 71

ABSTRACT

The development of alternates to the internal combustion (IC) engine is

widespread these days. Electric vehicles and hybrid electric vehicles are

promising alternatives to the IC engine. Electric vehicles are completely battery

operated while HEVs consists of at least one auxiliary source of electrical

energy that drives a set of motors, apart from the battery. Proton exchange

membrane fuel cell is one such auxiliary source.

This thesis is focused on developing a simple model for the proton

exchange membrane hydrogen-oxygen fuel cell. This report deals with the

theory behind the fuel cell from electrochemical, thermodynamic and

mathematical point of view. And documents the simple model of the fuel cell

developed using Simulink®. This is followed by a detailed analysis of this

model and comparison with the practical fuel cell.

VI

LIST OF TABLES

2.1 Electrochemical Reactions in fuel cells 11

3.1 Gibbs free energy of formation of different materials 24

3.2 Intemal Resistivity versus relative humidity 33

3.3Typical values of over voltage parameters 36

5.1 Values of Parameters in Load Limited Mode Analysis 56

5.2 Cell Voltages and Load Current 57

5.3 Values of Parameters in Reactant Limited Mode Analysis 63

5.4 Reactant Limited Mode analyses 65

Vll

LIST OF nOURES

1.1 Fuel cell market in future 3

1.2 Saving by fuel cell powered vehicles 4

1.3 Reduction in Emissions 4

1.4 Projected Emissions Reduction by use of Fuel cell vehicles 5

2.1 Proton Exchange Membrane (PEM) Fuel Cell 14

2.2 Electrode structure of fuel cell stack 17

2.3 Operation of fuel cell 19

3.1 Cell voltage versus current density plot of typical PEMFC 27

3.2 Overvoltage versus Log of Current density for different reactions 29

3.3 Effect of activation over voltage 32

3.4 Effect of mass transportation losses, for two values of 'B' 35

4.1 Top Level Block 41

4.2 Fuel Cell Systems 43

4.3 Determination of least available gas 44

4.4 Mass flows - Current density block 45

4.5 Voltage, Efficiency and Power Loss block 47

4.6 Humidity - Intemal Resistance Block 48

4.7 Mass Flow - Temperature Block 52

4.8 Power Loss Block 53

5.1 Variation of Cell Voltage ( V ) , Load Limited Mode Analysis 58

viu

5.2 Variation of Load Current (A), Load Limited Mode Analysis 58

5.3 Variation of Cell Voltage 59

5.4 Variation of Load Current 60

5.5 Comparison of actual values with values from Simulations 61

5.6 Variation of Cell Voltage with Time 64

5.7 Variation of Load Current with Time 64

5.8 V-I Plot of Reactant Limited Mode analyses 66

IX

CHAPTER 1

INTRODUCTION

Hydrocarbons have always been one of the most efficient fuels. The

combustion of hydrocarbons in a combustion chamber enables the conversion

of the potential energy of the hydrocarbons into mechanical energy. In general,

the chamber and mechanism involved in this conversion is called an internal

combustion engine. In the combustion engine, energy of the fuel is released in

the form of heat, which is then converted into mechanical or electrical energy.

This double conversion leads to multiple losses. There are a number of other

limitations on the efficiency of this process. In general the efficiency of this

process ranges between 15% and 25%. The combustion, also, is not always

efficient. The fuel does not burn completely if there is an insufficient supply of

oxygen. The un-bumt fuel is a major unwanted by product of this process.

Another important limitation of the internal combustion engine is the usage of

fuels that are nonrenewable. These constraints call for research in alternative

processes for the conversion of energy [5].

1.1 Hybrid Electric Vehicles

Electric vehicles and hybrid electric vehicles (HEVs) are promising

alternatives to the IC engine. A lot of research has been going on in the

development and improvement of these vehicles. These, alternatives to the IC

engine, have many advantages. They are more efficient and mostly use

renewable fuels. In addition, emissions are reduced.

Electric vehicles (EVs) include completely battery-operated vehicles. These

are not as reliable as IC engine powered vehicles. EVs are used for trips of

relatively short distances. Batteries have to be recharged or replaced frequently.

Hybrid electric vehicles are more promising and can be as reliable as IC engine

powered vehicles. HEVs can give the general "full tank" feeling [4].

A hybrid electric vehicle consists of at least one auxiliary source of

electrical energy that drives a set of motors. These vehicles are more

complicated than the IC engine or pure electric vehicles. The IC engine in the

HEV operates along with an electric motor to provide optimum usage of the

fuel and reduced emissions.

Another class of vehicles that may be a good alternative to the IC engine

powered vehicle and HEV are fuel cell vehicles. Basically this class of vehicles

is similar to battery-operated vehicles. These vehicles use a fuel cell stack as a

source of electrical energy. The fuel cell stack drives the electric motor. Fuel

cell powered vehicles can be either HEVs or pure electric vehicles.

Battery operated vehicles and fuel cell vehicles are considered zero

emission vehicles. Moreover they reduce the use of non-renewable fuels and

save fuel. Thus, these vehicles may have a huge market in the near future. The

following figures show the statistics concerned with fuel cell vehicles and their

usage in the near future. According to projection in Figure l.I, the percentage of fuel

cell vehicles in the market will be negligible prior to the year 2010.

30 ^ « > 25 11.

o a» 15 -.5 5 10 -o o 9 5 -Q.

n

Projected Fuel Cell Vehicle Market Penetration

^ ^ y/^

^^^^ y^

^y^ 2005 2010 2015 2020 2025 2030 2035

Year

Figure 1.1 Fuel cell market in future [1].

The percentage is expected to rise gradually and by the year 2030 fuel cell

vehicles may reach a significant proportion, if the many problems associated with them

can be overcome. Thus, there Ues a good opportunity in the development of fuel cell

vehicles. The following figures depict the projected figures in savings with fuel cell

vehicles in the future. Figurel.2, the projected savings shows the savings with EVs. In

Figure 1.3, the major reduction in the emissions is in carbon dioxide emission, which is a

good incentive.

Projected savings by the use of Light Fuel Cell powered vehicles in the future

70 1 ° 60 (A o 50

I s 40 £ o 30 « "o g» 20 a 10 (0

0 J

1 64 1

^^^^^^^^^^^^^^^^^H ^ ^ ^ 1 ^^^^H ^ ^ ^ ^ 1 ^^^^^^^^^^^H ^HmHmi

^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ m " :'••'•• " . ' < ' ' . " ]

^ ^ ^ ^ ^ ^ ^ ^ ^ K ^ 1

I Year 2020

I Year 2030

Year

Figure 1.2 Savings by fuel cell powered vehicles [1]

Emission Reduction by use of Fuel Cell vehicles

^ 40 I 35 c 30 I M 25 E i 20 c o - 5 15 •2 10

I ^ ^ 0

37 1

3.5 5.8

• Nonmethyl Organic gases

D N O X

laCO

^Corbondiaoxide in Billion metric tonnes

Gases

Gases

Figure 1.3 Reductions in Emissions [1]

Figure 1.4 shows the projections with fuel cell powered vehicles and

indicates why the development of fuel cell vehicles is important. The U.S.

4

Department of Energy (DOE) in collaboration with the three major U.S.

automobile companies is one of the forerunners in the research in alternative

fuel vehicles.

lars

B

illio

ns o

f D

o

Emissions Reduction by use of Fuel Cell Vehicle

25 ^

20

15 -

10-

5 -

n -

y^ y^

y^ y ^ • Series 1

^ w

2018 2020 2022 2024 2026 2028 2030 2032

Year

Figure 1.4 Projected Emissions reduction by use of Fuel cell vehicles [1]

The Departments of Electrical Engineering and Mechanical Engineering at

Texas Tech University have been developing alternative fuel vehicles, mostly

Hybrid Electric Vehicles, for some time. The departments have recently

focused on developing a complete Electric Vehicle powered by a hydrogen-

oxygen fuel cell. They are working towards optimization of the performance of

the vehicle. A wide range of tests is conducted to optimize the performance of

the fuel cell. This process involves an amount of risk as the fuel cell has certain

limitations. The current drawn from it should not exceed a certain limit and

voltage of the cell should not fall below a certain limit. There are limitations on

the temperature and humidity of the cell. As the limits vary from cell to cell

there is a possibility of break down of the fuel cell in the process of testing.

This can be avoided by performing these preliminary tests on a model of the

fuel cell. Application software like Advisor® and PSAT® provide environment

for simulation of performance of different vehicles. But these do not give a

good picture of the behavior of the fuel cell. A customized model of the fuel

cell is needed, especially in case of fuel cell-vehicles to predict the behavior of

the vehicle accurately. Modeling of the fuel cell is the first step in development

of the model of the over all vehicle.

