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Modelling, Simulation and Control of a Solid Oxide Fuel Cell System: A Bond Graph Approach

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MODELLING, SIMULATION AND CONTROL OF A SOLID OXIDE FUEL CELL SYSTEM: A BOND GRAPH APPROACH P. Vijay
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Page 1: Modelling, Simulation and Control of a Solid Oxide Fuel Cell System: A Bond Graph Approach

MODELLING, SIMULATION AND CONTROL OF A

SOLID OXIDE FUEL CELL SYSTEM: A BOND GRAPH

APPROACH

P. Vijay

Page 2: Modelling, Simulation and Control of a Solid Oxide Fuel Cell System: A Bond Graph Approach

MODELLING, SIMULATION AND CONTROL OF A

SOLID OXIDE FUEL CELL SYSTEM: A BOND GRAPH

APPROACH

Thesis submitted to

Indian Institute of Technology, Kharagpur

For the award of the degree

of

Doctor of Philosophy

by

P. Vijay

Under the guidance of

Dr. Arun Kumar Samantaray

and

Dr. Amalendu Mukherjee

DEPARTMENT OF MECHANICAL ENGINEERING

INDIAN INSTITUTE OF TECHNOLOGY, KHARAGPUR DECEMBER 2009

© 2009, P. Vijay. All rights reserved.

Page 3: Modelling, Simulation and Control of a Solid Oxide Fuel Cell System: A Bond Graph Approach

This thesis is dedicated to

My Parents

Page 4: Modelling, Simulation and Control of a Solid Oxide Fuel Cell System: A Bond Graph Approach

CERTIFICATE OF APPROVAL

Date: / / Certified that the thesis entitled Modelling, Simulation and Control of a Solid Oxide Fuel Cell System: A Bond Graph Approach submitted by P.Vijay to Indian Institute of Technology, Kharagpur, for the award of the degree of Doctor of Philosophy has been accepted by the external examiners and that the student has successfully defended the thesis in the viva-voce examination held today. Signature Signature Signature Name Name Name (Member of the DSC) (Member of the DSC) (Member of the DSC) Signature Signature Name Name (Supervisor) (Supervisor)

Signature Signature Name Name (External Examiner) (Chairman)

Page 5: Modelling, Simulation and Control of a Solid Oxide Fuel Cell System: A Bond Graph Approach

DECLARATION I certify that

a. the work contained in this thesis is original and has been done by me under the guidance of my supervisor(s).

b. the work has not been submitted to any other Institute for any degree or diploma.

c. I have followed the guidelines provided by the Institute in preparing the thesis. d. I have conformed to the norms and guidelines given in the Ethical Code of

Conduct of the Institute. e. whenever I have used materials (data, theoretical analysis, figures, and text)

from other sources, I have given due credit to them by citing them in the text of the thesis and giving their details in the references. Further, I have taken permission from the copyright owners of the sources, whenever necessary.

(P. Vijay)

Page 6: Modelling, Simulation and Control of a Solid Oxide Fuel Cell System: A Bond Graph Approach

CERTIFICATE

This is to certify that the thesis entitled Modelling, Simulation and Control of a

Solid Oxide Fuel Cell System: A Bond Graph Approach, submitted by Mr. P.

Vijay to Indian Institute of Technology, Kharagpur, is a record of bona fide research

work carried under our supervision and is worthy of consideration for the award of

the degree of Doctor of Philosophy of the Institute.

Dr. Arun Kumar Samantaray Dr. Amalendu Mukherjee Associate Professor Professor Mechanical Engineering Department Mechanical Engineering Department I.I.T. Kharagpur, India, 721 302 I.I.T. Kharagpur, India, 721 302 Email: [email protected] Email: [email protected] Date: Date:

Page 7: Modelling, Simulation and Control of a Solid Oxide Fuel Cell System: A Bond Graph Approach

Acknowledgement

Foremost, I would like to express my sincere gratitude to my supervisors Dr. Arun Kumar Samantaray and Prof. Amalendu Mukherjee. Dr. Samantaray with his seemingly inexhaustible energy and enthusiasm, his amazing intellect and a keen eye for detail, gave excellent guidance to me throughout the course of the work. He urged me to communicate my work to the scientific community which ensured that the work was properly recognised. The freedom that he gave me to pursue my own way of work helped me to gain confidence in my abilities. My independent research works, which are not included in this thesis, have been communicated to different journals. Prof. Mukherjee with his immense knowledge, excellent intellect and great intuitive prowess gave direction to the work and is a person whom I would always look up to and wish to emulate. He gave the initial momentum to the work, which was high enough to carry it through to completion, overcoming all the resistances it had to encounter. His uncompromising intellectual rigour and his insistence on thorough understanding of the basic physical concepts instilled in me values which will guide me throughout my career.

I would like to express my gratitude to Prof. Ranjit Karmakar, Prof. Ranjan Bhattacharyya, and Prof. Kingshook Bhattacharyya, who as members of the Doctoral Scrutiny Committee provided constructive criticism and advice during various stages of the work. Prof. Ranjan Bhattacharyya as the research coordinator took care to accommodate me into the lab environment and encouraged me by keeping a tab on my progress. Prof. Ranjit Karmakar taught the bond graph modelling course with such lucidity that it left a lasting impression on me.

I am thankful to Prof. Ajay Kumar Chattopadhayya, Head of Mechanical Engg. Department, for providing a congenial environment in the Department leading to smooth completion of my research work.

I would also like to make special mention of the research scholars of this department and my fellow colleagues Dr. Nilotpal Banerjee, Dr. Karali Patra, Dr. Sanjoy Kumar Ghosal, Dr. Sharifuddin Mondal, Dr. Sukhamoy Pal, Mr. Sovan Sundar DasGupta, Dr. Somnath Sarangi, and Mr. Tarun Kumar Bera who helped me in a lot of ways and maintained a lab environment conducive for research. I am also thankful to Mr. Birendra Nath Ghosh and Mr. D.K. Chakraborty for their good housekeeping of the machine dynamics laboratory.

Page 8: Modelling, Simulation and Control of a Solid Oxide Fuel Cell System: A Bond Graph Approach

I would like to express my sincere gratitude to all who directly or indirectly helped me for the successful completion of my thesis work.

Finally I will always remember the best wishes and blessings of my parents concerning this venture. Without their encouragement and moral support it would not have been possible for me to complete my Ph.D.

Campus of I I T, Kharagpur (P. Vijay)

July, 2009

Page 9: Modelling, Simulation and Control of a Solid Oxide Fuel Cell System: A Bond Graph Approach

Abstract A non-linear bond graph model of a Solid Oxide Fuel Cell (SOFC) is developed in this work by using the concepts of network thermodynamics, which features several improvements over the models of electrochemical reaction systems available in the literature. The constitutive relations of a C-field for modelling the energy storage in a mixture of two gas species are formulated and an existing R-field model is extended to represent the entropy convection in multi species gas flows. The developed model is energetically consistent and it clearly illustrates the physical structure of the system. Moreover, this model is useful in designing integrated model-based control strategies for the SOFC system.

The relationship between the partial pressures of the gas species in the channel and the fuel and oxygen utilisations (FU and OU, respectively) are established for a given set of input parameters and used for plotting the characteristic curves of the SOFC for various operating conditions. From the dynamic response studies, it is found that the developed model captures all the essential features of SOFC’s dynamics. A physical model-based control strategy is then formulated and it is found from the closed loop system simulations that all the control objectives are achieved at the same time by the proposed control strategy.

Finally, an algorithm has been developed to optimise the operational efficiency of the SOFC system (consisting of a cell, an after-burner and two recuperators) under varying load conditions. For this purpose, a comprehensive steady state model of the SOFC is derived and used inside the optimisation loop. The results indicate that constant FU operation of the fuel cell at a particular value of FU (which depends upon the specific system) can closely approximate the maximum efficiency operation of the fuel cell in terms of the cell operating conditions as well as the energy and exergy efficiencies. It is concluded from the dynamic response studies that the constant FU and constant temperature operating strategy (achieved through a secondary control loop) is suitable for the SOFC system because it gives good system efficiency and lower temperature transients, which leads to longer cell life.

Keywords: Solid Oxide Fuel Cell, Bond graph, Network thermodynamics, Electro-chemical reaction, Fuel Utilisation, SOFC control strategy, Operational efficiency optimisation.

Page 10: Modelling, Simulation and Control of a Solid Oxide Fuel Cell System: A Bond Graph Approach

Abstract

ii

Page 11: Modelling, Simulation and Control of a Solid Oxide Fuel Cell System: A Bond Graph Approach

List of Acronyms AC Alternating Current

AFC Alkaline Fuel Cell

DC Direct Current

FU Fuel Utilisation

LHV Lower Heating Value

LSM Lanthanum Strontium Manganite

MCFC Molten Carbonate Fuel Cell

MEA Membrane Electrode Assembly

NTU Number of Transfer Units

OU Oxygen Utilisation

PAFC Phosphoric Acid Fuel Cell

PEMFC Proton Exchange Membrane Fuel Cell

PI Proportional Integral

PID Proportional, Integral and Derivative

SOFC Solid Oxide Fuel Cell

TPB Triple Phase Boundary

YSZ Yttria Stabilised Zirconia

1D One Dimensional

2D Two Dimensional

Page 12: Modelling, Simulation and Control of a Solid Oxide Fuel Cell System: A Bond Graph Approach

List of Acronyms

iv

Page 13: Modelling, Simulation and Control of a Solid Oxide Fuel Cell System: A Bond Graph Approach

Nomenclature A Area (m2)

Ac Effective cell area (m2)

AF Forward affinity (J mol-1 )

AR Reverse affinity (J mol-1)

cp, cv Specific heat capacity at constant pressure and volume (J kg-1 K-1)

D Diffusion coefficient (m2 s-1)

E Activation energy (J mol-1)

F Faraday’s constant (C mol-1)

G Gibbs free energy (J)

h Specific enthalpy (J kg-1)

H Enthalpy (J)

i Current (A)

j Current density (A cm-2)

k Thermal conductivity (J m-1 s-1 K-1)

K Valve coefficient (m s)

m Mass (kg)

m Mass flow rate (kg s-1)

M Molar mass (g)

n Number of moles (mol)

ne Number of electrons participating in the reaction

p Pressure (N m-2)

Page 14: Modelling, Simulation and Control of a Solid Oxide Fuel Cell System: A Bond Graph Approach

Nomenclature

vi

q Heat flow rate (J s-1)

R Specific gas constant (J kg-1 K-1)

R Universal gas constant (J mol-1 K-1)

s Specific entropy (J kg-1 K-1)

S Entropy (J K-1)

S Entropy flow rate (J K-1 s-1)

T Temperature (K)

U Internal energy (J)

u Specific internal energy (J kg-1)

v Specific volume (m3 kg-1)

V Volume (m3)

V Volume flow rate (m3 s-1)

w Mass fraction

x Valve stem position (m)

ν Stoichiometric coefficient η Over-voltage (V) μ Chemical potential (J kg-1)

σ Stefan-Boltzmann constant (J m-2 s-1K-4)

ε Emissivity

τ Thickness (m) ψ Pre-exponential coefficient (A m-2)

ξ Reaction advancement coordinate (mol)

,f oζ ζ Fuel and oxygen utilisations

β Charge transfer coefficient

λ Convection heat transfer coefficient (J m-2 s-1 K-1) γ Adiabatic index

enη Energy efficiency

exη Exergy efficiency

Page 15: Modelling, Simulation and Control of a Solid Oxide Fuel Cell System: A Bond Graph Approach

Nomenclature

vii

Subscripts ai Anode side inlet

an Anode

ao Anode side outlet

act Activation

amb Ambient state

ABO After -burner outlet

AS Air source

b Bulk

c Cell

ca Cathode

ci Cathode side inlet

co Cathode side outlet

conc Concentration

d Downstream side

eq Equilibrium

ENV Environment

FCO Fuel cell outlet

gen Generated

HT Heat transfer

H Hydrogen gas

HS Hydrogen source

I1 Interconnect on the anode side

I2 Interconnect on the cathode side

L Limiting

M Membrane electrode assembly

N Nitrogen gas

ohm Ohmic

O Oxygen gas

PL Polarisation losses

r Reaction

Page 16: Modelling, Simulation and Control of a Solid Oxide Fuel Cell System: A Bond Graph Approach

Nomenclature

viii

TPB Triple phase boundary

u Upstream side

W Water Vapour

0,a Anode exchange

0,c Cathode exchange

1 Gas species #1

2 Gas species #2

Superscripts abi After-burner inlet

abo After-burner outlet

fco Fuel cell outlet

HX1 Hydrogen inlet pre-heater

HX2 Air inlet pre-heater

i Inlet

o Outlet

r Reaction

ref Reference state

0 Initial state

Page 17: Modelling, Simulation and Control of a Solid Oxide Fuel Cell System: A Bond Graph Approach

Table of Contents

Abstract i

List of Acronyms iii

Nomenclature v

Chapter 1: Introduction 11.1. Background and Motivation 1

1.2. Contributions of the Thesis 6

1.3. Organisation of the Thesis 7

Chapter 2: Literature Review and Fundamental Concepts 92.1. Introduction 9

2.2. Concept of Chemical Equilibrium 13

2.3. Bond Graph Formulation of Chemical Reaction Kinetics 15

2.4. Bond Graph Modelling of Electrochemical Systems 20

2.5. Simulation and Control of the SOFC System 25

2.6. Objectives of the Present Work 28

Chapter 3: Bond Graph Model of a Solid Oxide Fuel Cell 293.1. Introduction 29

3.2. A Preliminary Bond Graph Model of the Hydrogen Oxidation Electrochemical Reaction 31

3.3. A Bond Graph Model of the SOFC Using a C-field for the Mixture of Two Gas Species 41

3.3.1. Process Description and Modelling Approach 42

3.3.2. Modelling Assumptions 44

Page 18: Modelling, Simulation and Control of a Solid Oxide Fuel Cell System: A Bond Graph Approach

x

3.3.3. Word Bond Graph 44

3.3.4. Storage of a Two Species Gas Mixture Represented in a C-Field 46

3.3.5. Bond Graph Model of the SOFC 51

3.4. Formulation of the True Bond Graph Model of the SOFC 61

3.4.1. Convection of a Two Species Gas Mixture Represented in an R-Field 61

3.4.2. True Bond Graph Model of the SOFC 65

3.5. Conclusions 71

Chapter 4: Simulation of the Open and Closed Loop Dynamics of SOFC 73

4.1. Introduction 73

4.2. Model Initialization and Open Loop System Simulations 75

4.2.1. Static Characteristics 81

4.2.2. Selection of the Operating Regime 86

4.2.3 Dynamic Response 86

4.3. Detailed Bond Graph Model for Dynamic Simulation 88

4.4. Model Validation and Control Strategy Formulation 94

4.5. Open and Closed Loop Dynamic Responses 100

4.6. Conclusions 104

Chapter 5: Optimisation of Operational Efficiency Under Varying Loads 107

5.1. Introduction 107

5.2. Steady State Model of the SOFC System 110

5.2.1. The Dynamic Equations of the SOFC Bond Graph Model 110

5.2.2. The Steady State Model of the SOFC 121

5.2.3. Validation of the Steady State Model of the SOFC 127

5.2.4. Steady State Models of the After-burner and the Heat Exchangers 129

5.3. Optimisation of the SOFC Operating Conditions 131

5.3.1. Optimisation Algorithm 134

5.3.2. Discussions on the Results of the Efficiency Optimisation Study 136

5.4. Simulation of the Dynamic Responses 140

Page 19: Modelling, Simulation and Control of a Solid Oxide Fuel Cell System: A Bond Graph Approach

xi

5.5. Conclusions 146

Chapter 6: Conclusions 149

References 153

Appendices 163Appendix A Bond Graph Notations Revisited 163

A.1. Introduction 163

A.2. Causality 165

A.3. Activation 167

A.4. Sensors, Actuators and Instrumentation Circuits 167

A.5. Systems with Differential Causality 169

A.6. Field Elements 172

Appendix B Derivations of Some Important Relations 174

B.1. Chemical Potential of a Constituent in an Ideal Gas Mixture 174

B.2. Gibbs-Duhem Equation 176

B.3. Isentropic Nozzle Flow Equation 177

B.4. Nernst Equation 179

B.5. Butler-Volmer Equation 182

Appendix C The Nelder-Mead Simplex Algorithm 187

Appendix D List of Files and Programs Given in the Compact Disc 191

Curriculum Vitae 193

Page 20: Modelling, Simulation and Control of a Solid Oxide Fuel Cell System: A Bond Graph Approach

xii

Page 21: Modelling, Simulation and Control of a Solid Oxide Fuel Cell System: A Bond Graph Approach