1.2 Fuel Cell

A fuel cell is a reactor in which fuel and an oxidizer react electrochemically

producing electricity and a number of byproducts. A proton exchange

membrane (PEM) hydrogen-oxygen fuel cell uses hydrogen as fuel and oxygen

as oxidizer. The by products of it include heat and water.

Sir William Grove made the first successful operational attempt of the hydrogen -

oxygen fuel cell in 1839. The "gaseous voltaic cell" built by him generated electricity and

water upon reaction of hydrogen and oxygen. The fuel cell used porous platinum

electrodes and sulfuric acid as an electrolyte.

William White Jaques, who substituted phosphoric for sulfuric acid as the

electrolyte, coined the term "Fuel Cell." There have been many significant attempts

to build fuel cells for many years.

Hydrogen-oxygen fuel cells may be a good alternative for the internal

combustion engine since they can provide clean efficient energy conversion

unlike the IC engine. The efficiency in the conversion of chemical potential

energy into electrical energy is very high compared to chemical to mechanical

conversions. Efficiencies in the order of 50% to 60% are easily achieved by

fuel cells in contrast to Carnot cycle limited efficiencies of IC engines. The

conversion itself involves no moving parts and hence reduces wear and tear

compared to IC engines. Also, this enables relatively noise free conversion [2].

1.3 Course of the Report

This thesis is focused on developing a simple model for the proton

exchange membrane hydrogen-oxygen fuel cell. This report deals with the

theory behind the fuel cell from electrochemical, thermodynamic and

mathematical point of view. A simple general model for the fuel cell that could

help understand the function of the fuel cell is developed using Simulink.

The first chapter of the report introduces fuel cell vehicles. The second

chapter gives a brief description of the types of fuel cells used in fuel cell

vehicles and describes in detail the operation of proton exchange membrane

fuel cell (PEM fuel cell). The third chapter describes in detail quantitative

relations of parameters of the PEM fuel cell. This is followed by the description

of the Simulink model of the PEM fuel cell in fourth chapter. The fifth chapter

includes analysis of the Simulink model in detail. The final chapter is the

conclusion of the modeling of the fuel cell.

CHAPTER 2

FUEL CELL

A variety of fuel cells are at various stages of development today. Generally Fuel

Cells can be classified, depending on die combination of fuel, oxidant, electrolyte,

temperature of operation and so on. The most convenient classification is based on the

type of electrolyte used. The operating temperatures and all other factors depend on the

type of electrolyte, thus making it the important factor. A brief description of the various

types of fuel cells follows [3].

2.1 Types of Fuel Cells

2.1.1 Alkahne Fuel Cells

The electiolyte in alkahne fuel cells is KOH. These fuel cells are operated at

temperatures as high as 250 °C with concentiated (85 wt%) KOH and at lower

temperatures (<120 °C), with less concentiated (35-50% KOH). The electiolyte is

retained in a matrix usually made of asbestos. A wide range of catalysts are used in these

fuel cells.

2.1.2 Phosphoric Acid Fuel Cells

The electrolyte in this fuel cell is concentiated phosphoric acid and it operates at

150-200 ' C. The matrix used to retain the electiolyte is silicon carbide and the catalyst

used in both the anode and cathode is platinum. The use of concentrated acid (100%)

minimizes water vapor pressure. Hence, water management is not difficuU. The relative

stability of concentiated phosphoric acid is high compared to other electiolytes, but at

low temperatures it is a poor conductor.

2.1.3 Molten Carbonate Fuel Cells

The electrolyte in this cell is a combination of alkaline carbonates of Na and K. It

is retained in a ceramic matrix of LiA102. The fuel cell operates at 600-700 °C, where

alkali carbonates form a highly conductive salt, with carbonate ions providing ionic

conduction. At high temperatures, the Ni anode and nickel oxide cathode can operate

with no catalysts, unlike other cells, which require catalysts.

2.1.4 Sohd Oxide Fuel Cells

SoUd nonporous metal oxides Uke Y2O3 stabihzed in Zr02 are the electrolytes.

The cell operates at 650-1000 °C, where ionic conduction by oxygen takes place.

Typically, the anode in this fuel cell is Co-ZrOi or Ni-Zr02 and the cathode is Sr-doped

LaMn02.

2.1.5 Polymer Electiolyte fuel cell

A polymer can be used as an ion exchange membrane electrolyte in fuel cells.

These membranes are good conductors of protons. The byproduct in this type of fuel cells

is pure water. The operating conditions of these fuel cells are not broad due to the

constraints imposed by the membranes and due to problems with the byproduct, water.

10

Polymer membranes are usually used when operating temperatures are below 120 °C.

Hydrogen and Oxygen rich gases are the inputs to this fuel cell. Catalysts like platinum

are used to adsorb hydrogen. These are called Proton Exchange Membrane Fuel Cells,

PEMFCs. These are the most favored fuel cells, if byproducts are considered, but

pressure equilibrium across tiie membrane and humidity in the cell require critical

consideration. A wide range of other fuel cells has also been developed.

2.2 Typical Electrochemical Reactions in Fuel Cells

The following is a tabulation of the electrochemical reactions that take place in

different types of fuel cells considered in this chapter. The half-cell reactions are

tabulated in Table 2.1.

Table 2.1 Electrochemical Reactions in fuel cells [3]

Fuel CeU

Alkaline

Phosphoric Acid

Molten Carbonate

Proton Exchange

SoUd Oxide

Anode Reaction

H2 + 2(0H)' => 2H2O -1- 2e"

H2 =>2i r+2e"

H2 + CO3" => H2O + CO2 + 2e" CO + CO3" =>2C02 + 2e'

H2 =>2ir"+2e'

H2 -1- 0" =>1 H2O + 2e' CO + 0 ' => CO2 + 2e'

CH4 + 4 O' => 2H2O + CO2 + 8e"

Cathode Reaction

1/2 O2 + 2H2O + 2e" => 2(0H)"

i/2 0 2 + 2 i r + 2 e = > H 2 0

V2 O2 + CO2 + 2e" => CO3"

»/2 02+2i r+2e"=>H20

V2 O2 + 2e" => 0 '

II

2.3 Characteristics of fuel cells

Fuel cells have many favorable characteristics of all the energy conversion devices as

mentioned before. Some of these are sunmiarized as follows [3]:

• High energy-conversion efficiency relatively independent of size or load. The

conversion efficiency of 40-60 % is achievable with ease.

• Modular design of the fuel cell enables stacking them to achieve required voltage

and currents.

• Cogeneration capability, due to the continuous supply of heat from these cells as

these are operated at constant temperature, emitting continuous heat.

• Size flexibility, i.e., fuel cells of all sizes can achieve the same efficiency,

enabUng small size cells with the same efficiency as huge size stacks.

• Very low environmental intmsion due to the friendly byproducts produced.

• Flexibility of fuel is a major feamre of fuel cells. A redox reaction is all that a fuel

cell needs. It is apparent from the above discussion that a number of oxidants and

reducants that serve this purpose are available.

• Rapid load following capabihty

2.4 Hydrogen Fuel Cell

Fuel cells use a variety of fuels, but hydrogen is one of the best choices since it is an

abundant resource available and the by-product produced in hydrogen fuel cells is pure

water. Moreover, water can be the source of hydrogen. Thus, the process appears more or

less repeatable, but involves external energy if hydrogen is obtained from water. In a

12

hydrogen fuel cell, the fuel is hydrogen and the oxidizer is oxygen with by­

products of water and heat. A hydrogen fuel cell converts chemical energy

directly into electrical energy via a modified oxidation process [6].

There are different realizations of the hydrogen fuel cell each differing

basically at the interface between hydrogen and oxygen. The electrolytes used

are either in a free flowing state or immobilized in inorganic or organic

supports. The flowing state offers less electrolyte resistivity but requires well-

defined and controlled electrodes to maintain the liquid phase within the

electrode structure. The immobilized electrolytes do not require a well-defined

electrode, since the electrolyte is maintained within a matrix, but it offers more

electrolyte resistivity.

Ion exchange membranes are solid electrolytes containing dissociated

functional groups capable of transporting either cations or anions. These make

use of immobilized electrolytes, nondependent capillary forces, but chemically

attached to non-soluble, thermally and chemically stable polymeric materials.

The fuel cell under consideration falls into this category.

2.5 Proton Exchange Membrane Fuel Cell

The Proton exchange membrane fuel cell is a simple system made up of a cathode, an

anode and a conducting medium between these two. This is similar to the conventional

fuel cells except that the electrodes and the electrolyte in this are all porous membranes.

13

The major difference between PEM fuel cells and other fuel cells is the electrolyte,

which is a proton-conducting medium polymer. The mobile ion in the conducting

polymer is H . The electrolyte is an ion-conducting polymer with either sides of it

bonded with catalyzed porous electiodes. The basic operation of the PEM fuel cell is

same as the acid electrolyte fuel cells. But this fuel cell can operate at lower temperatures

enabhng it to start quickly. It is compact and could be used in any inclination. Figure 2.1

gives an idea of the basic blocks of the PEMFC.