Chapter 1

Introduction

1.1. Background and Motivation

The ever increasing energy requirements of mankind and rapidly depleting natural

resources combined with the detrimental effects of increased atmospheric pollution

have motivated scientists and engineers to develop cleaner and more efficient energy

conversion mechanisms. The efficiency of the heat engine, which is predominantly

employed today to convert energy to a useful form, is limited by the Carnot efficiency.

The heat engine is also highly polluting and responsible for effects such as ozone

layer depletion and green house effect. In this context, fuel cells, which are efficient

and environmentally friendly power-generating systems that produce electrical energy

by combining fuel and oxygen electro-chemically, are alternatives worthy of

consideration. Fuel cell research is attracting much greater effort and attention today

than ever before in its long history.

Page 22: Modelling, Simulation and Control of a Solid Oxide Fuel Cell System: A Bond Graph Approach

Chapter 1

2

A battery is an energy storage device, which contains the reactant chemicals. The

electrodes in a battery take part in the chemical reaction. A battery must be discarded

once the reactants are depleted (unless the battery is rechargeable). On the other hand,

a fuel cell is an energy conversion device where the reactants are continuously

supplied and the products are continuously removed. The electrodes and electrolyte

do not participate in the chemical reaction but they provide the surfaces on which the

reactions take place and they also serve as conductors for the electrons and ions.

Therefore, a fuel cell can be defined as a thermo-electrochemical device, which

converts chemical energy from the reaction of a fuel with an oxidant directly and

continuously into electrical energy.

The basic components of a general fuel cell are two porous electrodes, i.e. anode

and cathode, which are separated by a solid or liquid electrolyte. The electrolyte is

impervious to gases. Fuel is supplied to the anode side and air is supplied to the

cathode side. The oxidation reaction is made possible by conduction of ions through

the electrolyte. Although the basic principle behind the operation of a fuel cell is quite

simple, many challenges have to be overcome before its successful implementation.

Many issues regarding suitable materials for the electrolyte, interconnects, gas seals

and electrodes have to be addressed. There are also issues regarding cell stack design

and life span improvement that warrant immediate attention.

Once the above issues have been taken care of, the remaining big challenges are

computer controlling of fuel cell stack for load variation with minimum response time,

better stack design (improved heat integration, less fuel flow resistance, by-products

removal), lowering of stack mass per unit volume and cyclic endurance. Other

challenges related to successful implementation of fuel cell technology are: DC-to-DC

conversion through electronic transformer and power conditioner, i.e. DC to AC

conversion, for utility services and domestic appliances [Basu, 2007]. Developing

robust controls for integrated fuel cell systems is also a major challenge.

There are many types of fuel cells currently under research and development. Fuel

cells are classified according to the electrolyte used. Among many types, the major

ones are:

Page 23: Modelling, Simulation and Control of a Solid Oxide Fuel Cell System: A Bond Graph Approach

Introduction

3

Molten carbonate fuel cells (MCFCs): Electrolyte is a mixture of molten alkali

carbonates that conducts carbonate ions. Operate at 600 – 700 oC.

Proton exchange membrane fuel cells (PEMFCs): A polymer membrane that conducts

protons (or hydrogen ions) is used as an electrolyte. Operate at 80 – 100 oC.

Phosphoric acid fuel cells (PAFCs): Phosphoric acid is used as electrolyte and it

conducts protons. Operate at 180 – 210 oC.

Alkaline fuel cells (AFCs): Electrolyte is an aqueous solution of alkaline hydroxide

(e.g. KOH), which readily conducts hydroxyl ions. Operate at 50 – 100 oC.

Solid oxide fuel cells (SOFCs): Electrolyte is a ceramic that conducts ions at high

temperatures. Operate at 800 – 1000 oC.

SOFC is of considerable interest since it has considerably high system efficiency

in comparison to other fuel cell systems with cogeneration. The high efficiency of

SOFC systems is a result of high operating temperatures and negligible deterioration

in performance over several years. This is the type of fuel cell that is considered in

this work. Expensive catalysts, which are needed in the case of proton-exchange fuel

cells (platinum) and most other types of low temperature fuel cells, are not needed in

SOFCs due to the high operating temperatures. Moreover, light hydrocarbon fuels,

such as methane, propane and butane, can be internally reformed within the anode

because of the high operating temperature.

An SOFC is made up of four layers, three of which are ceramics (refer to Fig. 1.1).

A single cell consisting of these four layers stacked together is typically only a few

millimeters thick. Hundreds of these cells are then stacked together in series to form a

stack. The ceramics used in SOFCs do not become electrically and ionically active

until they reach very high temperature.

The electrolyte represents the media through which ions migrate from one

electrode to the other; thus causing a voltage difference between the anode and the

cathode, and consequently an electric current through an external load. For this

Page 24: Modelling, Simulation and Control of a Solid Oxide Fuel Cell System: A Bond Graph Approach

Chapter 1

4

reason, the electrolyte must meet the requirements such as high ionic conductivity,

high electronic resistivity, thermal expansion compatible with those of the other cell

components, chemical stability in contact with the two electrodes, resistance to

thermal cycling and low cost. It must also be fully dense to prevent short circuiting of

reacting gases through it and it should also be as thin as possible to minimize resistive

losses in the cell. Yttria stabilized zirconia (YSZ), which is a ceramic material, is

usually used as the electrolyte material [Singhal, 1994].

Fig. 1.1: Schematic representation of an SOFC

The requirements of physical properties of the anode are high electronic

conductivity and ionic conductivity, porous structure optimized for the mass transport

of the gas species, thermal expansion compatible with those of the other cell

components, chemical stability in contact with the two electrodes, resistance to

thermal cycling and high catalytic activity. The reaction takes place at the so-called

triple phase boundary (TPB), where electrons, ions and gas phase coexist. The typical

material of the anode is Nickel-Yttria stabilized Zirconia cermet [Singhal, 1994].

The requirements of physical properties of the cathode are similar to those of the

anode. The cathode must be porous in order to allow oxygen molecules to reach the

electrode/electrolyte interface. The most commonly used cathode material is

Page 25: Modelling, Simulation and Control of a Solid Oxide Fuel Cell System: A Bond Graph Approach

Introduction

5

lanthanum manganite. It is often doped with strontium and referred to as lanthanum

strontium manganite (LSM). The interconnect serves the purpose of connecting cells

together to form a stack and it also acts as a collector of the electrical current. It

functions as the electrical contact to the cathode while protecting it from the reducing

atmosphere of the anode.

The design and handling of complex SOFC systems require efficient control

strategies to promote safe and reliable operation. The development of powerful

control algorithms is based on an exact knowledge of the operating behaviour, which

can be obtained from dynamic system models. The fuel cell system involves multiple

energy domains such as chemical, thermal, electrical, hydraulic, mechanical etc.

Furthermore, it involves several phenomena with radically different time scales.

The reactions at the basis of the fuel cell operation are in the domain of

electrochemistry. The energy and mass balances implied in continuously feeding

reactants to the cell and pre-treating of reactants belong to chemical engineering

domain. The need to maintain the SOFC’s temperature during operation by means of

taking away the heat energy released due to the exothermic reaction falls into the

domain of thermal engineering. Electrical engineering is involved in the conversion

and transmission of the cell’s power output. Modelling and applying control theory to

such a multidisciplinary system is a challenging task. The bond graph technique is

ideally suited for modelling such systems that involve multiple energy domains. The

bond graph model of the fuel cell system can also be easily coupled to the dynamic

model of the system it is intended to power, i.e. to construct hardware in the loop

simulation system for design of efficient control strategies. Bond graph modelling of

SOFC systems ensures that the models are energetically consistent as the conservation

laws are built into the bond graph junction structure. Moreover, subsequent

modifications to the model can be easily incorporated and the causal structure of the

bond graph aids in computer simulation of the system. Causal studies on the bond

graph model show directly how element constitutive relations should be combined to

produce computer simulation of the system [Karnopp, 1990]. The simulations are

valuable in helping to understand the competing physical processes that are

Page 26: Modelling, Simulation and Control of a Solid Oxide Fuel Cell System: A Bond Graph Approach

Chapter 1

6

responsible for controlling the cell performance. Such understanding can assist in the

cell design and optimization as well as interpreting the experimental observations.

SOFCs have a wide variety of applications from use as auxiliary power units in

vehicles to stationary power generation with outputs ranging from 100 W to 2 MW.

SOFCs are also coupled with gas turbines in order to improve their efficiencies by

means of utilising the heat energy released due to the exothermic reaction. The SOFC

is also expected to be applied for combined heat and power systems, which produce

electricity and heat energy in a single integrated unit, and for certain other military

applications. In all these applications, the SOFC has to undergo transient operation

during start-up, shutdown and even during load fluctuations. Therefore, the study of

the transient operation of the SOFC is very important and so is the need to develop

suitable operating and control strategies to improve its dynamic performance.

One of the important advantages of the SOFC system over the heat engine is its

improved fuel efficiency. Attaining high efficiency from a power source is necessary

to achieve the goals of sustainable energy production in the future. Due to operational

constraints, the maximum possible efficiency may not be achieved under different

operating conditions of an SOFC system. Hence, the optimum energy efficient

operating conditions of the SOFC system have to be studied under varying load and

other inputs. This knowledge will help us in developing suitable operating strategies

for the SOFC system so as to achieve maximum fuel efficiency and minimum harm to

the environment.

1.2. Contributions of the Thesis

The main contributions of this work are summarised as follows.

• A true bond graph model of an SOFC has been developed for the first time.

For this purpose, the constitutive relations of a C-field for energy storage by

two species of gases in a control volume are formulated and an existing R-

field model has been extended for modelling of forced convection of a mixture

Page 27: Modelling, Simulation and Control of a Solid Oxide Fuel Cell System: A Bond Graph Approach

Introduction

7

of two gas species. The developed true bond graph model represents the

couplings between the various energy domains in a unified manner and

captures all the essential dynamics of the SOFC.

• A physical model-based control strategy for the SOFC is developed, which

satisfies the common SOFC control objectives.

• A comprehensive steady state model of the SOFC has been derived from the

bond graph model. A cascaded optimization algorithm has been developed,

which utilizes the steady state SOFC model and steady state models of the

after-burner and heat exchangers in order to investigate the implication of the

common SOFC control strategies on the energy and exergy efficiency of the

system. Based on this investigation, a suitable operating strategy that

maximises the efficiency of the SOFC system has been suggested.

1.3. Organisation of the Thesis

This thesis is organised into five other chapters other than this chapter. Overviews

of the contributions made in those chapters are given in the following.

A comprehensive survey of the literature pertaining to dynamic modelling of the

SOFC and the bond graph modelling of electrochemical reaction systems are

presented in Chapter 2. Literature on the simulation and control aspects of the SOFC

systems are also discussed in the same chapter.

Chapter 3 details the development of the true bond graph model of an SOFC. The

ideas involved in advancing the already available bond graph models of

electrochemical systems to model the SOFC are presented in a gradual manner before

finally presenting a true bond graph model.

The simulation studies based on the developed bond graph model of the SOFC are

presented in Chapter 4. A control strategy to improve the dynamic performance of the

SOFC is also presented therein.

In Chapter 5, a comprehensive steady state model is derived from the previously

developed bond graph model. This model, along with the steady state models of the

Page 28: Modelling, Simulation and Control of a Solid Oxide Fuel Cell System: A Bond Graph Approach

Chapter 1

8

after-burner and the heat exchangers, is used to optimise the operational efficiency of

the SOFC system under varying load conditions. Three different operating strategies

are simulated to compare their dynamic performances.

The conclusions drawn from this thesis and the scope for future research work are

given in Chapter 6.

Thereafter, a list of relevant references is presented. All these references are cited

in the main chapters and appendices of this thesis.

Finally, the thesis ends with appendices, which describe the basic bond graph

theory and alternative forms of the models given in the thesis by following

international convention, derivations of some important relations and the Nelder-

Mead simplex algorithm used for the optimisation studies.

Page 29: Modelling, Simulation and Control of a Solid Oxide Fuel Cell System: A Bond Graph Approach

Chapter 2

Literature Review and Fundamental Concepts

2.1. Introduction

The history of the fuel cell starts with William Grove who discovered it in 1839

[Grove 1839; Grove 1842]. Grove found that by arranging two platinum electrodes

with one end of each immersed in a container of sulphuric acid and the other ends

separately sealed in containers of oxygen and hydrogen, constant current flowed

between the electrodes. The sealed containers held water as well as the gases, and he

noted that the water level rose in both the tubes as the current flowed. By combining

several sets of such electrodes in a series circuit, he created the first fuel cell. Detailed

discussions on the history of fuel cells can be found in [Liebhafsky & Cairns, 1968].

The first SOFC’s emerged in 1930’s due to Emil Baur and H. Preis. The materials

they used were zirconium, yttrium, cerium, lanthanum, and tungsten oxide [Stambouli

& Traversa, 2002]. More recently, rising energy prices and advances in materials

technology have contributed to extensive research work on SOFCs.

Page 30: Modelling, Simulation and Control of a Solid Oxide Fuel Cell System: A Bond Graph Approach

Chapter 2

10

A number of dynamic models of SOFC are available in the literature [Aguiar et al.,

2004; Aguiar et al., 2005; Mueller et al., 2006; Qi et al., 2005; Singhal, 2000a;

Singhal, 2000b; Wang et al., 2007]. Most of them are distributed parameter, cell level,

multidimensional models [Fahrenthold & Venkataraman, 1997; Franco et al., 2005a;

Franco et al., 2005b; Franco et al., 2006]. Some other models excessively simplify the

system by representing them as equivalent electrical circuits [Choi et al., 2006]. The

physical processes that contribute to the local phenomena, or the occurrence of

concentration over-voltage at anode-cathode interface, are mainly due to the diffusion

of the reactants/products through the electrolyte to/from the reaction sites [Aguiar et

al., 2004]. The characteristic time constant of the diffusion process is in the order of

several milliseconds to a few tenths of a second [Singhal & Kendall, 2003]; whereas

the time constant of the thermal dynamics is of the order of a few minutes, which

necessitate a multi-scale simulation approach for the solution of detailed distributed

parameter models [Franco et al., 2005a; Franco et al., 2005b; Franco et al., 2006].

Though detailed models are suitable for configuration design and performance

evaluation, they are unsuitable for real time control applications as the computational

load is high. On the other hand, the equivalent circuit models use empirical relations

and cannot provide sufficient information (instantaneous pressures, temperatures,

mass flow rates etc.) for control strategy design. There is a need for a control oriented

model which captures all the necessary dynamics of the fuel cell system and, at the

same time, is not computationally too intensive. As the fast dynamics are irrelevant

from control perspective, they can be neglected in a control oriented model.

There is abundant literature on the bond graph modelling of thermo-fluid systems.

Applications of bond graph modelling to hydraulic systems can be found in [Borutzky,

1999; Borutzky et al., 2000, Mukherjee et al., 2006; Thoma & Bouamama, 2000], to

thermo-fluid systems in [Bouamama, 2003; Bouamama et al., 2006; Brown, 2002a;

Brown, 2002b; Hubbard & Brewer, 1981; Karnopp, 1979; Shoureshi & McLaughlin,

1984] and to 1D and 2D heat conduction in [Gawthrop 1999; Granda & Tang, 1989;

Nakrachi & Dauphin-Tanguy, 2003].

Works on the bond graph modelling of chemical reactions and electrochemical

reaction systems such as batteries and fuel cells are also available in the literature

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11

[Auslander, et al., 1973; Bruun & Pedersen, 2007; Karnopp, 1990; Oster et al., 1971;

Oster et al., 1973; Thoma & Bouamama, 2000; Wyatt, 1978]. Some ideas from the

models presented in those works are adopted in the present work. Therefore, the

salient features of the models presented in them are discussed in the following and

their advantages and drawbacks are pointed out.

Network or bond graph modelling of the kinetics of the isothermal and isobaric

chemical reactions were discussed in [Auslander, et al., 1973; Oster et al., 1971; Oster

et al., 1973; Wyatt, 1978]. It was indicated that the energy storage in a volume of

matter can be represented by a C-field element having three ports: a heat port, a

mechanical port and a chemical port. The constitutive relations of this capacitive field

element represented the thermo-statics or the equilibrium thermodynamics of that

volume. Assuming that all changes occur slowly enough so that the gas is

approximately at internal equilibrium at each moment, the change in the internal

energy of that volume of matter was given by

d d d dU U UU V S mV S m∂ ∂ ∂

= + +∂ ∂ ∂

, (2.1)

where V, S and m are the volume, entropy and mass respectively. Using

thermodynamic relations, the partial derivative terms on the right hand side of Eq.

(2.1) are given by the intensive variables pressure, temperature and chemical potential

as U V p∂ ∂ = − , U S T∂ ∂ = and U m μ∂ ∂ = , respectively. Therefore, Eq. (2.1)

becomes

d d d dU p V T S mμ= + + . (2.2)

From Eq. (2.2), we can see that the change in the internal energy of the volume

can be represented as a sum of three distinct energy exchanges. The Equation (2.2)

can be written in terms of time derivatives as

U pV TS mμ= + + . (2.3)

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Chapter 2

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From Eq. (2.3), it is evident that the internal energy of the volume changes due to

three distinct power exchanges which can be represented by the products of the

corresponding effort and flow variables. Therefore, the energy storage in the volume

can be represented as a 3 port C-field as shown in Fig. 2.1. This C-field has three

power ports: the flow and effort variables for the mechanical port are V and p,

respectively; those for the thermal port are S and T, respectively; and those for the

material transport port are m and μ, respectively.

Fig. 2.1: Thermodynamic C-field With Three Power Ports

A thermodynamic capacitor by itself gives no information about rates, available

energy loss, entropy production or any other non-equilibrium phenomenon. As in

most cases in experimental and biological chemistry, only isothermal and isobaric

chemical reaction kinetics was considered in [Wyatt, 1978] by keeping the

temperature and pressure constant through contact with large reservoirs. Therefore,

the thermal (T – S ) and the mechanical ports ( p –V ) were not included in the model.

The development of the network model of the chemical reactions requires formulation

of a thermodynamic force that drives the reaction flux. Such a force can be derived

from the reaction thermodynamics and it is called the affinity.

The non-equilibrium part of the chemical reaction was represented by a resistive

field element in [Wyatt, 1978]. Non-equilibrium thermodynamics represents the

relations between forces (intensive variables) and flows (the rate of change of

extensive variables). The first and second laws of thermodynamics impose restrictions

on the possible constitutive relations of the resistive element. The first law requires

that the power flow through the omitted T – S port will be exactly the sum of the

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13

power flows into the other ports. In an isothermal situation the second law says

roughly that other forms of energy can be converted to heat, but that conversion in the

other direction is impossible. In terms of the resistors without explicit T – S ports, the

second law manifests itself as the requirement that the sum of the power flows into

the remaining ports be always positive.

2.2. Concept of Chemical Equilibrium

For any isolated thermodynamic system undergoing a general process which does not

have any particular restrictions, the sign of the change in entropy of the universe

(system plus its surroundings), given by the second law of thermodynamics, is a

convenient criterion for determining its spontaneity [Callen, 1985; Holman, 1988;

Zemansky & Dittman, 1997]. But, many practical processes occur under the

conditions of constant temperature and pressure. For reactions occurring at isothermal

and isobaric conditions, the sign of the change in Gibbs function of the reaction gives

the information about the spontaneity of the reaction.

A derivation illustrating the relation between the sign of the Gibbs free energy

change and the spontaneity of the process [Zemansky & Dittman, 1997] is given in

the following. Consider a hydrostatic system in mechanical and thermal equilibrium

but not in chemical equilibrium. Suppose that the system is in contact with a reservoir

at temperature T and undergoes an infinitesimal irreversible process involving an

exchange of heat δQ from the reservoir. The process may involve a chemical reaction.

Therefore, the entropy change of the universe is 0d dS S+ , where the entropy change

of the reservoir is 0dS and the entropy change of the system is dS . According to the

second law of thermodynamics, an irreversible process leads to the increase in the

entropy of the universe. Thus, we may write 0d d 0S S+ > .

Since 0d δS Q T= − , we have

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Chapter 2

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d d 0Q ST

− + > or d d 0Q T S− < . (2.4)

During the infinitesimal irreversible process, the internal energy of the system

changes by an amount dU , and an amount of work dp V is performed by the system.

Therefore, according to the first law,

d d dQ U p V= + . (2.5)

Substituting Eq. (2.5) into the inequality (2.4), we get

d d d 0U p V T S+ − < . (2.6)

By definition, the Gibbs function G is given as

G U pV TS= + − , (2.7)

which on differentiation gives

d d d d d dG U p V V p T S S T= + + − − . (2.8)

When the conditions of constant temperature and pressure are imposed on Eq.

(2.8), it becomes

d d d dG U p V T S= + − . (2.9)

From Eq. (2.6) and Eq. (2.9), it is clear that the condition for the process to be

irreversible and occur spontaneously is d 0G < . Conversely, the non-spontaneity of an

isothermal and isobaric process can be identified by the condition d 0G > . Therefore,

if dG for a isothermal and isobaric chemical reaction system is negative, it means

that the process will occur irreversibly until the equilibrium is reached at which, the

Gibbs function G will become minimum, i.e. d 0G = (refer to Fig. 2.2).

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15

Fig. 2.2: Variation of the Gibbs Function of the Reaction with the Reaction Advancement

2.3. Bond Graph Formulation of Chemical Reaction Kinetics

The network formulation of the chemical reaction kinetics [Wyatt, 1978] is described

here with respect to the following example reaction.

2A+B 2Ck

k+

−, (2.10)

where k+ is the forward reaction rate coefficient and k− is the reverse reaction rate

coefficient. The progress or advancement of a reaction can be represented in terms of

mole numbers as

( ) ( ) ( )0i i in t n tν ξ= + , (2.11)

where ( )tξ is the reaction advancement coordinate with ( )0 0ξ = and iν is the

stoichiometric coefficient of the thi species. The stoichiometric coefficients of the

species in this example reaction (refer to Eq. (2.10)) are A 2ν = , B 1ν = and C 2ν = .

The time variation of the number of moles of the three species are related as

A Ad dn ν ξ= − , B Bd dn ν ξ= − and C Cd dn ν ξ= . As these quantities are perfect

differentials, their integrations give the following:

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Chapter 2

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( )A A A( ) (0)n t n tν ξ= − , (2.12)

( )B B B( ) (0)n t n tν ξ= − , (2.13)

and ( )C C C( ) (0)n t n tν ξ= + , (2.14)

where ( )0 0ξ = .

The reaction advancement coordinate may be thought of as a generalized

displacement whose time derivative defines the reaction rate. The corresponding

effort, which drives the reaction, is called affinity. The Gibbs free energy is the

maximum amount of non-expansion work that can be extracted from a closed system

through a completely reversible process. The change in Gibbs free energy of a system

which is maintained at constant temperature and pressure is given as [Wyatt, 1978;

Zemansky & Dittman, 1997]

C A BC A B

d d d dG G GG n n nn n n∂ ∂ ∂

= − −∂ ∂ ∂

, (2.15)

which gives

( )C C A A B Bd dG μ ν μ ν μ ν ξ= − − . (2.16)

In a general form, Eq. (2.16) is given as

P R

d di i i ii i

G μν μν ξ∈ ∈

⎛ ⎞= −⎜ ⎟⎝ ⎠∑ ∑ , (2.17)

where subscript i is used to represent the sums over the product and the reactant

components. In this example, [ ]P= A B and [ ]R= C .

The affinity of the reaction is given by [Thoma & Bouamama, 2000; Wyatt, 1978]

R Pi i i i

i iA μν μν

∈ ∈

= −∑ ∑ , (2.18)

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Literature Review and Fundamental Concepts

17

where the quantity FR

i ii

A μν∈

=∑ is defined as the forward affinity and the quantity

RP

i ii

A μν∈

=∑ is defined as the reverse affinity.

Therefore, Eq. (2.17) can be written as

ddG Aξ= − . (2.19)

This affinity may be defined as a generalized effort. At the equilibrium, the

affinity of the reaction is zero, i.e. d d 0G ξ = , which implies that the Gibbs free

energy G is minimum as shown in Fig. 2.2. Thus, at the equilibrium, 0A = and

F RA A= .

The reaction rates are defined by the law of mass action. The law of mass action

states that the rate of an elementary reaction (a reaction that proceeds through only

one mechanistic step) is proportional to the product of the concentrations of the

participating molecules. In modern chemistry, this is derived using statistical

mechanics [Benson, 1960]. However, the concentration dependence of the rate can be

qualitatively understood from the collision theory of chemical reactions [Benson,

1960; Butt, 2000; Houston, 2001; Pannetier & Souchay, 1967; Wright, 2004],

according to which, molecules must collide in order to react together. As the

concentration of the reactants increases, the frequency of the molecules colliding

increases, striking each other more frequently by being in closer contact at any given

point in time. Therefore, according to this law [Butt, 2000], the net rate of the

reversible chemical reaction considered in Eq. (2.10) is given as

2 2A B Ck c c k cξ + −= − , (2.20)

where c refers to the concentration of the species ‘i’. The concentration of a specific

component is defined as the ratio of the number of moles of that component to the

total number of moles of products and reactants.

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Chapter 2

18

The bond graph model for the kinetics of the reaction (refer to Eq. (2.10)) is

shown in Fig. 2.3. Note that the transformer elements shown in Fig. 2.3 have arrows

above them. This notation, used in [Mukherjee et al., 2006] is also used in this thesis.

It is used because it makes the constitutive relation of the transformer very clear. The

constitutive relations do not depend on the power direction but only on the direction

of the arrow above the transformers (refer to Appendix A).

In true bond graph representation, the effort and flow variables are chemical

potential and mole flow rate, respectively. As the model represents isothermal and

isobaric reactions, the T – S port and the P –V ports are omitted.

The chemical potential of the component ‘i’ is given as (refer to Appendix B)

( ),0 , R ln ii i

cT p Tc

μ μΣ

⎛ ⎞= + ⎜ ⎟

⎝ ⎠, (2.21)

where ic is the concentration of the component ‘i’, cΣ is the reference concentration

(it is usually taken as unity) and ,0iμ is the standard chemical potential. Equation

(2.21) can be written in terms of number of moles ‘n’ as

( ),0 , R ln ii i

nT p Tn

μ μΣ

⎛ ⎞= + ⎜ ⎟

⎝ ⎠, (2.22)

which forms the constitutive relation for the capacitive element of the bond graph

model in Fig. 2.3.

From Eq. (2.21), the concentrations of the reacting species can be represented in

terms of their chemical potentials as

( ),0 ,exp

Ri i

i

T pc

Tμ μ⎛ ⎞−

= ⎜ ⎟⎝ ⎠

. (2.23)

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Literature Review and Fundamental Concepts

19

The effort in the bond number 7 shown in Fig. 2.3 denotes the forward affinity of

the reaction and is given by 7 F A B2e A μ μ= = + . Similarly, the effort in bond number

8 denotes the reverse affinity of the reaction and is given by 8 R C2e A μ= = .

Fig. 2.3: The Bond Graph Model of an Isothermal and Isobaric Chemical Reaction

The non-equilibrium part of the chemical reaction kinetics is represented as an R-

field whose inputs are the forward and reverse affinities and the output is the reaction

rate. The constitutive relation of the R-field is formulated using the mass action law.