Figure 2.1 Proton exchange membrane (PEM) fuel cell [3]

14

2.5.1 Electix)lvte

The electi-olyte is a polymer made up of sulphonated fluoro-hydrocarbons, usually

sulphonated fluoro-ethane. The perfluorination of ethane results in the formation of

polyteti-afluoro-ethane also called PTFE or Teflon®. The most common polymer

electix)lyte used in PEMFCs is Nafion®. The PTFE polymer is sulphonated by

substitution of a fluorine atom with sulphonic acid, HS03'. This group is ionically bonded

and hence the side chain consisting of tiiis group ends in a SO3. This results in the

presence of SO3" and H*, a strong attracting pair of ions. The polymer consists of two

regions with entirely different properties. The C-F bond is very strong and hydrophobic

while the HSO3" bond is highly hydrophyllic. Nafion is thus a hydrophobic compound

with hydrophylUc regions around the sulphonated side chains. These regions absorb large

quantities of water. In these hydrated regions the H^ ions are weakly bonded to the SOs'

group and hence are more or less mobile. The mobility of the hydrogen ions within this

hydrated region enables the transfer of protons, provided the hydrated regions are large

enough. This is taken care of, to some degree, with the attiaction of water molecules that

are by products in the fuel cell reaction [6]. But in the initial sages and also at later stages

when operating at higher temperatures, humidification of the gases ensures enough

hydrated region.

The main characteristics of the Nafion® membranes that are of importance in

enabUng fuel cell reactions are:

• They are acidic and form dilute acid when hydrated;

• They absorb large quantities of water;

15

• H* ions are free to move in die dilute acids or in the hydrated regions;

• These membranes are highly chemically resistant.

2.5.2 Electiodes

The basic stiiicture of the electi-odes in different designs of PEM fuel cells is

almost the same. In most PEM fuel cells anode and cathode electiodes are similar. These

are made of carbon electiodes with small particles of catalyst coated on the surface. The

best choice of catalyst for both anode and cathode is platinum.

Generally two methods are used in the construction of the electrode - electrolyte

stmcture of PEM fuel cells. In first method called the 'separate electiode method,' the

fine particles of catalyst supported on large particles of carbon are fixed to a porous

conductive material. Carbon cloth and carbon paper are frequently used porous materials

in this method. The porous conductive material forms the gas diffusion layer through

which gas diffuses and reaches the catalyst. Polytetrafluoro-ethane is added to expel

water particles to the electiode surface where it evaporates easily. This whole structure

forms a single electrode. Two such electiodes are fixed on either side of the polymer

exchange membrane. This electrode-electiolyte structure can also be realized by directly

fixing the carbon- supported catalyst on either side of a polymer membrane. The gas

diffusion layer is then applied above the catalyst [6].

The gas diffusion layer has different functions. It allows the diffusion of gas on to

the catalyst and forms an electrical connection between the electrodes and the metal

plates that form the extemal circuitry. It expels water formed at the electiolyte surface

16

and also acts as a protective layer over die catalyst. Figure 2.2 shows a PEM fuel cell

stack.

Connectious between each cell

Hydrogen fedro each anode

e^Uiode

Figure 2.2 Electrode stmcture of a fuel cell stack [3]

2.5.3 Operation of PEM fuel cell

The operation of the fuel cell is best described by the half-cell reactions at the

anode and cathode electrodes. The role of electrodes and electrolyte in enabling this

reaction is also as important as the reactions themselves.

At the anode, hydrogen dissociates at the catalyst producing H* ions and

electrons. The half-cell reaction is

17

2H2 => 2H^ + 2e-

The protons thus formed move through the electrolyte to react with oxygen at the cathode

producing water. The electi-ons formed at the anode that are transferred by the extemal

circuitiy to the cathode supply the elections required in this reaction. The cathode half-

cell reaction is

V2O2 -I- 2e" + 2 i r => H2O

The transfer of the electrons through the extemal circuitiy constitutes the current supplied

by the fuel cell. The current depends on the amount of hydrogen dissociating at the

anode, provided there is a continuous flow of Oxygen. Figure 2.3 shows the operation of

the PEM fuel ceU.

18

Hydrogen fual

LOAD e.g. electric motor

Electrons flow rourKi the extemal circuit

Figure 2.3 Operation of fuel cell [3].

Hydrogen gas is fed at the anode end of the fuel cell where it passes through the

gas diffiision membrane on the anode side to reach the anode. The anode is basically a

number of small carbon particles coated with a catalyst that absorbs hydrogen molecules.

The hydrogen absorbed at the anode is in direct contact with the electrolyte. High

humidity is maintained in the fuel cell. The hydrophyllic HS03' absorbs water and acts as

a dilute acid. In this state, the H* ions in sulphonic chain are weakly bonded to the SO3".

The oxygen at the cathode reacts with the H* provided by the electrolyte to form water.

Thus the electrolyte absorbs H* ions from the anode and supplies them to the cathode

reaction. The electrons generated by the dissociation of hydrogen at the anode supply the

electrons required for the cathode reaction. The elections tiaverse through an extemal

path from anode to cathode. The extemal path could include the load, which the fuel cell

drives.

19

The EMF or voltage between the anode and cathode is a complex phenomenon

and is addressed in the following chapter. The EMF of the cell depends on the different

parameters. The following chapter gives a quantitative description of the fuel cell. It

describes the voltage, current and efficiency achieved by the fuel cell in mathematical

terms.

20

CHAPTER 3

MATHEMATICAL MODEL OF A PEM FUEL CELL

The operation of a PEM fuel cell has been discussed in the previous chapter.

Modeling of a fuel cell is concemed with understanding the fuel cell and predicting its

behavior. This chapter discusses the fuel cell quantitatively. The parameters addressed

include current, voltage, power loss and efficiency of the fuel cell.

3.1 Electrochemical Reaction in the Fuel cell

The basic reaction in the fuel cell is

2H2 + O2 > 2H2O.

Each hydrogen molecule loses two electrons and each oxygen molecule gains four

elections. Thus current produced in this reaction is given by [3]

, 4.F.mf^>.A000 2.F.m/M~,».1000 ,^ , .

32 2.02

where 'F ' is Faraday's constant, 'mf oxygen- is the mass flow rate of oxygen in grams per

second, 'mf hydrogen- is the mass flow rate of hydrogen in grams per second and T is the

current produced, in Amperes.

The actual current produced is less than this current since there never is a total

utihzation of either oxygen or hydrogen. The utilization factor 'X' is the fraction of either

21

oxygen or hydrogen utilized in the reaction. This depends on the geometry of the

electrolyte gas interface and also the catalytic activity of the catalyst.

In general, the mass flow rate of hydrogen is considered in estimating the current

produced in the reaction. The major reason for this is the assumption of continuous

supply of oxygen to the fuel cell. But an unlimited supply of oxygen is not available for

the fuel cells in the real world.

3.2 Voltage produced by the Fuel Cell

The voltage produced by the fuel cell depends on a number of parameters. The

maximum cell voltage of the fuel cell reaction at a specific temperature and pressure is

determined by the relation between the change in Gibbs free energy of formation (Ag f)

and the number of moles of elections (n) involved in the reaction per mole of hydrogen.

3.2.1 Gibbs free energy of formation

The energy released in a chemical reaction is given by the change in Gibbs free

energy of formation. The Gibbs free energy of formation is analogous to mechanical

potential energy. Thus, a change in this energy is the energy released.

As mentioned earher, the basic operation of the fuel cell involves the tiansfer of

two electrons per mole from the anode to the cathode producing a water molecule. The

work done in this process is the same as the change in the Gibbs free energy of formation.

That is, mathematically

22

Agf = - (2NeE) = - ( 2 F E ) (3.2)

where 'Ag r is the change in the Gibbs free energy of formation, 'E' is the EMF or

reversible open circuit voltage of the PEM fuel cell, 'N' is Avagadro's number (the

number of molecules in one mole), 'F' is Faraday's constant and 'e' the charge of the

electron.

Thus the open circuit voltage of the fuel cell is given by

E" = ^ . (3.3)

2F

The change Gibbs free energy of formation changes with temperature and state. The

value of change in the Gibbs free energy of formation is dependent on the temperature of

the fuel cell, change in the reaction enthalpy and entropy as

Ag, = Ah-TAs (3.4)

The change in enthalpy (Ah) of the reaction is -285,800 J. The change in entropy of the

reaction (As) is -163.2 J/K. 'T' is the temperature of the fuel cell, assuming the reaction

takes place at the same temperature as the temperature of the fuel cell [3].