However, other more complex rate laws can also be used for some reactions. From

Eqs. (2.20) and (2.23), the flows in the bonds numbered 7 and 8 are given as

A A,0 B B,0 C C,07 8

2 2 2 2exp exp exp

R R Rf f k k

T T Tμ μ μ μ μ μ

ξ + −

− − −⎛ ⎞ ⎛ ⎞ ⎛ ⎞= = = −⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠, (2.24)

where, ξ has the unit of mol s-1. The constitutive relation of the R-field element can

also be written in terms of the forward and reverse affinities, which are the efforts in

the bonds numbered 7 and 8, respectively, as

A,0 B,0 C,0F R2 2exp exp exp exp

R R R RA Ak k

T T T Tμ μ μ

ξ + −

− − −⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠

. (2.25)

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Chapter 2

20

The direction of the two flows are opposite as denoted by the power directions of

the bonds numbered 7 and 8 in Fig. 2.3. This resistance field satisfies the Onsager

reciprocity requirements [Onsager, 1931a; Onsager, 1931b]. The network

representation of the chemical reactions helps in integrating the kinetic (rate laws

defined by Eq. (2.20)) and thermodynamic (affinities as defined in Eq. (2.18) as

driving forces) points of view.

2.4. Bond Graph Modelling of Electrochemical Systems

Bond graph models for chemical kinetics were extended to electrochemical systems

such as electrical vehicle batteries including the electrical, chemical and thermal

effects in [Karnopp, 1990]. Such energy storage systems undergo drastic change in

temperature during operation. In [Karnopp, 1990], it was assumed that the

temperatures and pressures were imposed on the phases representing the reactants and

the products. Under such conditions, the Gibbs function was found to be an

appropriate thermodynamic potential for representing the energy storage in the system.

A C-field with the Gibbs function as the thermodynamic potential is shown in Fig. 2.4.

Note that the thermal and the mechanical ports of the C-field are in differential

causality. Even if the temperature imposed on the system is not constant, the entropy

flow rate in the thermal bond of the C-field can be calculated as

ii

S S SS T p nT p n∂ ∂ ∂

= + +∂ ∂ ∂

. (2.26)

Fig. 2.4: The C-field with the Gibbs Function as the Thermodynamic Potential

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21

In [Karnopp, 1990], the bond graph model for a Daniell cell consisting of a copper

electrode in a copper sulphate solution and a zinc electrode in a zinc sulphate solution

with the two solutions separated by a diaphragm was given. The reactions at the

anode, the cathode and the total reaction, respectively, are

2+Cu Cu 2e−⎯⎯→ + (2.27)

2+Zn 2e Zn−+ ⎯⎯→ (2.28)

2e2+ 2+Zn Cu Cu Zn−

+ ⎯⎯→ + (2.29)

In an electrochemical reaction, the reaction rate is proportional to the current or

the charge flow rate in the external circuit. The current (i) and the reaction flux (J) are

related as

ei n Fξ= , (2.30)

where ne is the number of electrons participating in the reaction and F is the Faradays

constant. As the temperature and pressure are imposed on the system, the change in

the Gibbs free energy of the reaction is given by i inμ∑ . Under reversible operation

of the cell, the change in the Gibbs free energy of the reaction is converted into

electrical energy. Equating these two energies, the relation between the change in the

Gibbs free energy of the reaction and the reversible cell voltage is obtained as

re

GVn FΔ

= . (2.31)

If some useful current has to be drawn from the cell, it is accompanied by several

irreversibilities due to the rate limiting or loss elements in the system. Therefore, the

actual cell voltage, V , is less than the reversible cell voltage, rV . The over-voltage is

defined as

rV Vη = − . (2.32)

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Chapter 2

22

With these considerations, the bond graph model for the electrochemical reaction

mechanism of the Daniell cell (Fig. 2.5) was given in [Karnopp, 1990]. The thermal

and the mechanical bonds of the C-field are in differential causality, which means that

the temperature and the pressure are imposed on the C-field. If the model represents

an isothermal system, then there is no necessity to include the thermal and the

mechanical ports. For including the thermal effects, an anonymous compliance

element was introduced in [Karnopp, 1990] as shown in Fig. 2.5, which imposes the

temperature of the system. The resistance field element models the entropy generated

due to the polarisation losses. This generated entropy was given by the power balance

as

geniS

= . (2.33)

Fig. 2.5: The Bond Graph Model of a Daniell Cell

The entropy flow rate output of the C-field element is given according to Eq.

(2.26). This gives rise to numerical problems in simulation due to the differential

causality unless the first and the second terms in Eq. (2.26) are neglected [Karnopp,

1990]. The second term in Eq. (2.26) could be neglected if the reaction is assumed to

occur at constant pressure. However, for neglecting the first term, we must assume

that the temperature does not change much, in which case an isothermal model may

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Literature Review and Fundamental Concepts

23

be good enough. We would still get a non-isothermal model by neglecting the first

and the second terms in the Eq. (2.26), as the entropy change due to the reaction is

accounted for in the third term. But even then there is the problem of the unknown

parameter of the C-element.

The problem of differential causality for the thermodynamic C-field with the

Gibbs function as the potential may be avoided by considering the temperature and

the pressure of the system as state variables. In this case, the change in the Gibbs free

energy of the matter represented by the C-field is given as follows. The Gibbs

function G is defined as

G U pV TS= + − . (3.34)

Differentiating Eq. (3.34) and substituting the expression for dU , which is given as

d d d dU T S p V mμ= − + , (3.35)

we get

d d d dG V p S T mμ= − + . (3.36)

By using Eq. (3.36), an integrally causalled C-field may be constructed as shown

in Fig. 2.6, for which, the Gibbs function would be the thermodynamic potential.

However, such usage of the extensive variables as efforts and intensive variables as

flows will not be convenient for constructing the other parts of the model and

therefore not followed in this work.

Fig. 2.6: An Integrally Causalled C-field with the Gibbs Function as the Thermodynamic Potential

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Chapter 2

24

Though the bond graph model of the electrochemical cell proposed in [Karnopp,

1990] is a significant improvement over the isothermal models given earlier, it suffers

from the problem of differential causality and the lack of knowledge of the parameter

of the compliance element. Moreover, in a fuel cell, the reactants are continuously

replenished and the products are continuously removed so as to produce electrical

energy continuously. In the case of high temperature fuel cells like the SOFC, the

temperatures of the reactants are raised by pre-heating before being supplied to the

cell. This requires the inclusion of the phenomena of convection in the model wherein

the flowing masses carry energy along with them. Though the model given in

[Karnopp, 1990] can represent the non-isothermal electrochemical reaction dynamics

of a Daniell cell, it is unsuitable for modelling fuel cells.

Bond graph model of electric batteries and fuel cells were given in [Thoma 1999],

where some newly defined elements were used. Bond graph modelling of chemical

reactions were also discussed in [Thoma & Bouamama, 2000]. These models also

suffer from the problem of differential causality and do not include the convection

processes which are essential for modelling fuel cells.

A bond graph model of a high temperature fuel cell was presented in [Bruun &

Pedersen, 2007]. In that work, a mixed approach was taken, where the electrical and

the electrochemical parts were modelled using true bond graphs whereas all other

parts were modelled by using pseudo bond graphs. The coupling between the pseudo

bond graph and the true bond graph parts can only be done using ad hoc elements that

do not obey the rules of normal bond graph elements [Heny et al., 2000]. Therefore,

that approach was not adopted for this work.

Bond graph models of lead acid batteries were also given by [Esperilla et al.,

2003; Esperilla et al., 2005; Esperilla et al., 2007a; Esperilla et al., 2007b]. The

isothermal models given by them were merely bond graph equivalents of electrical

circuits and the chemical reaction dynamics were not modelled. They also proposed

simple non-isothermal models that included the thermal effects. The bond graph

model of a polymer electrolyte membrane (PEM) fuel cell was developed in [Saisset

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Literature Review and Fundamental Concepts

25

et al., 2006]. They used a mixture of pseudo and true bond graphs with block diagram

components for performing the thermo-chemical calculations.

The primary aim of this work is to develop a true bond graph model of the high

temperature fuel cell (SOFC) based on the concepts of network thermodynamics,

which will be an improvement over the existing bond graph models of the

electrochemical cell. The developed model should be able to capture all the essential

dynamics of the cell so as to give a comparable performance with other detailed

SOFC models available in the literature.

2.5. Simulation and Control of the SOFC System

Many emerging applications of the SOFC will require their ability to follow very fast

load transients [Mueller et al., 2006]. The transients in SOFC dynamics occur during

load changes (due to variable power demand) and also during start-up and shut down.

Dynamic response studies are important for analysing the SOFC transient operation

and its interactions with the distributed energy network. SOFC stack dynamics are

identified to have three characteristic response times governed by different

mechanisms. One of these is due to power electronics and the electrochemical

dynamics with characteristic response time in the order of milliseconds; another is due

to hydrogen composition changes within the cell with characteristic response time in

the order of seconds, and the third is due to the cell temperature change with

characteristic response time in the order of minutes to hours [Dicks & Martin, 1998].

A one dimensional model of an anode supported intermediate temperature, direct

internal reforming SOFC was presented in [Aguiar et al., 2004]. The model was

obtained by applying mass and energy balances to the air channel, the fuel channel

and the solid structures. The steady state characteristic curves and the spatial

distribution of several important variables such as temperatures, voltage and mass

fractions were plotted. The dynamic characteristics of this model were studied in

[Aguiar et al., 2005] by subjecting it to step disturbances. A control strategy was also

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Chapter 2

26

formulated and tested for achieving the control objectives of constant temperature and

constant fuel utilisation (FU) operation of the fuel cell.

The fuel utilisation and air utilisation are two of the most important control

variables of the fuel cell [Aguiar et al., 2005]. Fuel utilisation ( fζ ) is defined as the

ratio of the mass flow rate of the fuel taking part in the reaction to the mass flow rate

of the fuel supplied to the cell. Air Utilisation is defined as the ratio of mass flow rate

of oxygen consumed by the reaction to the mass flow rate of oxygen supplied to the

cell. As it is the oxygen in the air that is being utilised, oxygen utilisation (OU) will

be a more appropriate term than air utilisation and is used in some works [Zhang et al.,

2006]. Therefore, in this work it will be referred to as OU ( oζ ). Some common

objectives of the SOFC control system include: Controlling the average stack

temperature; maintaining constant fuel utilisation (FU) for all power outputs; and

ensuring that the OU is always less than a maximum specified value. Stack

temperature control is normally provided by varying the air ratio, i.e., the supply of

the air for cooling [Aguiar et al., 2005]. Another requirement for the operation of the

SOFC, which is often neglected by most researchers, is that the difference of pressure

between the anode and the cathode channels has to be small [Serra et al., 2005;

Wachter et al., 2006]; the adequate value being dependent on the membrane electrode

assembly (consisting of the electrodes and the electrolyte, abbreviated as MEA)

support (allowable stress) and on the age of the fuel cell (fatigue considerations).

Some authors [He, 1998; Wang et al., 2007; Zhu & Tomsovic, 2002] state that the

pressure difference between the hydrogen and oxygen must be kept below 8kPa under

transient conditions and below 4kPa during steady state conditions in order to prevent

damage to the electrolyte. Therefore, formulation of a suitable physical model-based

control strategy for the SOFC, which satisfies all the above-mentioned control

objectives, is considered as an objective of the present work.

The aim of the last part of this work is to formulate a suitable operating strategy of

the SOFC system (including the heat exchangers and the after-burners), which

maximises the energy efficiency of the overall system. Literature on the energy and

exergy analysis based optimisation of the fuel cell system is abundant. In [Kazim,

2004], exergy analysis is performed on a Polymer Electrolyte Membrane (PEM) fuel

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Literature Review and Fundamental Concepts

27

cell at variable operating conditions. Energy and exergy analyses are performed on

simple hydrogen fed SOFC power system in [Chan et al., 2002] and it was reported

that the second law efficiency is higher than the first law efficiency. Exergy analysis

was performed in [Saidi et al., 2005] in order to optimize the operating conditions of a

PEM fuel cell. Other studies on exergy analysis of fuel cell systems can be consulted

in [Akkaya et al., 2007; Calise et al., 2006; Monanteras & Frangopoulos, 1999; Song

et al., 2005]. Several works in the literature deal with various aspects of the

optimization of the fuel cell system. In some works, a very detailed and specialized

model of the fuel cell is adopted for such optimization studies [Lin et al., 2006; Wu et

al., 2006], whereas in others the model is over-simplified [Chen et al., 2006; Cownden

et al., 2001; Douvartzides et al., 2003; Frangopoulos & Nakos, 2006; Hussain et al.,

2005; Na & Gou, 2007; Palazzi et al., 2007; Saidi et al., 2005; Subramanyan &

Diwekar, 2007; Yeh & Chen, 2008]. The former is computationally expensive, while

the latter does not sufficiently represent the fuel cell system dynamics for

development of control laws. There is a need for a comprehensive static model of the

fuel cell system that can calculate the states of the system given the inputs so that it

can be used for studying the optimum operating conditions of an SOFC system.

In [Aguiar et al., 2005; Mueller et al., 2006; Singhal & Kendall, 2003; Stiller et al.,

2006], maintaining constant FU and constant cell temperature is advocated as the

desirable operating condition for an SOFC. Constant FU operation of the fuel cell is

recommended because it minimises the dynamics during load changes [Mueller et al.,

2006] and also avoids uneven distribution of voltage and temperature within the cell

[Stiller et al., 2006]. The constant temperature operation is recommended in order to

avoid thermal cracking [Stiller et al., 2006]. However, the implications of maintaining

these operating conditions on the efficiency of the SOFC system have not been

investigated in any work. In this work, the influence of cell operating conditions on

the efficiency and dynamic performance are studied.

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Chapter 2

28

2.6. Objectives of the Present Work

Based on these discussions, it is found that bond graph method is ideally suited for

modelling the multi-disciplinary SOFC system. Clearly, there is a lack of a

comprehensive bond graph model of the fuel cell system in the literature, which

includes the chemical, electrical, thermal and pneumatic dynamics of the system. The

availability of such a comprehensive bond graph model will aid in the analysis and

design of the SOFC system and will also be useful in designing integrated model-

based control strategies for the overall system. There is also a lack of suitable physical

model-based control strategies for achieving the control objectives of the SOFC

system and improving its dynamic performance.

Although constant FU and constant temperature operation is advocated as the

desirable operating condition of an SOFC for several reasons, the implications of such

an operation of the cell on the system efficiency and transient dynamics have not been

investigated so far. Also, there is a lack of a comprehensive steady state model of the

SOFC system, which can be utilised in an optimisation algorithm for investigating the

optimal efficient operating condition of the SOFC system. Such a static model, if

available, can be used to investigate the implications of various operating strategies

on the SOFC system’s efficiency.

Therefore, the objectives of the present work are summarised as follows:

1. To develop a true bond graph model of an SOFC from the principles of network

thermodynamics.

2. To validate the model by comparing its static characteristics and dynamic

responses with other models from literature and to develop a physical model-

based control strategy to achieve the common SOFC control objectives.

3. To develop a comprehensive steady state model of the SOFC system and use it to

investigate the implications of various operating strategies on the energetic and

exergetic efficiencies of the SOFC system.

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Chapter 3

Bond Graph Model of a Solid Oxide Fuel Cell

3.1. Introduction

Many of the applications of SOFC systems require their ability to follow very fast

load transients. Therefore, static models of SOFC systems are unsuitable for their

design. The choice of a proper modelling method, which is suitable for the analysis of

the multidisciplinary system's behaviour, controller synthesis and at the same time,

clearly exposes the interplay between various subsystems, is very important because it

can greatly reduce the further development time. The purpose of the work presented

in this chapter is to develop a non-linear model of an SOFC using the bond graph

approach [Gawthrop, 1991; Granda, 2002; Karnopp et al., 2006; Mukherjee et al.,

2006; Thoma & Bouamama, 2000], which can be utilised for system dynamic studies.

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Chapter 3

30

As fuel cell system is inherently multidisciplinary, being composed of interacting

subsystems or parts from different engineering disciplines, it requires an integration of

chemical engineering, process engineering, mechanical engineering, electrical

engineering, and control engineering. Bond graphs provide a unified approach to the

modelling and analysis of dynamic systems and are ideally suited for modelling fuel

cell systems because the coupling between different domains can be easily modelled.

Moreover, a fundamental property of a bond graph model relates to power

conservation in the junction structure, which in the context of a fuel cell system

ensures that the various constraints, e.g. conservation of mass and energy, are always

correctly represented. Moreover, subsequent modifications to the model can be easily

incorporated and the causal structure of the bond graph aids in computer simulation of

the system.

Although a number of control relevant models of SOFC are available in the

literature, only few deal with bond graph modelling of fuel cells. An early effort in

this direction was made in [Karnopp, 1990] where a bond graph model for

electrochemical energy storage in batteries including the electrical, chemical and

thermal effects was given and a C-field for multiple gas species was suggested. Bond

graph modelling of lead acid batteries and fuel cells was discussed in [Bruun &

Pedersen, 2007; Esperilla et al., 2003; Esperilla et al., 2005; Saisset et al., 2006;

Thoma, 1999]. In those works, pseudo bond graph approach was used. This is because

chemical processes are usually described by using a control volume in which matter

flows across boundaries carrying energy with it in several forms, which is considered

to be difficult to represent in true bond graphs [Heny et al., 2000]. However, pseudo

bond graphs suffer from the disadvantage that they cannot be readily coupled to other

energy domains, except by means of some ad hoc elements that do not obey the rules

of normal bond graph elements [Heny et al., 2000]. In the work presented in this

chapter, a true bond graph approach has been taken in order to model the fuel cell.

This is because, a true bond graph representation assures that the model is

energetically consistent and it depicts the couplings between the energy variables in

the system clearly. The entropies generated due to various processes in the system are

taken into consideration in a true bond graph model, whereas these are neglected in a

pseudo bond graph representation. The contributions of the cell irreversibilities to the

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Bond Graph Model of a Solid Oxide Fuel Cell

31

transient responses are included in a true bond graph representation. Thus, a true bond

graph model not only gives more accurate results, but also helps to calculate the

exergetic efficiency of the system.

In this chapter, a true bond graph model of the SOFC is systematically developed

starting from the previous ideas about electrochemical reaction dynamics modelling

from literature, which were presented in Chapter 2. A preliminary model of the

hydrogen oxidation electrochemical reaction is first developed which is an

improvement over the earlier model by [Karnopp, 1990]. This model uses a C-field

with the internal energy as the thermodynamic potential for representing the energy

storage in the reactants and the products, thereby avoiding the problem of differential

causality present in the model proposed by [Karnopp, 1990].

This model is then improved to represent a fuel cell by formulating and using C-

fields capable of representing energy storage in two species gas mixtures. These C-

fields are used for modelling the anode and the cathode channel control volumes of

the fuel cell. This results in a model in which some parts of are constructed by using

pseudo bonds and entropy convection processes are not sufficiently represented in the

model.

In order to rectify these deficiencies, a true bond graph model of the SOFC is

finally presented, which uses an extended R-field for representing the convection of

mixture of two gas species.

3.2. A Preliminary Bond Graph Model of the Hydrogen

Oxidation Electrochemical Reaction

The bond graph models of electrochemical reaction systems given in the literature

[Karnopp, 1990; Thoma & Bouamama, 2000] model the energy storage in the

reacting species using a C-field, in which the Gibbs free energy serves as the

thermodynamic potential. These models suffer from the drawback that two of the

three power ports of the energy storing C-field are differentially causalled (refer to

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Chapter 3

32

Section 2.4 of Chapter 2). A preliminary bond graph model of the electrochemical

hydrogen oxidation reaction is developed in this section, which avoids the problem of

differential causality. This is achieved by using a C-field [Breedveld, 1984;

Mukherjee et al., 2006] in which the internal energy serves as the thermodynamic

potential for modelling the energy storage in the reactant and product gas species. In

this representation, the C-field element is integrally causalled and hence the

differential causality is avoided.

The electrochemical reaction of hydrogen oxidation is the basic reaction in any

fuel cell. In an SOFC, hydrocarbons such as methane may also be used as fuel instead

of pure hydrogen. In such a case, the hydrocarbons must undergo a reforming reaction

resulting in the formation of hydrogen, which then undergoes the oxidation reaction

with oxygen. In this work, only pure hydrogen is considered as the fuel. The hydrogen

oxidation reaction is given as follows.

22Ο 4e 2Ο− −+ → . (3.1)

22 22H 2Ο 2H Ο 4e− −+ → + . (3.2)

2 2 22H Ο 2H Ο+ → . (3.3)

The formulation of the thermodynamic C-field is given in the following. This C-

field represents a volume of ideal gas contained in a collapsible chamber. This

chamber allows heat and mass transfer with the surroundings. Mechanical work can

also be done on the system or extracted from the system. The change in the internal

energy of the gas volume represented by the C-field is given by

d d d dU U UU V S mV S m∂ ∂ ∂

= + +∂ ∂ ∂

, (3.4)

which can be written in terms of time derivatives as

U U UU V S mV S m∂ ∂ ∂

= + +∂ ∂ ∂

. (3.5)

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Bond Graph Model of a Solid Oxide Fuel Cell

33

From well-known thermodynamic relations U V p∂ ∂ = − , U S T∂ ∂ =

and U m μ∂ ∂ = , it is evident that the internal energy of the volume of the gases

change due to three distinct power exchanges which can be represented by the

products of the corresponding effort and flow variables.

Therefore, the energy storage in the gas mixture can be represented as a 3 port C-

field as shown in Fig. 3.1. This C-field has three power ports: the flow and effort

variables for the mechanical port are V and p, respectively; those for the thermal port

are S and T, respectively; and those for the material transport port are m and μ,

respectively. The entropy flow into the control volume is due to two kinds of

phenomena: heat transfer and mass transfer. Therefore, if S ms= then the entropy

flow rate is given by

HTS ms ms ms S= + = + . (3.6)

Fig. 3.1: A Thermodynamic C-field with Internal Energy as the Thermodynamic Potential

In Eq. (3.6), the component ms is the convected entropy due to mass transfer and

HTS ms= is the entropy growth due to heat transfer to/from the gas volume.

The enthalpy of the gas volume is given as

h Tsμ= + . (3.7)

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Chapter 3

34

An outer layer is added to the C-field as shown in Fig. 3.1 which simultaneously

enforces the thermodynamic relationships given in Eqs. (3.6) and (3.7).

Thermodynamic relations or constitutive relations for the C-field can be derived

as follows by assuming that ideal gas law applies. Using the first law of

thermodynamics and the definition of entropy, we get for a particular mass of gas in a

control volume,

d ds du T p v= − , (3.8)

which using the ideal gas law and the definition of internal energy becomes

vd d dRTc T T s vv

= − . (3.9)

Integration of Eq. (3.9) by assuming constant cv leads to

v0 0

0 0 0lnRcS T Vms

m T V m

⎧ ⎫⎛ ⎞⎪ ⎪⎛ ⎞− = ⎨ ⎬⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎪ ⎪⎩ ⎭

. (3.10)

Multiplying Eq. (3.10) with m we get

v0 0

0 0 0lnmRmcS m T VmS

m T V m

⎧ ⎫⎛ ⎞⎪ ⎪⎛ ⎞− = ⎨ ⎬⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎪ ⎪⎩ ⎭

, (3.11)

which can be rearranged to obtain the expression for temperature as a function of the

three state variables (S, V and m, which are integrations of the flow variables in bonds

connected to the three port C-field) as

v0 0

0 0v v

exp

Rc

oVm S ST TV m mc m c

−⎛ ⎞⎛ ⎞

= −⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠

. (3.12)

The expression for internal energy of the gas volume may now be obtained as

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Bond Graph Model of a Solid Oxide Fuel Cell

35

v0 0

0 0v v

exp

Rc

v oVm S SU mC TV m mc m c

−⎛ ⎞⎛ ⎞

= −⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠

. (3.13)

The expression for pressure is obtained making use of the ideal gas law

( /p mRT V= ) or alternatively by taking the partial derivative of Eq. (3.13) with

respect to the volume ( p U V= −∂ ∂ ). In its final form, the pressure is expressed as

0 0 0

0 0 0 0v v

expm RT V m S SpV V m mc m c

γ γ− − ⎛ ⎞⎛ ⎞ ⎛ ⎞= −⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

. (3.14)

The chemical potential of the gas volume is the partial derivative of internal

energy with respect to the mass. This leads to a very cumbersome formula. An

alternative derivation which leads to a simple expression is presented in the following.

The chemical potential of the gas volume is also given as

vu pv Ts c T RT Tsμ = + − = + − . (3.15)

By substituting the expressions for specific entropy and temperature given,

respectively, in Eq. (3.10) and Eq. (3.12), into Eq. (3.15), we get the expression for

chemical potential in terms of the three state variables. However, due to the

assumption of constant specific heat capacities, the chemical potential obtained is not

very accurate. In order to get a more accurate expression for chemical potential, the

standard procedure given in [Zemansky & Dittman, 1997] is followed. From Eq.

(3.15), we obtain

h Tsμ = − . (3.16)

Substituting 0pdh h c T= + ∫ and ( ) ( )0 0

p d lns s c T T R p p= + −∫ in Eq. (3.16),

we get

( )00ln pT RT

pμ μ

⎛ ⎞= + ⎜ ⎟

⎝ ⎠, (3.17)

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Chapter 3

36

where ( )0 Tμ is purely a function of temperature only. The values of ( )0 Tμ for the

individual gases are calculated using the following relation [Benson, 1977]:

( ) ( )2 43

0 3 541 1 2 6ln

2 3 4 oa T a Ta TT RT a a T a T aμ μ

⎡ ⎤⎛ ⎞= − − + + + − +⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦, (3.18)

where the values of experimentally obtained coefficients 1 6 a a… and oμ for various

gases are given in [Benson, 1977]. The pressure and the temperature in Eq. (3.17) are

written in terms of the state variables by using the earlier expressions (refer Eqs.

(3.12) and (3.14)). The Equations (3.12), (3.14) and (3.17) are the constitutive

relations of the 3 port C-field as they give the effort variables ( and T , p μ ) in terms

of the three state variables ( , and m V S ), which are obtained by integrating the flow

variables. An alternative way of deriving the expression for the chemical potential of

a constituent gas in an ideal gas mixture (Eq. (3.17)) is given in Appendix B.

A preliminary bond graph model of the electrochemical hydrogen oxidation

reaction is shown in Fig. 3.2. The mechanical ports are not shown for the C-fields in

Fig. 3.2 because the volume is considered to be constant. The two R-field elements

shown in Fig. 3.2 are included to facilitate the temperature equilibration of the

reactant and product gas species as they occupy the same volume.

The transformation of energy from the chemical to the electrical domain is given

in the following. The change in the Gibbs free energy of the system is given as (refer

to Section 2.4 in Chapter 2)

d d d dG V p S T mμ= − + . (3.19)

Using Eq. (3.19) and the assumption of constant temperature and pressure, the

change in the Gibbs free energy of the reaction is obtained as

W H OW H O

d d d dG G GG n n nn n n∂ ∂ ∂

= − −∂ ∂ ∂

. (3.20)

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Bond Graph Model of a Solid Oxide Fuel Cell

37

Fig. 3.2: A Preliminary Bond Graph Model of the Hydrogen Oxidation Electrochemical Reaction

Note that the temperature and the pressure of the system may changes during the

system’s dynamics. However, Eq. (3.20) is assumed to be valid for each instantaneous

values of pressure and temperature of the system.

A quantity called the reaction coordinate (ξ ) is defined such that H Hd dn ν ξ= − ,

O Od dn ν ξ= − and W Wd dn ν ξ= . Using these relations and the definition of the

chemical potential, Eq. (3.20) becomes

( )W W H H O Od dG μ ν μ ν μ ν ξ= − − . (3.21)

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Chapter 3

38

As the quantities in Eq. (3.21) are state functions, the equation can be written as

( )W W H H O OG μ ν μ ν μ ν ξΔ = − − Δ . (3.22)

If unit mole of fuel (hydrogen) is considered then 1ξΔ = . Therefore, the change

in the Gibbs free energy per mole of fuel is given by

W W H H O OG μ ν μ ν μ νΔ = − − , (3.23)

Note that the chemical potentials are in J.mol-1 in Eq. (3.23). Under reversible

conditions, this change in the Gibbs free energy is converted entirely into electrical

energy. Therefore, from the energy balance, the reversible cell voltage can be

obtained as

re

GVn FΔ

= − , (3.24)

where the denominator gives the charge of the total number of electrons participating

in the reaction per mole of the fuel. The Equation (3.24) can further be written in

terms of the partial pressures of the reactant and the product gases and is called the

Nernst equation (refer to Appendix B). The Nernst equation is used to calculate the

effect of the change in the partial pressures of the reacting species on the reversible

cell voltage. Note that the minus sign in Eq. (3.24) is required to obtain a positive

value of voltage because the change in the Gibbs free energy per mol as defined in Eq.

(3.23) is negative (as the free energy of the products is less than the free energy of the

reactants).

The chemical potentials are calculated in J.kg-1 in the anode and cathode channel

C-fields of the model. The three transformers shown in the effort activated bonds

around the 1ξ junction have factors of ‘1000/Mi’ in order to convert the chemical

potentials to J.mol-1. The 1ξ junction shown in the Fig. 3.2 enforces the following

relationship, which defines the negative of the change in Gibbs free energy per mol of

fuel for the reaction.