The following is a tabulation of the change in Gibbs free energy of formation for

PEM fuel cell reaction at different temperatiires (Table 3.1).

23

Table 3.1 Gibbs free energy of formation of different materials [3]

Temperature (°C) 25 80 80 100 200 400 600 800 1000

State of Water Liquid Liquid

Gas Gas Gas Gas Gas Gas Gas

^g f (kJ/mole) -237.2 -228.2 -226.1 -225.2 -220.4 -210.3 -199.6 -188.6 -177.4

At 25 °C, the change in the Gibbs free energy of formation is obtained from the Table 3.1

and the open circuit voltage of the fuel cell is calculated as

Agy = 237200

237200

2*96485 1.229 V. (3.5)

The open circuit voltage of the fuel cell changes with a change in temperature.

Thus tiiis has to be recalculated as the temperamre of the fuel cell changes in modeling

the fuel cell.

3.2.2 Nemst Equation

The Nemst equation provides the relationship between standard potential for a

cell reaction and tiie actual voltage produced at various temperatures and pressures of the

reactants and products.

24

Nemst Equation:

:?0 Kl , r reactant E = E" + i l l . In. IF P product

(3.6)

where 'Preactam' is die product of partial pressures of the reactants and 'Pproduct' is the

product of die partial pressures of tiie products produced in the cell reaction. All

pressures are expressed in bars. 'F' is Faraday's constant, 'R' is the universal gas

constant and 'T' die operating temperatiire of the fuel cell.

In the case of tiie PEM fuel cell tiiis reaction is

E T O . RT Phydrogen RT . rr = E + In + In V P oxygen

2F Pwater 2F

Ti T-.0 ^i 1 Phydrogen / v i , / _ \ E = E ° + — l n - ^ + — I n ( P oxygen) ( 3 . 7 )

The pressures on either side of the proton exchange membrane in the PEM fuel

cell should be equal to keep the membrane intact. Too large a difference in pressure

across the membrane can result in a mpture of the membrane. Thus the partial pressures

of hydrogen and oxygen are frequently considered equal in this case. That is

P hydrogen — " oxygen ( j . o a )

25

Thus, the following relation gives the Nemst Equation for the PEM fuel cell.

E = E ° + — In (Phydrogen) In (P water) (3 .8b) 4 F 2 F

Another factor to be considered here is that the partial pressure of water on either

side of the membrane should be maintained constant and equal. This should also consider

the humidity of hydrogen and oxygen and also water produced at the cathode. Thus the

following relation gives the change in the voltage of the fuel cell due to a change in

pressure of hydrogen and oxygen fi-om Pi to P2 [3].

•.. ^ 3/vi , P2 ,„ „^

AV = In— (3.9) 4F Pi

Generally the theoretical voltage calculated from the above relations is not achieved

practically, by the fuel cell. The actual voltage produced is less than the theoretical value

as a result of losses associated with some irreversibihty inherent in the PEM fuel cell.

Figure 3.1 depicts the difference between the theoretical and actual voltages of a PEM

fuel cell.

3.3 Causes of the Voltage drop

There are a number of causes for the difference in the actual voltage and the ideal

reversible open circuit voltage of the fuel cell. This difference in the voltage is called by

26

different terms that imply different causes for it. Electiochemists call it the overvoltage

that opposes the ideal voltage of the fuel cell. It is also called polarization, irreversibility

and losses. Major causes of this drop are activation losses, intemal currents, also called

fuel crossover, ohmic losses and mass transportation losses, also called concentration

losses. These losses are discussed quantitatively and their influence on the voltage of the

fuel cell is explained in detail in the course of this chapter [9].

0

No loss" voltace of 1.2 volts

Even the open circuit voltage is less than the theoretical no loss value

Rapid initial fall in voltage

Region where voltage falls more slowly and graph is fairly linear

Rapid fall at higher currents

200 400 600 800 1000

Current density/ mA.cm'

Figure 3.1 Cell voUage versus current density plot of typical PEMFC [3]

27

3.3.1 Activation Losses and Tafel's Equation

Tafel observed tiiat the overvoltage at the surface of an electrode followed a

similar pattern in a great variety of electiochemical reactions. The general pattem is

shown in Figure 3.2. It is apparent Uiat the plot of overvoltage against log of current

density is almost a sti-aight line. This plot is called a "Tafel plot" [3].

The Tafel equation gives the values of overvoltage for different currents. One

form of expression of the Tafel equation is given by Equation 3.10a.

V = A l n - ^ | (3.10a)

The constant 'A' is the slope of the Tafel plot. It determines the speed of the

electrochemical reaction. The constant io is the current density at which the overvoltage

begins to move from zero. This current density, io is called the exchange current density.

Figure 3.2 shows the Overvoltage versus Current density plot for different reactions. The

Tafel equation is valid only when the current density is greater than the exchange current

density io, that is, overvoltage is given by Equation 3.10b

^ V = A In -^

\ioj ; i> io (3.10b)

28

Equut ion o f l>i.-si-rli line is

reaction

•* 5

l..t>B(ciirre»»t d c n s i t y V n i A

Bi!«t''^t line intcrccixs »he currfsnt dcntiity ax.it

Figure 3.2 Overvoltage versus Log of Current density for different reactions [3]

The exchange current density is a cmcial factor in controlling the performance of

the electrode and the fuel cell. The value of this current is made as high as possible to

reduce the overvoltage. If the fuel cell has no losses except the activation overvoltage on

one electrode, then 3.11 gives voltage of the fuel cell

f V = E°-A In (3.11)

v'"/

29

where 'E° is die reversible open circuit voltage, 'A' is the Tafel constant and the slope of

die Tafel plot. The theoretical grounds that explain the Tafel plot reveal that Equation

3.12 relates the Tafel constant to temperature.

A = ( ^ | (3.12)

The constant ' a ' is called the "charge ti-ansfer coefficient". Its value is in the

range '0' to '1.0'.It depends on tiie reaction involved and the material of the electrode.

'R' is the gas constant and 'F' is Faraday's constant.

3.3.2 Fuel Crossover and Intemal Currents

The electrolyte in the PEM fuel cell is an ion-conducting polymer. Transfer of

electrons through the membrane results in a loss in the current produced by the cell.

There is always a significant amount of electron current through the membrane as the

membrane allows small amounts of election conduction apart from ionic conduction.

Moreover the membrane, being porous, cannot inhibit the diffusion of hydrogen through

it. A small amount of hydrogen diffuses through the membrane and combines with

oxygen on the other side. This is called the intemal current. And the major causes of this

are 'fuel crossover' and election conduction through the membrane. This results in a

reduction in the current provided to the extemal circuit. These two losses have the same

effect on the current drawn. They result in small currents in the fuel cell even under open

30

circuit condition, due to the intemal currents they induce. These currents actually can

result in finite activation over voltage, if they are significant even under open circuit

conditions. Thus, in closed circuit these intemal currents increase the effect of extemal

current on the activation overvoltage.

The activation over voltage without die intemal currents is given by Equation

(3.10b). In tiie presence of the intemal currents the relation holds well, but with an

increase in tiie current by an amount equal to intemal currents. Thus, Equation (3.11)

representing activation voltage of die fuel cell, including die effect of intemal currents

and activation overvoltage can be represented by Equation 3.13

V = E° - A In ''i + in

^ lo (3.13)

where 'in' is the intemal current density and 'io' is the exchange current density.

Figure 3.3 shows the effect of activation overvoltage and intemal current on the fuel cell

voltage, without any effect of other over voltages.

3.3.3 Ohmic losses

The ohmic losses are the losses in the voltage of the cell due to the resistance of

electrodes to flow of electrons and the resistance of the electrolyte to the flow of ions.

The product of current density and area specific resistance corresponding to the current

density give the ohmic loss given by Equation (3.14).

31

ir (3.14)

em

o >

1.20

1.00

0.80 -

0.60

0,40

0.20 -

0.00

"No los l ' r eve is ibbvo l tage ^jl

; "P - 5 > j

Current d&sity / mA-crrv , j

Figure 3.3 Effect of activation over voltage [3]

where 'i ' is the current density through certain area considered and 'r' the resistance

corresponding to this area. The resistance is obtedned by considering resistances of

electiodes and the electrolyte. But generally, the resistance of the membrane is more

significant, compared to resistance of the electrodes. Thus, the area specific resistance 'r'

is considered equivalent to the resistance of the membrane. The resistance of the

membrane depends on the humidity of the fuel cell. Thus, the ohmic losses vary with

change in humidity of the fuel cell membrane.

32

Table 3.2 tabulates typical values of resistance of the membrane of a PEM fuel

cell at various values of relative humidity. Curve fitting of values of resistance and

relative humidity is done. The resistance of the membrane at certain relative humidity is

obtained from the resulting curve.