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Bond Graph Model of a Solid Oxide Fuel Cell

39

H H H O O O W W W

1000M M MG ν μ ν μ ν μ+ −

−Δ = . (3.25)

The reversible cell voltage, which is defined by the Nernst equation [Liebhafsky

& Cairns, 1968], is realised by means of a transformer element (with modulus en F )

in Fig. 3.2. When the reaction system is in equilibrium, the change in the molar Gibbs

free energy ( GΔ ) is zero. Therefore, the reversible voltage as predicted by the Nernst

equation is also zero. When the reaction system is forced out of equilibrium (i.e.,

when the concentrations of the reactants and the products differ from the equilibrium

concentrations), the reversible open circuit voltage (Vr) can be calculated by using the

Nernst equation (refer to Appendix B). However, the reaction cannot proceed as the

circuit is not closed. But once the circuit is closed (as we try to draw current from the

cell), the irreversibilities come into play and result in voltage losses.

The mole flow rate of the reaction (ξ ), which can be considered as the reaction

rate, is related to the mole flow rates of consumption and production of the reactants

and products, respectively, as

r rrW OH

W H O

n nnξν ν ν

= = − = − . (3.26)

The reaction mole flow rate and the current (i) are related as

ei n Fξ= . (3.27)

Therefore, the relations between the mass-flow rates (in kg.s-1) of hydrogen,

oxygen and water vapour taking part in the reaction and the current drawn by the load

are given as

r r re W e H e O

W W H H O O

1000 1000 1000n Fm n Fm n FmiM M Mν ν ν

= = − = − (3.28)

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Chapter 3

40

and they are realised through the 1ξ junction and the set of transformers in the flow

activated bonds surrounding it as shown in Fig. 3.2. The current, i , drawn by an un-

modelled external load is represented by a source of flow.

Note that in Fig. 3.2, transformers have been used in between two activated bonds

(signal ports). They simply act as a gain which amplifies the effort or the flow signal.

This kind of usage is adopted from [Mukherjee et al., 2006]. This convention is

followed here because it clearly reveals the structure of the electrochemical process.

Note that the bond graph model in Fig. 3.2 is also given using the conventional

representation in Appendix A.

The theoretical open-circuit voltage ( rV ) is the maximum voltage that can be

achieved by an electrochemical cell under specific operating conditions. However, the

voltage of an operating cell, which is equal to the voltage difference between the

cathode and the anode, is generally lower than this. As current is drawn from a fuel

cell, the cell voltage falls due to internal resistances and over-voltage losses. The

actual cell voltage is generally obtained by subtracting all the voltage losses from the

open circuit voltage. The field element RPL shown in Fig. 3.2 models these voltage

losses or over-voltages (η). The entropy generated due to these irreversible processes

are obtained from the power balance as

PLW

iSTη

= , (3.29)

and it is introduced into the zero junction representing the water vapour temperature

as shown in Fig. 3.2.

This bond graph model redresses several shortcomings of the previously proposed

bond graph models of electrochemical reactions in the literature. The use of internal

energy as the thermodynamic potential for the energy storing C-fields helps to avoid

the problem of differential causality encountered in the earlier models. In the model

given in Fig. 3.2, the temperature is not imposed on the system. Rather the system

temperature is decided by the exothermic reaction and the irreversible polarization

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Bond Graph Model of a Solid Oxide Fuel Cell

41

phenomena. This is very relevant for modelling fuel cell systems as the temperature of

such systems are dependant on the heat released due to the reactions and the over-

voltage phenomena. Further, the relations between the enthalpies of the gas species

and their entropies are clearly represented in the bond graph. This facilitates the direct

measurement of the enthalpy of the reaction, which is given by the difference between

the enthalpy of the products and the enthalpy of the reactants, from the bond graph

model. Note that the enthalpy of the considered reaction is positive because it is an

exothermic reaction (the enthalpy of the products is more than the enthalpy of the

reactants). This results in the increase of the temperature in the reaction system.

However, some more issues have to be addressed so as to obtain a bond graph

model which adequately represents the process dynamics of the fuel cell. One of the

problems faced while using the model shown in Fig. 3.2 concerns the two R-fields

which are added to facilitate temperature equilibration between the gas species. As

this process of temperature equilibration is extremely fast, the resistance needs to be

made very small, which makes the model very stiff (the simulation time is too large).

Another drawback of this model is that the thermal part and the electrochemical part

of the model are not connected by true bonds carrying powers but rather by activated

bonds or signal bonds which are signal carriers. Further, in the SOFC, the hydrogen

and the water vapour are present in the anode channel and are separated from the

oxygen in the cathode channel by the electrolyte. The model presented in Fig. 3.2

does not correctly represent this situation and hence cannot as such be used to

represent the SOFC dynamics. Therefore, further improvements are needed which are

made to the model in the subsequent sections.

3.3. A Bond Graph Model of the SOFC Using a C-field for

the Mixture of Two Gas Species

In the SOFC, a mixture of hydrogen and water vapour is present in the anode side and

a mixture of oxygen and nitrogen (which are considered to be the constituents of air)

is present in the cathode side. Therefore, it is justified to assume constant

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Chapter 3

42

temperatures for each of these two mixtures. In this section, the basic process

occurring in an SOFC is described, the modelling assumptions are stated and the word

bond graph model of the SOFC is given. Later, the formulation of a C-field for

representing the energy storage in a mixture of two gas species is given. Finally, the

developed C-field is used to construct a bond graph model of the SOFC. The various

over-voltages or voltage losses and the entropy generation due to those over-voltages

are also represented in the bond graph model.

3.3.1. Process Description and Modelling Approach

The basic components of the SOFC are the anode, the cathode and the electrolyte, as

are shown in Fig. 3.3. They are together referred to as the MEA.

Fuel (hydrogen) is supplied to the anode side and air is supplied to the cathode

side. At the cathode–electrolyte interface, oxygen molecules accept electrons coming

from the external circuit to form oxide ions as given in Eq. (3.1).

The solid electrolyte allows only oxide ions to pass through. At the anode–

electrolyte interface, hydrogen molecules present in the fuel react with oxide ions to

form steam, and electrons get released as given in Eq. (3.2). As a result of the

potential difference set up between anode and cathode due to the resultant excess and

scarcity of electrons at anode and cathode respectively, electric current flows in the

external circuit through which they are connected and thus the circuit is closed.

Because the reaction is exothermic, heat is evolved as a by-product. The overall

reaction is given in Eq. (3.3).

The current drawn from the fuel cell and the temperatures of the anode and cathode

channel exhaust gases are the variables that are measured for controlling the valves

shown in Fig. 3.3. The pressures in the inlet and the outlet sides are assumed to be

known constants and the inlet side temperature is also assumed to be a known constant.

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Bond Graph Model of a Solid Oxide Fuel Cell

43

The model developed in this work is physics based zero-dimensional bond graph

model of a single cell, which is suitable for system dynamics studies and will be

helpful in developing control strategies. In line with [Aguiar et al., 2005], the cell is

considered to be at the centre of the stack such that no edge effects are present. The

anode channel volume, through which the supplied hydrogen and the produced water

vapour flow, is represented as a single volume and is referred to as the anode channel

volume. Similarly the cathode channel volume through which the air flows is

represented as a single volume and is referred to as the cathode channel volume. The

intensive variables; temperature, pressure and chemical potential, for each gas are

assumed to be uniform throughout the control volume. The interconnect plates form

the interface between the cells in a stack and also form the channels through which

the gases flow. The thermal capacitances of the solids (anode, cathode, electrolyte and

interconnect) are lumped. The convective heat transfer taking place between the gases

and the MEA and also between the gases and the interconnect plates are modelled.

The radiation heat transfer between the MEA and the interconnect plates and the

convection of heat energy due to the gas flow are also considered in the model. The

supply and the removal of gases to and from the channel volumes are through four

different controlled valves as shown in Fig. 3.3.

Fig. 3.3: Schematic of an SOFC

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Chapter 3

44

3.3.2. Modelling Assumptions

The following assumptions are made in order to create a bond graph model of the

system.

1. The water formed due to the reaction is in the vapour form. All the gases

involved are assumed to be ideal. This assumption is valid because of the low

pressure and high operating temperatures.

2. Because it is a zero-dimensional model, the spatial variation of intensive

variables such as temperature, pressure and chemical potential are neglected.

3. The fuel considered in this model is pure hydrogen. The oxidant is air with

oxygen and nitrogen as its primary constituents.

4. The pressures of the gases are assumed to be governed by the Dalton’s law of

partial pressures.

5. As the pressure difference across the valves is small, the flows through the

valves are assumed to be governed by linear nozzle flow equations [Thomas,

1999].

6. As the cell is well insulated, the heat loss to the surrounding is neglected.

7. As the fast dynamics are irrelevant from control perspective, the diffusion

process is modelled through an approximation.

3.3.3. Word Bond Graph

A word bond graph represents the technological level of the model where the global

system is decomposed into its subsystems. It is the first step towards building a bond

graph model in which words define the components. The word bond graph for the

open-loop SOFC system is given in Fig. 3.4, which shows its various components.

The interaction between various subsystems is through the exchange of power. The

power variables for different domains are labelled in Fig. 3.4.

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Bond Graph Model of a Solid Oxide Fuel Cell

45

Fig. 3.4: Word Bond Graph Model of the SOFC

The gases in the anode and the cathode channel volumes interact with other

components of the system by exchanging energy in thermal, chemical and mechanical

domains. The power variables for the thermal domain are the temperature and the

entropy flow rate. The power variables for the chemical domain are the chemical

potential and the mass flow rate and those for the mechanical domain are the pressure

and the volume flow rate. The MEA heat transfer part models the thermal phenomena

due to heat transfer between the two control volumes (channels) and also accounts for

the heat generated due to chemical reactions and current flow through the MEA. The

interaction between the electro-chemical part and the two control volumes are through

the exchange of chemical energies. The exchange of electrical energy takes place

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Chapter 3

46

between the electro-chemical and the electrical parts. The voltage and the current are

the power variables for the electrical domain. The thermal energy is also exchanged

between the channel volumes and the interconnects. The inlet and exhaust gases flow

through the valves which are initially modeled using pseudo bonds ( p and m as

power variables) but later they are represented using true bonds (μ and m as power

variables) in a refined true bond graph model.

3.3.4. Storage of a Two Species Gas Mixture Represented in

a C-Field

In this section, the constitutive relations for a bond graph C-field representing the

energy storage by two species of gases in a control volume are derived. The relations

for a C-field representing a single gas species already exists in the literature

[Breedveld, 1984; Breedveld & Beaman, 1985; Feenstra, 2000; Mukherjee et al.,

2006] and were presented in Section 3.2. Although the constitutive relations are given

by considering a mixture of two gas species, it can be extended for a mixture of any

number of gas species. The two gas species are represented by subscripts 1 and 2. The

C-field for the two species gas mixture proposed here models the following scenario.

A mixture of two gases is contained in a collapsible chamber, which allows heat

transfer from and to the surroundings. In the general scenario modelled here, it is

assumed that individual gases can independently flow either into or out of the

chamber. Allowing the individual gas mass flow rates in proportion to their mass

fractions in the mixture can also incorporate the mass flow of the mixture as a whole.

The change of internal energy of the two gases in the mixture is given by

1 21 2

d d d d dU U U UU V S m mV S m m∂ ∂ ∂ ∂

= + + +∂ ∂ ∂ ∂

. (3.30)

The above equation may be written in terms time derivatives as

1 21 2

U U U UU V S m mV S m m∂ ∂ ∂ ∂

= + + +∂ ∂ ∂ ∂

. (3.31)

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47

From well-known thermodynamic relations U V p∂ ∂ = − , U S T∂ ∂ = ,

1 1U m μ∂ ∂ = and 2 2U m μ∂ ∂ = , it is evident that the internal energy of the volume

of the gases change due to four distinct power exchanges which can be represented by

the products of the corresponding effort and flow variables.

The energy storage in the gas mixture can be represented as a 4 port C-field as

shown in Fig. 3.5. This C-field has four power ports: the flow and effort variables for

the mechanical port are V and p, respectively; those for the thermal port are S and T,

respectively; and those for the material ports are 'sm and μ’s, respectively.

The growth of entropy inside the control volume can be given as

1 1 2 2 HTS m s m s S= + + , (3.32)

where the first two terms are due to the mass transfers of the two species and the third

term is due to the heat transfer through the chamber walls.

By making use of the relations, 1 1 1h Tsμ= + and 2 2 2h Tsμ= + , an outer layer can

be added to the C-field as shown in the Fig. 3.5. The thermodynamic relationships

required for constructing the C-field are derived by assuming that both the species of

gases in the chamber are in thermal equilibrium, i.e., they have a common

temperature (T), and that they occupy a common volume (V).

Fig. 3.5: Two Species of Gases Represented in a C-field

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Chapter 3

48

Note that SOFC systems operate at near atmospheric pressure and high

temperature. This is why one can assume ideal gas model. According to the

fundamental thermodynamic relation [Zemansky & Dittman, 1997], the change in the

specific entropy of an ideal gas (gas species #1) in terms of the specific internal

energy, the specific volume, the partial pressure and the equilibrium temperature is

given by,

1 1 11

d dd u p vsT T

= + . (3.33)

Using the ideal gas equation of state ( pv RT= ) and the definition of specific heat

capacity at constant volume ( vd du c T= ), Eq. (3.33) may be rewritten as

v1 1 11

1

d dd c T R vsT v

= + . (3.34)

Integrating Eq. (3.34) from an initial state (indicated by superscript 0) to a final

state with the assumption of constant specific heat capacities and then writing the

specific quantities in terms of the absolute quantities ( s S m= and v V m= , where S,

V and m are state variables) gives

1v10 01 1

1 0 0 01 1

lnRcS VmTs

m T V m

⎧ ⎫⎛ ⎞⎪ ⎪⎛ ⎞= + ⎨ ⎬⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎪ ⎪⎩ ⎭

. (3.35)

Similarly, for gas species #2, which occupies the same volume and is at same

temperature, we obtain

2v20 02 2

2 0 0 02 2

lnRcS VmTs

m T V m

⎧ ⎫⎛ ⎞⎪ ⎪⎛ ⎞= + ⎨ ⎬⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎪ ⎪⎩ ⎭

. (3.36)

Multiplying Eq. (3.35) with 1m and Eq. (3.36) with 2m , we get the following

expressions for the entropies of the gas species #1 and #2.

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49

1 11 v10 01 1 1

1 0 0 01 1

lnm Rm cS m VmTS

m T V m

⎧ ⎫⎛ ⎞⎪ ⎪⎛ ⎞− = ⎜ ⎟⎨ ⎬⎜ ⎟ ⎜ ⎟⎝ ⎠⎪ ⎪⎝ ⎠⎩ ⎭ (3.37)

and 2 22 v20 0

2 2 22 0 0 0

2 2ln

m Rm cS m VmTSm T V m

⎧ ⎫⎛ ⎞⎪ ⎪⎛ ⎞− = ⎜ ⎟⎨ ⎬⎜ ⎟ ⎜ ⎟⎝ ⎠⎪ ⎪⎝ ⎠⎩ ⎭. (3.38)

The total entropy of a mixture of gases is given by the sum of the entropies of the

individual gases:

1 1 2 21 1 2 2 1 v1 2 v20 0 0 01 1 2 2 1 2

0 0 0 01 21 2

lnm R m Rm R m R m c m cS m S m m mV TS

m mm m V T

+ +⎧ ⎫⎛ ⎞ ⎛ ⎞⎪ ⎪⎛ ⎞ ⎛ ⎞− − = ⎜ ⎟ ⎜ ⎟⎨ ⎬⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠⎪ ⎪⎝ ⎠ ⎝ ⎠⎩ ⎭(3.39)

where ( )1 2S S S= + .

From Eq. (3.39), the temperature of the gases is given as a function of the four

state variables ( 1 2, , and m m V S ):

1 1 2 21 1 2 21 v1 2 v2 1 v1 2 v21 v1 2 v2

0 00 1 1 2 2

0 0 0 01 v1 2 v2 1 1 v1 1 2 v2 1 2 v1 2 2 v2

1 20 0 0

1 2

exp

.

m R m Rm R m Rm c m c m c m cm c m c

m S m SST Tm c m c m m c m m c m m c m m c

m mVV m m

⎛ ⎞ ⎛ ⎞⎛ ⎞+ ⎜ ⎟ ⎜ ⎟−⎜ ⎟ + +⎝ ⎠ ⎝ ⎠+⎝ ⎠

⎛ ⎞= − −⎜ ⎟⎜ ⎟+ + +⎝ ⎠

⎛ ⎞ ⎛ ⎞⎛ ⎞× ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

(3.40)

The internal energy of the gas mixture is obtained as the sum of the internal

energies of both the gases in the chamber, i.e. 1 v1 2 v2U m c T m c T= + , or

( )1 1 2 21 1 2 2

1 v1 2 v2 1 v1 2 v21 v1 2 v20 1 21 v1 2 v2 0 0 0

1 2

0 01 1 2 2

0 0 0 01 v1 2 v2 1 1 v1 1 2 v2 1 2 v1 2 2 v2

exp

m R m Rm R m Rm c m c m c m cm c m c m mVU m c m c T

V m m

m S m SSm c m c m m c m m c m m c m m c

⎛ ⎞ ⎛ ⎞⎛ ⎞+ ⎜ ⎟ ⎜ ⎟−⎜ ⎟ + +⎝ ⎠ ⎝ ⎠+⎝ ⎠ ⎛ ⎞ ⎛ ⎞⎛ ⎞= + ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎛ ⎞

× − −⎜⎜ + + +⎝ ⎠.⎟⎟

(3.41)

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Chapter 3

50

The total pressure in the chamber is then obtained by summing the partial

pressures, i.e. VTRmVTRmp // 2211 += , where T is given by Eq. (3.40). The same

result can also be obtained by taking the partial derivative of the total internal energy

in Eq. (3.41) with respect to the total volume.

1 1 2 21 1 2 21 v1 2 v2 1 v11 v1 2 v2

0 00 1 1 2 2 1 1 2 2

0 0 0 01 v1 2 v2 1 1 v1 1 2 v2 1 2 v1 2 2 v2

1 20 0 0

1 2

exp

m R m Rm R m Rm c m c m cm c m c

m R m R m S m SSp TV m c m c m m c m m c m m c m m c

m mVV m m

⎛ ⎞⎛ ⎞+ ⎜ ⎟−⎜ ⎟ +⎝ ⎠+⎝ ⎠

⎛ ⎞+⎛ ⎞= − −⎜ ⎟⎜ ⎟ ⎜ ⎟+ + +⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞⎛ ⎞× ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

2 v2.

m c⎛ ⎞⎜ ⎟+⎝ ⎠

(3.42)

Likewise, the chemical potentials of the gases can be obtained by taking the

partial derivative of U with respect to their corresponding masses. That leads to a

cumbersome formula. Alternatively, the chemical potential of gas #1 can be given as

1 1 1 1 1 v1 1 1u p v Ts c T RT Tsμ = + − = + − . (3.43)

Equation (3.43) can be written as

1 1 1h Tsμ = − . (3.44)

Substituting 01 1 pdh h c T= + ∫ and ( ) ( )0 0

1 1 p 1 1 1d lns s c T T R p p= + −∫ in Eq.

(3.44), we get

( )0 11 1 1 0

1

ln pT RTp

μ μ⎛ ⎞

= + ⎜ ⎟⎝ ⎠

, (3.45)

where ( )01 Tμ is purely a function of the temperature. The values of ( )0

1 Tμ for the

individual gas molecules are calculated using the relation given in Eq. (3.18). The

partial pressure of the gas species #1 and the temperature of the mixture in Eq. (3.45)

are written in terms of the state variables by using the earlier expressions (refer to Eqs.

(3.40) and (3.42)). The chemical potential of gas #2 is calculated in a similar fashion:

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Bond Graph Model of a Solid Oxide Fuel Cell

51

( )0 22 2 2 0

2

ln pT R Tp

μ μ⎛ ⎞

= + ⎜ ⎟⎝ ⎠

. (3.46)

The Equations (3.40), (3.42), (3.45) and (3.46) are the constitutive relations of the

four port C-field as they give the effort variables ( 1 2 and , , p Tμ μ ) in terms of the

four state variables ( 1 2, , and m m V S ), which are obtained by integrating the flow

variables in the bonds connected to the four port C-field.

3.3.5. Bond Graph Model of the SOFC

The bond graph model of the SOFC, which utilises the C-fields presented in Section

3.3.4 for representing the energy storage in the gas mixtures in the anode and the

cathode channels, is shown in Fig. 3.6. The same bond graph in the conventional

representation may be referred to in Appendix A.

The electrochemical part of the bond graph (i.e., the calculation of the voltage and

the current) is the same as described in Section 3.2 (refer to Eqs. (3.24), (3.27) and

(3.28)). The theoretical open-circuit voltage ( rV ) is the maximum voltage that can be

achieved by a fuel cell under specific operating conditions. However, the voltage of

an operating cell, which is equal to the voltage difference between the cathode and the

anode, is generally lower than this. As current is drawn from a fuel cell, the cell

voltage falls due to the internal resistances and over-voltage losses. The electrode

over-voltage losses are associated with the electrochemical reactions taking place at

the electrode/electrolyte interfaces and can be divided into concentration and

activation over-voltages. These losses are common to all types of fuel cells and cannot

be eliminated; although temperature, pressure, gas flow rate and composition,

electrode and electrolyte materials, and cell design, all influence their magnitude. The

actual cell voltage is generally obtained by subtracting all the voltage losses from the

open circuit voltage.

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Chapter 3

52

Fig. 3.6: Bond Graph Model of the SOFC by Using the 4 Port C-fields

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Bond Graph Model of a Solid Oxide Fuel Cell

53

There are different kinds of voltage losses or over-voltages contributing to the cell

irreversibility. Activation over-voltage refers to the over potential required to exceed

the activation energy barrier so that the electrode reactions proceed at the desired rate.

The anodic and the cathodic activation over-voltages are governed by the Butler-

Volmer equation [Bockris et al., 1998; Campanari & Iora, 2004; Qi et al., 2005]

which in its general form (refer to Appendix B for derivation) is given as

( ) e acte act0

1exp exp

R Rn Fn Fi i

T Tβ ηβ η⎧ ⎫⎛ ⎞− −⎪ ⎪⎛ ⎞= −⎨ ⎜ ⎟⎬⎜ ⎟

⎝ ⎠⎪ ⎪⎝ ⎠⎩ ⎭. (3.47)

If the transfer coefficient (β ) is 0.5 (refer to Eq. (B.47) in Appendix B), which is

normally the case [Qi et al., 2005], the anodic and cathodic activation over-voltages

can be obtained from Eq. (3.47) as

1Mact,an

e 0,an

2R 0.5sinhT in F i

η − ⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠ (3.48)

and 1Mact,ca

e 0,c

2R 0.5sinhT in F i

η − ⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠, (3.49)

where the anodic and the cathodic exchange currents are given as

W anH0,an an c

amb amb M

expR

p Epi Ap p T

ψ⎛ ⎞⎛ ⎞ ⎛ ⎞−

= ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠⎝ ⎠

and 0.25

O ca0,ca ca c

amb M

expR

p Ei Ap T

ψ⎛ ⎞ ⎛ ⎞−

= ⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

.

It is clear from Eqs. (3.48) and (3.49) that the contribution of the activation over-

voltage to the overall voltage loss is significant at low currents.

The Ohmic over-voltage ( ohmη ) is due to the resistance to the transport of ions in

the electrolyte and to the flow of electrons through the electrodes and current

collectors. It is governed by the Ohm’s law:

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Chapter 3

54

ohm ohmiRη = , (3.50)

where ohmR is the resistance per unit area. The ohmic over-voltage comes into play

typically at the middle range of current densities within which the fuel cell is usually

designed to operate. However, due to load fluctuations during operation, the fuel cell

may have to be operated at low and high current density conditions, where other kinds

of over-voltages are predominant.

The reactants, i.e. hydrogen and oxygen, in the flow channels have to diffuse

through the porous anode and cathode, respectively, to reach the electrode electrolyte

interface where the reaction occurs. Similarly, the product of the reaction, i.e., water

vapour, which is formed at the anode electrolyte interface, has to diffuse through the

porous anode so as to reach the flow bulk in the anode channel. If the cell is

functioning reversibly, the partial pressures of the reactant and the product gas species

are same at the flow bulk in the gas channels and at the TPB where the actual reaction

takes place (called so because of the presence of three phases, viz. the solid nickel or

lanthium strontium manganite of the electrodes, the solid yttria stabilized zirconium

oxide phase of the ceramic electrolyte and the gas phase of either the reactants or the

products). But when current is drawn from the cell, the partial pressures of the gas

species at the TPB differ from their corresponding partial pressures in the bulk due to

limitations imposed by the diffusion process (refer to Fig. 3.7). The voltage lost due to

this pressure difference between the bulk and the TPB is called as the concentration

over-voltage.

Fig. 3.7: Schematic Showing the Variation of the Partial Pressures of Hydrogen and Water Vapour

Through the Anode, and Oxygen Through the Cathode

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Bond Graph Model of a Solid Oxide Fuel Cell

55

The physical processes that contribute to the occurrence of the concentration over-

voltage include gas species molecular transport in the electrode pores, solution of

reactants into the electrolyte, dissolution of products out of the electrolyte and

diffusion of the reactants/products through the electrolyte to/from the reaction sites.

By assuming that the pressure loss of one of the reactants, say hydrogen, determines

the concentration over-voltage and that the pressure drop between the bulk and the

TPB region is linear, a simple expression for the concentration over-voltage can be

derived as follows. If the system is at steady state, then the absolute rate of diffusion

of the gas for a unit of geometric area must be equal to the current. Therefore

H,b H,TPB( )i k p p= − . (3.51)

Moreover, H,TPBp tends to zero as the current ‘i’ tends to a limiting value ‘ Li ’. So

we can write

LH,b

ipk

= . (3.52)

Substituting Eq. (3.52) into Eq. (3.51), we get

( )LH,TPB

i ip

k−

= . (3.53)

The difference in the Nernst voltages calculated by using the partial pressures of

the reacting gas species at the bulk and the TPB gives the concentration over-voltage.

A simplified model for the concentration over-voltage in terms of the partial pressures

can be derives as follows.

By using Eqs. (3.23), (3.24) and (3.45), and taking the reference pressure as unity,

the reversible Nernst voltage of the cell is represented in terms of the species partial

pressures and temperature as (refer to Appendix B)

W

OH

0W

re H O

R ln pGV Tn F p p

ν

νν

⎛ ⎞Δ= − − ⎜ ⎟

⎝ ⎠, (3.54)

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Chapter 3

56

where 0 0 0 0W W H H O OG μ ν μ ν μ νΔ = − − , which is the change in the Gibbs free energy of

the reaction at the reference state.

The concentration over-voltage is obtained by subtracting the Nernst voltage (Eq.

(3.54)) obtained by using the partial pressures at the flow bulk and those at the TPB as

W W

O OH H

W,b W,TPBconc

H,b O,b H,TPB O,TPB

R ln R lnp p

T Tp p p p

ν ν

ν νν νη⎛ ⎞ ⎛ ⎞

= − +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

. (3.55)

It is assumed that the pressure loss of hydrogen alone is significant and is

responsible for the concentration over-voltage. Imposing this assumption on Eq.

(3.55) results in

H,bMconc

e H,TPB

R lnpT

n F pη

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠. (3.56)

Substitution of Eq. (3.52) and Eq. (3.53) into Eq. (3.56) yields [Mueller et al.,

2006]

M L Mconc

e L e L

R Rln ln 1T i T in F i i n F i

η⎛ ⎞ ⎛ ⎞

= = − −⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠. (3.57)

The concentration over-voltage is significant only at high currents. From Eq.

(3.57), it can be understood that the concentration over-voltage is very less when

Li i<< . It becomes significantly high when the value of the current approaches the

limiting current. Note that Eq. (3.57) is not valid for Li i= .

All these over-voltages are modelled by the RS-field shown in Fig. 3.6. The effort

output (for the port with current as the flow input) of the RS-field is given as

1 1Mohm

e 0,a 0,c L

R 0.5 0.52sinh 2sinh ln 1T i i i iRn F i i i

η − −⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞

= + − − +⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠ (3.58)

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Bond Graph Model of a Solid Oxide Fuel Cell

57

and the flow output (for the port with temperature as the effort input), i.e. the entropy

flow rate which goes to the heat transfer part of the model, is given as

21 1 ohm

PLe 0,a 0,c L M

R 0.5 0.52sinh 2sinh ln 1 i Ri i i iSn F i i i T

− −⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞

= + − − +⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠. (3.59)

The M

0T junction shown in Fig. 3.6 represents the temperature of the MEA solid.

Convection is an important means of heat transfer in an SOFC as the gases flow

through the anode and the cathode channels. Due to the ideal gas assumption and the

low velocities, the flow in a fuel cell is usually laminar. The bond graph model shown

in Fig. 3.6 includes the convective heat transfers between the anode and cathode

channel gases, the MEA and the interconnects. The R-fields, Rcv2 and Rcv4, model the

convective heat transfers between the gases and the MEA and the R-fields, Rcv1 and

Rcv3, model the convective heat transfers between the gases and the interconnects

denoted by I1 and I2, respectively, in Fig. 3.6. As the solid parts of the SOFC have

high thermal capacities and the cell is well insulated from the surroundings, the

temperatures of the interconnects are represented by the two sources of effort shown

in Fig. 3.6. The constitutive relations of the R-field, Rcv1, is given as [Mukherjee et al.,

2006; Thoma & Ould Bouamama, 2000]

( )an c I1 an3

an

A T TS

Tλ −

= (3.60)

and ( )an c I1 an4

I1

A T TS

Tλ −

= . (3.61)

The constitutive relations for the other R-field elements defining the convection

heat transfer (Rcv2, Rcv3 and Rcv4) are defined in a similar fashion. The thermal

capacity of the MEA is represented by the compliance element CM in Fig. 3.6. The

constitutive relation of thermal capacity [Karnopp et al., 2006; Thoma & Ould

Bouamama, 2000] of CM element is given as:

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Chapter 3

58

0

0 M MM M

M M

exp S ST Tm c

⎛ ⎞−= ⎜ ⎟

⎝ ⎠. (3.62)

The supply of fuel and air to the cell and the rejection of excess gases are through

four valves. Hydrogen gas is supplied to the anode channel through the anode channel

inlet valve and hydrogen-water vapour mixture goes out through the anode channel

outlet valve. The air is supplied to the cathode channel through the cathode channel

inlet valve and the depleted air exits through the cathode channel outlet valve. As the

pressure difference across the valves is small, a linearised nozzle flow equation is

used to calculate the mass flow rate through the valves [Thomas, 1999].