Table 3.2 Intemal Resistivity versus relative humidity

Relative Humidity 0 2

2.5 3 4 5 7 9 23 29

Intemal Resistivity 0

0.005 0.013 0.023 0.03 0.049 0.052 0.063 0.11 0.123

3.3.4 Mass Transportation or Concentration Losses

The concentration of the gases in the fuel cell decreases during the operation of

the fuel cell. The extent of the change in the concentiation of the gases depends on the

current being drawn from the fuel cell, the geometry of the electiodes and the supply of

the gases. The change in the concentration results in a reduction in the partial pressures of

the gases. The reduction in concentration of both hydrogen and oxygen depends on the

electric current drawn and the physical characteristics of the system. Equation 3.15

represents the effect of change in partial pressure of the fuel on the voltage of the fuel

cell.

33

AV = 15I,„fi (3.15) 4F Pi

The current density at which fuel is used up at a rate equal to the maximum

supply speed is called the limiting current density of the fuel cell. Current density cannot

rise beyond this value, as the fuel cannot be supplied at a greater rate. The pressure

reaches zero at this current density. Pi is the pressure at zero current density and iithe

current density at zero pressure. Assuming a linear fall in both pressure and current from

Pi and ii value to zero, the pressure at a current density 'i' is given by the Equation 3.16.

P2 = H'^'i) ^'^

Substituting this relation in Equation 3.16, the voltage drop due to mass transportation

losses is given by Equation 3.17a.

AV = — l n ( l - ^ | (3.17a)

RT The first term is ' — ' for oxygen. Adding both the losses, the concentration

4F

losses are given by the Equation 3.17b.

I AV = - ^ i " h - ^ ^ • ' ^

34

'B' is a constant witii a typical value of 0.05 V. Thus, the fuel cell voltage, considering

only die ti-ansportation losses, is given by

E° -h ^ ' " [ I - T ] (3.18)

Figure 3.4 shows tiie effect of mass tiansportation losses on fuel cell voltage for two

values of 'B'.

0 200 400 600 800 1000 1200 • {

Current density/ inA.cm^

Figure 3.4 Effect of mass transportation losses, for two values of 'B'

The actual fuel cell voltage is obtained by combining these irreversibilities, including the

intemal current density term in all the other over voltages [3]. The current density

through the extemal circuit is increased by an amount equal to the intemal current

density. And the area specific resistance 'r' in the ohmic losses is approximated to the

35

area specific resistance of tiie membrane (R) of the PEM fuel cell. Thus, the following

relation represents die actual voltage of die PEM fuel cell.

V = E O . ^ , „ £ i - ( , - . , . . ) . K _ , , i i ± ^ ^ + Bin 1 l + tn

ii ) (3.19)

In tills equation, 'E° is tiie open circuit voltage, ' in' intemal current density, 'A' is slope

of Tafel curve, io is exchange current density, 'B' is mass tiansfer over voltage constant,

' 1/' is the limiting current density at die electiode, 'R' is specific resistance. Typical

values of these constants for a PEM fuel cell are given in the Table 3.3.

Table 3.3 Typical values of overvoltage parameters [3]

Constant

E

in

r

io

A

B

ii

Typical Value

1.0 V

2 mAcm"^

0.03 Q.cm^

0.067 mAcm"^

0.06 V

0.05 V

900 mAcm'^

36

3.4 Efficiency of the Fuel Cell

The efficiency of the fuel cell is the representation of the actual energy produced

by the fuel cell in terms of the maximum possible energy. This could be represented as

the product of the voltage and current efficiencies of the fuel cell.

Maximimi energy produced by the fuel cell is equal to the maximum change of

Gibbs free energy of formation. The maximum EMF is attained when the enthalpy of

formation or calorific value of hydrogen is considered in the open circuit voltage relation

instead of Gibbs free energy of formation [3]. Thus

E max = - ^ ^ =1-48 V, considering the higher heating value (HHV). (3.20a) 2F

=1.25 V, considering the lower heating value (LHV). (3.20b)

Thus considering the HHV, voltage efficiency is given by

n. = v , M (3.21) ' 1.48

The current efficiency depends on the utilization factor of the fuel, 'X'.

m = X (3.22)

The total efficiency of the cell is thus

Tit AVc— (3.23) 1.48

37

3.5 Power Lost

If all die energy of the fuel cell is converted into electiical energy, the output

voltage of tiie fuel cell is, 1.48 V if the water produced is in liquid state or 1.25 volts if

water produced is in vapor state. The energy lost in this conversion is given by following

relation.

Conversion Loss = I*(1 .48-Vc) (3.24a)

The ohmic loss widiin the membrane of die fuel cell is also a significant loss. Thus die

total power lost is given by following relation.

Power Lost = I^ * R + 1 * (1.48-Vc) (3.24b)

3.6 Heat Produced

The power lost by the fuel cell manifests itself mostly as heat produced by the fuel

cell. Moreover the heat is liberated by the PEM fuel cell's electrochemical reaction also.

Thus the heat produced by the fuel cell is the sum of heat produced in these two ways.

The heat produced by the fuel cell is given by

Heat Produced per second = I ^ * R +1 * (1.48 - V c) + (285.83)* 10^*n (3.25)

38

where 'n' is the number of moles of hydrogen that react per second in the fuel cell, 'Vc' is

the actual fuel cell voltage, T is die current drawn from the fuel cell, 'R' is the resistance

of the membrane.

The relations derived in the above discussion explain the behavior of the PEM

fuel cell and determine the various output characteristics of the cell. These mathematical

relations are simplified in the next chapter and used to develop the Simulink model of the

fuel cell, which could be used in simulating the behavior of the cell corresponding to

certain values of the above parameters of the fuel cell.

39

CHAPTER 4

SIMULINK MODEL OF THE FUEL CELL

Simulink is the tool of choice for control system design, DSP design,

communications system design, and odier simulation applications. It is an interactive tool

for modeling, simulating, and analyzing dynamic systems. It enables the building of

graphical block diagrams, evaluation of system performance, and refinement of the

design. It integrates seandessly witii MATLAB and is tightiy integrated with State flow.

Thus it enables modeling of event-driven behavior.

The PEM fuel cell is modeled with Simulink. The Simulink model consists of a

hierarchy of blocks performing different functions. The hierarchy includes a top-level

block consisting of sub-blocks and sub-blocks consisting sub-sub-blocks. The function

and the logic involved in each block is discussed in this chapter.

4.1 TOP Level Block

The Simuhnk model developed is a hierarchal model. The top-level block is the

block at the top of the hierarchy. This block is analogous to the Front end in software

programming such as the Conti-ol Panel in Lab-View. The input parameters to the

Simuhnk model are defined and modified in this block. This block consists of die sources

and displays of the fuel cell model. The sources to the model are constant value blocks,

values of which are predefined. Some of the sources include variable blocks that vary as a

40

function of time. Figure 4.1 shows tiie top-level model. The values of input parameters

are varied by variation of the values of tiiese blocks.

. _

10

MF u i (.aLriwj

MF

Veloo •Matt

Relati ol

10

H2(SL

1500

PM)

1 tf of coo'ling rCMete

30

ve Hum i h e »

y-

ts/hc)

idity 1

2 RH vanation i 1

CAtmosph

Lo

y-idCOhn isl

k

| >

+

+

.P2 etes)

-. fe

—•

w

—•

02 MF

H2MF

Coolant Velocity

H2 Hum. in

P2

Load

Fuel Cell

1

Sys temp Otit

power loss

heat out

efficiency

voltage

Total Power

cument

System

1

w

L J Cell Te

1

mp(C i lei us)

Internal Pouuer loss per square cm ^

- • f = l

1 1

Heat Energy Out (Watts)

Cell Efficiency

w

Total

1

Powuei

1 1

Output

^ 1 1

Current (Amps)

Figure 4.1 Top Level Block

The inputs to the model include mass flow rate of oxygen, mass flow rate of

hydrogen, velocity of the coolant (water), relative humidity in the cell and load

resistance. The units of these parameters are shown in the braces following each

parameter. Load voltage, load current, efficiency, fuel cell temperature, heat produced in

41

die cell, total power lost and cell temperature are the outputs of the Simulink model. The

outputs of the fuel cell model are displayed in the time domain.

4.2 Fuel Cell System

Fuel Cell System is die next level, below the top-level block, in the hierarchy.

This block consists of the logic for determination of load current and electrochemical

currents of the cell. It also includes sub-blocks that determine cell voltage, efficiency,

heat produced, cell temperature and die total power lost. Figure 4.2 shows the 'Fuel Cell

System' block.

This block has feed back of die load current to the voltage block. This includes

the logic to restrict the load current and voltage beyond certain limits. The load current is

limited to the electrochemical current when the load current exceeds the electrochemical

current. In this case, the load limits the cell voltage as the load current has reached its

maximum value. The cell voltage is hmited to the drop across the load. The relational

operator determines if the load current exceeds the electiochemical current while the

switches, 'switchl' and 'switch2' determine the load current and voltages at the extreme

condition.

4.3 Electiochemical Current

Equation 4.1 gives the electiochemical current generated by the fuel cell. This is

the current generated by the flow of elections, which are generated in the electrochemical

reaction of the fuel cell, from Anode to Cathode. This current depends on the mass flow

42

rates of hydrogen and oxygen. It is proportional to the number of moles of hydrogen

oxygen, which ever is the least available. Equation 3.1 is rewritten below.

or

j _ 4.F.m/L,„„.1000 _ 2.F.m/»..„„.1000 32 " 2.02

(3.1)

H2MF

02 HF

Comljnt

Coolant Vtiteitf^ Aaa4 \i*^. Vt H a Out

© - -H2 Hum. ir

Figure 4.2 Fuel Cell Systems

The maximum utility of the hydrogen and oxygen is possible when their ratio is

equivalent to the stoichiometric ratio of the gases, in the electiochemical reaction of the

PEM fuel cell. The electiochemical current is determined in two steps. Initially the gas

with the lowest flow rate is determined. The mole flow rate of this gas is determined and

43

used in the 'mass flow - current' block to determine the electrochemical current. Figure

4.3 shows the logic used in determining the least available gas.

€>-M f(u)

H2 MF in Mole of H2/sec

& f(u)

0 2 MF In Mole of 02/sec M

2 -r

X J

Constant PfoducM

Relational Operator

u(1J'2.016 *tri Stwitch"^^^ ^l**'^ <** "2(g /sec)

Current

Figure 4.3 Determination of least available gas

This block consists of a multiplier, a relation block and other blocks that realize

mathematical relations. First step in this block is the conversion of mass flow rate of the

gases into moles. The function 'f(u)' converts mass flow rate to rate of flow of moles of

the gases as

Rate of flow of moles = f(u) (4.1)

massflow

2.02

massflow

32

for Hydrogen

for Oxygen

'u' is mass flow rate of hydrogen and oxygen in each 'f(u)' block.

44

The relational sub-block determines the number of moles of the least available gas. Each

mole of die gas generates current equivalent to the product of Faraday's constant and the

valency of die gas. The current generated by the electrochemical reaction of the fuel cell

ftx)m tiie mass flow rates of oxygen and hydrogen. The 'Mass Flow versus Current

Density' block determines die current generated in the electiochemical reaction. Figure

4.4 shows die determination of die current. The 'Mass Flow- Current Density' block

implements Equation 3.1. The block determines die maximum current density of the cell.

MF H2

- M 2 > current

H2 lAieight

Ared of Gel

Figure 4.4 Mass flows - Current density block

4.4 Load Current

The load current is limited to a maximum value by available reactants, as described in

the previous section. The actual load current is the cell voltage divided by the load

45

resistance, as long as the current is below the maximum. Once the current reaches the

maximum value, die voltage is determined by product of current and load resistance.

Ohm's law relates die load current, and cell voltage. The load connected to fuel cell is

considered purely resistive in tiiis analysis. Equation 4.2 relates cell voltage and load

current.

'• = T. ''•''

The 'Fuel Cell System' block shown in Figure 4.2 includes the logic to determine the

actual current drawn by the load. The load is a variable resistor. This resistance can be

varied to study the operation of the fuel cell.

4.5 Determination of Actual-Voltage of the Fuel Cell

The voltage of the fuel cell, combirung reversible open circuit voltage with the effects of

change in pressure and the over voltages, is given by Equation 3.19.

V = E ° + ^ ^ I n - - ( f + / . ) r -AIn 4F PI

/,-l + ln , „,

i lo

(• I+In 1.1^^1 (3.19)

\, ii

When the typical values given in Table 3.1 are substituted in to this relation, the voltage

of a typical fuel cell is obtained. The Simulink block that realizes this relation is shown in

Figure 4.5.

46

lnt.R

Cell Temp.T

Current density,! d

& •

Cell pressure

P - atmospherio f

Mux

MATLAB Funotlon

CcllVoltjge

Mux1

Mux

Mux

voltjge

MATLAB Function -*0 Effioienoy Eff

MATLAB Function -*0

Power Lost PoiAier Loss

Figure 4.5 Voltage, Efficiency and Power Loss block

The Cell Voltage function in this block determines the voltage of the fuel cell.

This function implements Equation 3.19 in the Simulink model. The inputs to this

function are fed as arguments, in order, by the multiplexer, Muxl. The Intemal

Resistance - Himiidity block shown in Figure 4.6, determines intemal resistance of the

cell. The 'Mass Flow -Temperature' shown in Figure 4.7, determines the cell

temperature. The current density is obtained from the 'Mass Flow-Current density block'.

The pressure 'P2' of the cell in atmospheres is a source to the Top Level Block while the

pressure 'PI ' is defined to be a constant. The values of the parameters A,B, in, ii, io

determine the characteristics of the cell.

47

4.6 Intemal Resistance

Intemal Resistance of the fuel cell is a function of the relative humidity in the cell.

The humidity of the cell determines the extent of humidification of the ion exchange

membrane. The intemal resistance of the cell is determined by the extent of

humidification of the membrane. This is thus a complex parameter. 'Humidity - Intemal

Resistance' block is shown in Figure 4.6.

(^ H2 Rel. Humidityr Look-Up

Table coductvs

3.8429e-4

X -K1

IntrR Producti

length / area

Figure 4.6 Humidity - Intemal Resistance Block [13]

The specific resistance of the cell for certain value of relative humidity is obtained

fi-om the look up table [13]. Equation 4.3 relates the intemal resistance and specific

resistance of the cell.

R = ^ (4.3)

4.7 Cell Temperature

Heat is liberated in a variety of forms in the fuel cell. The heat liberated by the

electi-ochemical reaction is the primary form of heat generated. This heat depends on the

48

number of moles of hydrogen reacting in the cell. The Figure 4.7 shows the 'Mass Flow

Temperature'. Equation 4.4 gives die amount of heat generated by the fuel cell reaction.

2Z5.S3*mfH2

2.02 < -*)

Heat is also Uberated by the losses witii in the fuel cell. These losses result in the drop of

the cell voltage from die ideal reversible voltage. Equation 4.5 gives the heat generated

by this loss.

H = I* (1 .23-Vc) (4.5)

The total heat produces with in the cell is thus the sum of Equation 4.4 and

Equation 4.5. The fiiel cell temperature increases due to this. The increase in temperature

has a variety of detrimental effects. Thus the cell temperature is maintained by circulation

of a coolant (water in this case) through the cell. The coolant circulates through the tubes

embedded in a heat sink. If no loss is assumed at the sink, Equation 4.6 gives heat

absorbed by the coolant.

Qa = hAAT (4.6)

'A', area of contact,

'AT', temperature difference between sink and coolant,

'h', heat transfer coefficient.

49

The coolant is forced tiirough the tube wound inside the heat sink. Thus the heat

tiansfer coefficient in this case is given as

h*D 0.023 *(Re)°'*(Pr)'" (4.7)

where

'h' is specific heat tiansfer coefficient,

'D' is diameter of the pipe, (0.1m, as assumed in this analysis),

'k' is thermal conductivity of the coolant,

'Re' is Reynolds number, and

'Pr' is Prandti number of the coolant.

This is a dimension less equation for the heating of water flowing in turbulent

motion in horizontal pipes. A simphfied form of this equation is represented in Equation

4.8. This relation is a function of temperature [14,16].

(884.6-h 0.38l*r)*Vc ,^ „, h =- TT — ^ ^ (4.8)

D ° ' * 10000

where Vc is velocity of the coolant.

Equation 4.9 gives the temperature rise due to the heat liberated by the electiochemical

reaction.

50

^T = , ^ ' ' , (4.9)

where

Qr is given by Equation 4.6,

Cp is specific heat of the heat sink material,

Mai is mass of the heat sink.

The temperature of the cell increases by the amount AT. The coolant is assumed

to be at room temperature (293°C). Equation 4.8 gives the heat absorbed by the coolant.

Heat is generated by the losses in the cell. Equation 4.7 gives the heat generated due to

losses. Equation 4.10 gives the net heat absorbed due to these two effects.

AQ = Q - H (4.10)

ATa gives the fall in temperature, due to this absorption.

ATa = , ^ ^ , (4.11) {Cp*Mj

51

The cell remains at room temperature if the coolant absorbs more heat than produced.

This is ensured by die relational elements.

^23

Open ddreveislble vatUa

0-0- Mux

cooling H20 uux

Hu)

0.01

ContJotArea

Actual Voltage. _ -—, Vc 0 [J] ^

Actual ourrenti Product

285 83*3

0 delta H

MF H2

X X H

n

2.01S

lolar mass K I

heatQ

1

Mass of Aluminum cooling plate

0.89

delta Q Sum8

r* initial temp of CW

203

delta Temp

> _f\LJ 1 SumO

delta Tempi

Sum7

Heat Out

Sum10

TSl!

Transport Delay

Swiitchi

Cp

Relational Operate r1

^

Sys Temp

293

initial tempi

Figure 4.7 Mass Flow - Temperature Block

4.8 Efficiency of the Fuel Cell

The relation that determines the efficiency of the fuel cell is given by the Equation

(3.21), which is repeated below. The block that determines the efficiency of the fuel cell

along with the voltage is shown in the Figure 4.3

„ 100 11

1.23 (3.21)

52

4.9 Power Lost by Fuel Cell

The power lost by the fuel cell depends on the resistance of the membrane, the

number of moles of Oxygen reacting in the fuel cell and the actual voltage of the fuel

cell. The difference in die voltage from die ideal voltage determines the amount of power

loss due to die over voltages produced in die fuel cell. The following relation is obtained

fixim die Equation (3.24b).

Power Lost = I ^ * R +1 * (1.23 - V c)

The power loss block is shown in the Figure 4.8.

(3.24b)

& potAier loss

voltage

ay-load

X Sum1

Product

total power

Figure 4.8 Power Loss Block

The Simuhnk model described in this chapter can be used for analysis under

different conditions for each parameter. The constants and values of various parameters

in the Simulink model have been described in this chapter. SI units have been observed

for values of almost all the parameter. The cell area considered while calculation of

53

current density is represented in square centimeters as current density has been

considered in Amp/sqcm in determining voltage.

The following chapter is concemed with the Analysis of this model with variation

of different parameters.

54

CHAPTER 5

ANALYSIS OF SIMULINK MODEL

The Simulink Model of the PEM fuel cell provides a good tool for simulation of

the fuel cell. Analysis of the Simulink model includes the study of the behavior of the

model under different conditions. The conditions include limited and unlimited supply of

either of hydrogen and oxygen gases. The load regulates the behavior of the fuel cell

when the supply of gases is unlimited. That is, the behavior of the cell is load limited.

This is not the case in practice, as reactants are limited. To study this behavior the load is

varied at constant mass flow rates of reactants.

This chapter includes the study of the Simulink model in both of these cases. The

simulation of the Simuhnk model gives the behavior in the time domain. The displays in

the top-level block of the model show the variation of each parameter with time. Cell

voltage, load current, efficiency and cell temperature are the parameters considered in the

analysis of the model. Variation of load in abundant supply of reactants is considered in

load limited analysis while a limited supply of reactant is considered in reactant limited

analysis.

5.1 Load Limited Mode

The flow rate of reactants is fixed at a large value in the load-limited mode. The

values of the parameters fixed in this analysis are hsted in Table 5.1. The variation of cell

voltage with load current is stressed in this analysis. The load resistance is assumed to be

55

completely resistive. It is varied through a wide range as shown in Table 5.1 and the

value of cell voltage and load current obtained at different load resistances is tabulated in

Table 5.2. Figure 5.1 and Figure 5.2 show the variation of cell voltage and load current in

the time domain. The load is varied from 500 Q to 50 mQ.

Table 5.1 Values of Parameters in Load Limited Mode Analysis

Parameter

Mass flow rate of oxygen

Mass flow rate of Hydrogen

Velocity of Coolant

Relative humidity of the Cell

Load Resistance

Value

lOSLPM

lOSLPM

1500 (mti/hr)

30%

500Q to 50mQ

Table 5.2 tabulates the fuel cell voltage and load current data obtained from

simulation of the Simulink model for ii = 200Amps, ii = 80Amps and data available from

the four-cell Proton Exchange Membrane fuel cell stack.

56

Table 5.2 Cell Voltages and Load Current

Simulation Data with

ii = 200A

Cell Voltage ( V )

1.23 0.881 0.877 0.874 0.87 0.865 0.86 0.855 0.849 0.842 0.834 0.825 0.813 0.799 0.778 0.76 0.743 0.712 0.676 0.593 0.544 0.495 0.445 0.396

Load Current ( A )

0 22 23 25

26.75 29

31.25 34.2 37.7 42.1 47.6 55 65

79.9 103.8 127.5 148 178 193

197.9 198 198 198 198

Simulation Data with

ii = 80A

Cell Voltage ( V )

0.99 0.967 0.945 0.926 0.911 0.894 0.888 0.883 0.876 0.868 0.859 0.847 0.831 0.807 0.792 0.768 0.698 0.546 0.39 0.33 0.3

0.26 0.17 0.14

Load Current ( A )

1.98 3.86 6.29 9.26 12.1 16.3 17.8 19.6 21.9 24.8 28.6 33.7 41.6 53.8 61

69.8 77.6 78 78 78 78 78 78 78

Practical Fou

PEM fue

Cell Voltage ( V )

0.9725 0.8925

0.86 0.835 0.815 0.735 0.5425

Data From r-cell cell stack

Load Current (A )

1.02 5.17 10.54 14.94 20.55 50.58 70.36

57

Figure 5.1 Variation of Cell Voltage ( V), Load Limited Mode Analysis

Figure 5.2 Variation of Load Current (A),Load Limited Mode Analysis

58

Figure 5.1 and Figure 5.2 show the voltage and currents vary by insignificant

amount compared to die variation of the load resistance. The load resistance variation by

a factor of few hundred ohms resulted in the variation of current by less than a few

amperes. This current resulted in variation of die cell voltage by a few 'mVs'. The load

resistance and cell voltage at any instant determine the load current at that moment.

While the load current is fed back to die fuel cell to determine the load voltage for the

next clock pulse.

Figure 5.3 and Figure 5.4 show the cell voltage and load current when the load is

varied ftx)m 50Q to 50 mQ.. The load resistance is reduced by fraction of an ohm in this

analysis.

Figure 5.3 Variation of Cell Voltage (V)

59

Figure 5.4 Variation of Load Current (A)

These figures show a wide variation in load current and a significant variation in

cell voltage. The data from above simulations is combined to give a good picture of the

behavior of die cell. The cell voltage and load current from the above two simulations are

listed. Table 5.2 shows die cell currents and corresponding load voltages obtained in the

above simulations. Thus these values of cell voltage and load current variation

correspond to the variation of load from 500Q to 50 mQ.

Figure 5.5 shows the plots of 'cell voltage versus load current' obtained from the

above simulations and from actiial data available. The acmal data has been collected from

die four-cell fuel cell stack, at Electrical Engineering Department, at Texas Tech

60

m University. The voltage obtained from this cell corresponds to four fuel cells arranged

a stack. The current obtained from this fuel cell stack is corresponds to the current drawn

fix)m a single cell as the stack is made up of combining fuel cells in series. The single cell

voltage however is a fraction of die voltage obtained from the stack. Assuming all the

cells to be similar, die voltage of a single cell is equivalent to 25% of the voltage of the

stack. Figure 5.5 shows die plot of die single fuel cell voltage and load current of the

four-cell fuel cell stack [13]. The other curves shown in this figure are obtained from the

simulation of tiie Simulink model. These curves correspond to different values of the

hmiting current, ii = 200Amps and ii = 80Amps, in the cell voltage relation given by

Equation 3.19. Figure 5.5 plots die cell voltage and load current of the fuel cell stack and

simulation data of Table 5.2.

Comparision of Practical values with values from Simulations

-Simulation results, 1

- Practical values

-Simulation values, 2

100 150

Cun-ent

250

Figure 5.5 Comparison of actual values with values from Simulations

61

4F " p , V u ^ = E°-^^ln^-0> '")*R-Alnp^U51n I-i±^ (3.19)

V I'

Figure 5.5 shows how, die behavior of the PEM fuel cell could be emulated by the

Simulink model. The abmptness of the fall of voltage of the Simulink model can be

reduced by die variation of die Tafel's slope coefficient of the model. This is apparent

fi-om the Figure 5.6. The fuel cell is not limited by load only. There can be constraints on

the supply of the reactants. The behavior of the cell in this case is shown by the analysis

of the Simuhnk model in the Reactant Limited mode.

5.2 Reactant Limited Mode

Reactant hmited mode is the most common mode of operation of the fuel cell.

The simulation of the Simulink model in this mode involves variation of the load

resistance with a hmited supply of reactant. The mass flow rate of hydrogen is fixed at a

smaller value compared to load limited mode. The electrochemical current produced by

the reaction of hydrogen and oxygen in this case is 11.366 Amps. The values of the input

parameters of the Simuhnk model in this model are listed in Table 5.3.

62

Table 5.3 Values of Parameters in Reactant Limited Mode Analysis

Parameter

Mass flow rate of oxygen

Mass flow rate of Hydrogen

Velocity of Coolant

Relative humidity of the Cell

Load Resistance

Value

lOSLPM

0.1 SLPM

1500 (mti/hr)

30%

500Q to 50mQ,

Table 5.3. Load resistance is varied from 500 Q to 50 i2 with time and the

corresponding variation of cell voltage and load current are shown in Figure 5.6 and

Figure 5.7. Thus these figures show the proportionate variation with change in load.

From Figure 5.6 it is clear that cell voltage decreases gradually with time due to a

gradual decrease of load resistance and a resultant increase of load current drawn,

initially. Once die voltage reaches the voltage corresponding to the electi-ochemical

current limit, it starts falhng rapidly, due to a gradual decrease of load resistance.

63

Figure 5.6 Variation of Cell Voltage with Time

Figure 5.7 Variation of Load Current with Time

64

Figure 5.7 shows that the load current increases gradually until it reaches the

electrochemical current limit. Load current remains constant once it reaches this limit.

This shows that the load current is insensitive to the load resistance beyond this point.

Table 5.4 tabulates the load current and corresponding cell voltage obtained in the

Reactant limited mode analysis of the Simulink mode. Figure 5.8 plots the cell voltage

against load current obtained from the Table 5.5. Practical data is not available in this

mode of application of the fuel cell.

Table 5.4 Reactant Limited Mode analyses

Cell Voltage (V ) 0.9905 0.989 0.9874 0.9855 0.9836 0.9815 0.9792 0.9766 0.9736 0.9668 0.9578 0.9453 0.9261 0.9151 0.909 0.8525 0.7388 0.682 0.5683

Load Current (A ) 1.98 2.08 2.19 2.32 2.46 2.62 2.8 3

3.25 3.87 4.79 6.3

9.26 11.366 11.366 11.366 11.366 11.366 11.366

65

Reactant Limited Mode

1 o 1 .c.

1

9 0.8

%, 0.6

^ 0.4

0.2

0

^^^^pi^^T^^^^T^^i^^Pisl^^

^ — - - ' ^ ^ ^ ' ^ J 1 1

HI ' ^ H' ^ ^ ^ ^ H

-^^^^^^^

^^^^^^^^Hr ^

0 5 10

Load Current

- • - V - I Plot

15

Figure 5.8 V-I Plot of Reactant Limited Mode analyses

The behavior of the cell in this mode is quite simple. The cell voltage and load

current in this mode are related in a manner similar to the Load Limited Mode until the

fuel cell can supply no more current to the load. The load current remains constant and

equivalent to the electrochemical current provided by the electrochemical reaction

between oxygen and hydrogen in the fuel cell. The product of load current and load

resistance now determines the cell voltage. In the above simulation load resistance is

decreased from 500 Q to 50 Q gradually witii time. The load current increases gradually,

due to the decrease in load resistance, according to Ohm's law. The cell voltage is

determined by the behavior of the fuel cell. Once the load current reaches the

66

electix)chemical current limit, (11.366Amps in this case) it becomes insensitive to any

further decrease in load resistance. The cell voltage in this region becomes indifferent to

fuel cell behavior. Further decreases of load results in dependence of the cell voltage on

the load current. This case is analogous to die behavior of an ideal power supply when it

reaches its current limit. The load current remains constant and the load voltage drops in

a power supply widi any further reduction of the load resistance once the current limit is

reached. This behavior is similar to the behavior of the PEM fuel cell once the

electrochemical current limit is reached.

5.3 Comparison with the Practical Fuel Cell Data

The practical data is obtained from the data available from the four-cell PEM fuel

cell stack, at Electrical Engineering Department at Texas Tech University. The data

available is limited to the Load Limited Mode of operation of the fuel cell. This data is

used in comparison of the performance of the Simulink model developed. Figure 5.5

shows the characteristics of the 'fuel cell voltage versus load current' plots obtained from

actual data and the data from Simulink model simulation in Load Limited Mode. This

figure shows how the Simuhnk model can emulate the behavior of the PEM fuel cell.

The curve obtained from actual data is slightiy lower than the curve obtained from

Simulink analysis. This discrepancy may due to the presence of losses in the

interconnections between the fuel cell electrodes and the load. There are a number of

other factors that result in this.

67

The above analysis shows that the Simulink model can be used to analyze the

behavior of the PEM fuel cell under different conditions. Moreover the effect of different

parameters on the behavior of the fuel cell can be studied with this model.

68

CHAPTER 6

CONCLUSION

Modehng of die PEM fuel cell has successfully addressed various aspects of the

fuel cell modeling. The madiematical background behind the fuel cell voltage, load

current, efficiency, power loss and cell temperature has been clearly understood. The

Simulink model of the PEM fuel cell has been developed according to it. The model

considers die effect of pressure, cell temperature, relative humidity, load current and over

voltages on the cell temperature. It doesn't address the flooding at the cathode and the

variation of relative humidity with change in temperature.

The Simulink model developed is a good tool in study of the behavior of a fuel

cell. It has the ability to emulate the behavior any PEM fuel cell by variation of certain

parameters. A trial and error approach or a curve fitting approach can be followed in

defining the values of the parameters of the model in achieving desired performance. The

values of parameters of this model have been obtained by a trial and error approach to

emulate the performance of a single cell of the four-cell fiiel cell stack at the Electrical

Engineering Department at Texas Tech University. The parameters are reahzed by

considering the data available from the Load Limited mode analysis of the fuel cell stack.

The model has been simulated, by hmiting the load resistance. The performance of it has

been compared to that of a single cell of the fuel cell stack. The model has been able to

emulate the behavior of the fuel cell satisfactorily. The model can be used as part of the

69

model of a hybrid electric vehicle or fuel cell vehicle. The simplicity of the model

enables good understanding and contiol of the resultant model.

For future work, the inclusion of effect of variation of humidity, capacitive effect

of interconnection between load resistance and electiodes of the fuel cell would enhance

the performance of the model significantiy. The inclusion of the effect of flooding at the

electrodes, especially at cathode relates to the contiol of the humidity of the ion exchange

membrane. This could ensure more complete optimization of the performance of the fuel

cell.

70

REFERENCES

I. Department of Energy. Hybrid vehicle propulsion- online resource center. http://www.hev.doe.gov

2. Mitchel Will, Fuel Cells. Academic Press Incorporated, San Diego, 1963 [6 - 8], [47 49].

3. Larminie James, Andrew Dicks, Fuel Cell Systems Explained. John Wiley and Sons, UK, 2000 [1-34], [37-53], [61-83], [293- 302].

4. Shrestha Nabin, "A Generic Interface System for Automatic Contiols." Master's Thesis, Electrical Engineering Department, Texas Tech University, Lubbock, TX [8].

5. Adams, David R., Pierre Yves Cathou, and Robert E. Gaynor, Fuel Cells, Power for die Futiire. Harvard University, Boston, 1960 [9 - 22].

6. Berger Carl, Handbook of Fuel Cell Technology. Prentice- Hall, Englewood Cliffs, N.J., 1968 [425 - 428], [451], [470 - 472].

7. Yang Yinghn, "Simulation model for a Fuel Cell powered vehicle." Master's Thesis, Electrical Engineering Department, Texas Tech University, Lubbock, TX [58 - 63].

8. H.A. Liebhafsky, E.J. Cauns, Fuel Cells & Fuel Batteries. Texas Ai&M University, College Station, TX, 1968 [137 -140].

9. McDougall, Angus, Fuel cells. John Wiley and Sons, Manchester, 1934 [31 - 52].

10. www.mathworks.com, Simulink.

11. Onhne fuel cell infonnation center- http://www.fuelcells.org/.

71

12. Turner, Wallace David Lysander, "Evaluating the power capabilities of a hydrogen Fuel cell." Master's Thesis, Electrical Engineering Department, Texas Tech University, Lubbock, TX, 1999.

13. "Four Cell Fuel Cell Model Data," Graduate students. Electrical Engineering Department, Texas Tech University, Lubbock, Texas.

14. McAdams William H., Heat Transmission. Chemical Engineering Department, Massachusetts Institute of Technology, Cambridge, MA, 1933 [159 -160], [169] [178 -185].

15. Fuel cells for transportation. Program Implementation Stiategy, US. Dept. of Energy. http ://www .eren .doe. gov/transportation.

16. Hirschenhofer J.H., D. B. Stauffer, R.R. Engleman and M.G. Klett, Fuel Cells Handbook. US. Department of Energy, Morgantown, WV, 1998.

17. Singh Jasbu-, Heat Transfer Fluids & Systems for process and energy applications. Marcel Dekker, Monticello, NY, 1985 [96 -102].

18. Aylor, Stephen Scott, 'Development of HEV,' Master's Thesis, Electrical Engineering Departinent, Texas Tech University, Lubbock, TX, 1996.

72

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