The models of these valves are shown separately in Fig. 3.8 and the connections

with the main bond graph in Fig. 3.6 are indicated by the numbered interface ports.

The pseudo bond graph models of the anode channel inlet valve, the anode channel

outlet valve, the cathode channel inlet valve and the cathode channel outlet valve are

shown in Fig. 3.8. The constitutive relations of the R-elements, which represent the

valve resistances in Fig. 3.8, are given as

( )u dm K p p= − . (3.63)

The values of the pressure variables are obtained from the main model in Fig. 3.6

through signal ports. Likewise, the calculated mass flow rates are exported through

signal ports to be used in the main model. The total mass flow rates, which are

represented by 1-junctions in Fig. 3.8 are converted into individual species mass flow

rates by multiplication with the upstream mass fractions. The mass fractions of the gas

species in the air source and those of the anode and cathode channels are represented

as gains in Fig. 3.8. The models of anode channel and the cathode channel outlet

valves need the information of the contemporary values of the mass fractions of the

individual gas species in the anode and the cathode channels. These mass fractions are

calculated from the state variables in the main model shown in Fig. 3.6 and used to

modulate the corresponding gains in the outlet valve models given in Fig. 3.8.

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Bond Graph Model of a Solid Oxide Fuel Cell

59

(a) (b)

(c) (d)

Fig. 3.8: Sub-models for (a) Anode Channel Inlet Valve, (b) Anode Channel Outlet Valve, (c) Cathode Channel Inlet Valve and (d) Cathode Channel Outlet Valve

The entropy of the reaction is accounted in the following way. The entropy

balance for the anode channel control volume is given by

an H H W W 1S m s m s S= + + , (3.64)

where the quantities Hm and Wm are given respectively (refer to Fig. 3.6) as follows.

i o rH H H Hm m m m= − − . (3.65)

i o rW W W Wm m m m= − + . (3.66)

Substituting Eqs. (3.65) and (3.66) into Eq. (3.64), we get

i o r i o ran H H H H H H W W W W W W 1S s m s m s m s m s m s m S= − − + − + + , (3.67)

which can further be written as

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Chapter 3

60

i o r i o ran H H H W W W 1S S S S S S S S= − − + − + + . (3.68)

Similarly the entropy balance equation for the cathode channel control volume

may be written as

i o r i oca O O O N N 8S S S S S S S= − − + − + . (3.69)

From Eqs. (3.68) and (3.69), it is apparent that this model accounts for the entropy

change of the reaction. This entropy change results in the enthalpy change of the

reaction which is positive for this case (the enthalpy of the products being more than

the enthalpy of the reactants). Note that the enthalpies of all the gas species can be

directly read from the bond graph model.

The bond graph model presented in this section avoids the stiff differential

equations of the system resulting from the earlier model presented in Section 3.2 (due

to the presence of the R-fields for temperature equilibration) by means of using a C-

field which assumes a constant temperature for two species gas mixtures. However,

some parts of the model are constructed by using pseudo bond graphs (the valves) and

the electrochemical part of the model is constructed by using activated or signal bonds.

This is not desirable because energy consistency of the system is not very clear from

the bond graph. Also, the entropy convection phenomena are not clearly represented

in this bond graph model. For example, in Eq. (3.67), the specific entropies of the

inlet side and outlet side must be used, respectively, for calculating the inlet and outlet

entropy flow rates, while the channel volume specific entropy must be used for

calculating the entropy flow due to the reaction. Due to the improper representation of

the entropy convection in the model, the heat released due to the exothermic reaction

is insufficiently removed. During simulations, the sources of efforts representing the

interconnect temperatures ( I1T and I2T ) act as heat sinks. If the sources of efforts are

replaced by capacitive elements representing the thermal capacitance of the

interconnect solids then the temperature of the cell will go on increasing as the

reaction proceeds. This restricts the usage of the model only for cases in which the

temperature of the cell does not vary much. All these deficiencies are rectified in the

true bond graph model of the SOFC presented in the next section in which the entropy

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Bond Graph Model of a Solid Oxide Fuel Cell

61

convection is represented in an R field element and the pseudo and activated parts of

the model are replaced with true bonds.

3.4. Formulation of the True Bond Graph Model of the

SOFC

In this section, a true bond graph model of the SOFC is formulated to rectify the

deficiencies of the preliminary models developed in the previous sections. The C-

fields for energy storage in the two species gas mixtures developed in Section 3.3.4

are used in this model. Further, an R-field element is formulated (based on the model

given in [Breedveld, 1984]) for modelling the entropy convection of a two species gas

mixture.

3.4.1. Convection of a Two Species Gas Mixture Represented

in an R-Field

Models for the irreversible convective flow of compressible fluids are available in the

literature. In [Brown, 2007], they are modelled by using convection bonds and the HS

element. A true bond graph formulation for the forced convection of a compressible

ideal gas was given in [Feenstra, 2000]. The energy storage in a control volume was

modelled as a C-field element and the gas enters and leaves the control volume

through two R-field elements. However, the formulation in [Feenstra, 2000] assumed

convection of a single gas species. In this work, the resistive field is extended to

handle the flow of a gas mixture, which will be used to represent the convection of

fluids between different control volumes in the SOFC model.

The details of the sub-model for modelling the convection of a two-component

gas mixture are given in Fig. 3.9. The most important element in the expanded model

of the MR element is the RS-field element (see Fig. 3.9). This element is modelled by

following the same true bond graph representation given in Breedveld’s original work

[Breedveld, 1984]. This element receives the downstream side temperature and the

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Chapter 3

62

information of the valve position (x), the upstream side chemical potentials and

temperature, and the downstream side chemical potentials to calculate the mass and

entropy flow rates. Note that all these variables are inputs to the MR element. To

maintain the clarity of the figure, the connections needed to explicitly show these

modulations are not drawn.

From the causal analysis, this sub-model receives six effort variables and

computes six flow variables without the use of integration and/or differentiation.

Therefore, this sub-model can be represented as an encapsulated R-field (a six port

element MR in Fig. 3.9). From the continuity equation, the mass flow rate of a

particular gas is the same for the inlet and the outlet side. This reduces the total

number of independent flow variables to four (see Fig. 3.9). Then the constitutive

relation of the non-linear resistive field element is given as

( ) ( ) T Tu d 1 2 R u d 1u 1d 2u 2d, , , , , , , ,S S m m T T μ μ μ μ= Φ , (3.70)

where, RΦ is a vector-valued function. The individual relations between the input and

the output variables are derived as follows.

Fig. 3.9: Bond Graph Representation For Convection of a Two-Component Gas Mixture

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Bond Graph Model of a Solid Oxide Fuel Cell

63

The overall mass flow rate ( m ) of the mixture is imposed at the 1m junction by

the modulated RS-field element in Fig. 3.9 and it is given by the linear nozzle

equation:

( )u dm K p p= − . (3.71)

Note that although Eq. (3.71) needs the total upstream and downstream side

pressures, they can indeed be calculated from the chemical potentials and

temperatures. These calculations are given later in this section (see Eqs. (3.76–3.79)).

The individual mass flow rates of the two gases are then realized through the

modulated transformer elements shown in Fig. 3.9 as 1 1um mw= and 2 2um mw= . The

upstream mass fractions 1uw and 2uw are obtained from the upstream side storage

element, i.e. ( )1u 1u 1u 2uw m m m= + , ( )2u 2u 1u 2uw m m m= + and 1u 2u 1w w+ = , where

1um and 2um are the contemporary masses (state variables) in the upstream side

control volume.

The entropy flow rate due to the mass flow rate is calculated by means of a

transformer element (between junctions 1S and 1m ), which is modulated by the

specific entropy of the upstream side gases. This information of the upstream side

specific entropy can either be obtained directly from the upstream side storage

element or if a standalone scheme is required, it can be calculated from the upstream

sideμ ’s and T ’s (which are inputs of the MR element) as

( )1u 1u 2u 2uu p1 1u p2 2u

u

w ws c w c w

Tμ μ+

= + − . (3.72)

The entropy flow rate from the upstream side is given as u uS ms= . The R-field

represents the change in the intensive variables between the upstream and the

downstream sides. The temperatures, pressures and the chemical potentials of the gas

mixture in the upstream and the downstream sides are imposed by the storage

elements on the corresponding sides. Due to this, there is an enthalpy difference

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Chapter 3

64

between the upstream and the downstream sides. This enthalpy difference can be

represented as the relation between the changes in the intensive variables by using the

Gibbs-Duhem equation [Callen, 1985; Denbigh, 1955] as (refer to Appendix B)

( ) ( ) ( ) ( )u d u u d 1u 1u 1d 2u 2u 2d v p p s T T w wμ μ μ μ− = − + − + − . (3.73)

This relation is enforced by the 1m -junction in Fig. 3.9. Due to the enthalpy

difference between the upstream and the downstream side gases, entropy is generated

in the resistive field. Using the principle of power conservation, the irreversible

entropy generated genS can be given as

( )u dgen

d

mv P PS

T−

= . (3.74)

Substitution of Eq. (3.73) into Eq. (3.74) gives

( ) ( ) ( )( )u u d 1u 1u 1d 2u 2u 2dgen

d

,m s T T w w

ST

μ μ μ μ− + − + −= (3.75)

where ( ) ( ) ( )u u d 1u 1u 1d 2u 2u 2ds T T w wμ μ μ μ− + − + − and dT are effort inputs to the RS-

element and m is calculated internally from the constitutive relation of the RS-

element (see Eq. (3.71)).

The downstream side entropy flow rate is the sum of the upstream side entropy

flow rate ( uS imposed at 1S -junction by the MTF element) and the irreversible

entropy generated ( genS in Eq. (3.75)). This sum is realized by means of the zero-

junction shown in Fig. 3.9.

The upstream and downstream pressures, which are needed in Eq. (3.71), can

either be read directly from the upstream and downstream side storage elements or

can be calculated as functions of μ ’s and T ’s (the input variables to the MR element)

as follows. The change in the upstream side specific entropy of a given mass of gas

species #1 is given by

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Bond Graph Model of a Solid Oxide Fuel Cell

65

v1 1

0 u u11u 1u 0 0

u u1

lnc R

T vs sT v

⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟− = ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

. (3.76)

Substitution of 1u p1 1u us C Tμ= − in Eq. (3.76) and rearrangement gives

v1

10

0 1u 1u uu1 u1 0 0

u 1 u 1 u

exp

cRTv v

T R T R Tμ μ

−⎛ ⎞⎛ ⎞

= − +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

. (3.77)

The upstream side partial pressure of gas species #1 is given as

v1

10

01 u 1u 1u u1u u1 0 0

u1 u 1 u 1 u

exp

cRR T Tp p

v T R T R Tμ μ⎛ ⎞⎛ ⎞

= = −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

. (3.78)

Similarly, the upstream partial pressure of gas #2 is

v2

20

02 u 2u 2u u2u u2 0 0

u2 u 2 u 2 u

exp

cRR T Tp p

v T R T R Tμ μ⎛ ⎞⎛ ⎞

= = −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

. (3.79)

The total upstream side pressure is u 1u 2up p p= + . The total downstream side

pressure can also be expressed similarly.

3.4.2. True Bond Graph Model of the SOFC

The true bond graph model of the SOFC system is given in Fig. 3.10. The same bond

graph model in the conventional representation may be referred to in Appendix A.

This model uses the 4-port C-field (presented in Section 3.3.4) for representing the

storage of the gases inside the anode and the cathode flow channels. It also uses the

R-field representation discussed in Section 3.4.1 for modelling the entropy convection

at the inlet and the outlet of the SOFC channels.

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Chapter 3

66

Representing a thermodynamic system in terms of true bond graph involves the

concepts of network thermodynamics, which depend on the following three postulates

[Perelson, 1975]:

1. The assumption of local phase equilibrium is made. That is, even though the

system as a whole may be far from the equilibrium, locally one can still

describe the state of the system by thermodynamic variables, such as the

temperature, the pressure, and the chemical potential.

2. The physical space is viewed as discrete rather than as a continuum. This will

generate finite dimensional models described by ordinary differential

equations as opposed to the partial differential equation description of

continuum theories.

3. Although complex processes are occurring on every volume element of the

physical system, we assume that one can conceptually separate these processes

into dissipative and non-dissipative parts.

The true bond graph model of the SOFC, shown in Fig. 3.10, is constructed by

using the concepts of network thermodynamics. As the volumes of both the channels

remain constant, the mechanical ports of the C-fields are not shown in Fig. 3.10. The

constitutive relations for the two C-fields shown in Fig. 3.10 were given in Section

3.3.4 (Eqs. (3.40), (3.42), (3.45) and (3.46)), where the subscripts 1 and 2 represent

the gas species in the anode and cathode channels. Note that the outer layer of the C-

field presented in Section 3.3.4 imposing the entropy balance (Eq. (3.32) and the

enthalpy-entropy relations) is not used in the model given in Fig. 3.10. Rather, in this

model the mass and entropy balances of the anode and cathode channel control

volumes are given by the corresponding zero junctions in Fig. 3.10. The an

0T and the

ca0T junctions give the entropy balances for the anode channel and the cathode

channel control volumes, respectively. The H

0μ , W

0μ , O

0μ and N

0μ junctions give the

mass balances for the hydrogen, water vapour, oxygen and nitrogen gases,

respectively, in the control volumes. The M

0T junction gives the entropy balance at the

MEA solid control volume.

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Bond Graph Model of a Solid Oxide Fuel Cell

67

The capacitive elements and fields in the model represent equilibrium

thermodynamics part of the model. As the simulation proceeds, the matter inside the

control volume represented by these elements change reversibly from one equilibrium

state to the next, which means that the process is assumed to be quasi-static. The R-

fields represent the non-equilibrium parts of the model, i.e. they introduce the

irreversibilities into the system. The R-field elements represented by ‘MR’ in Fig.

3.10 introduce the irreversibility due to mass convection into the system (refer to

Section 3.4.1). The R-field element represented by ‘RS’ in Fig. 3.10 introduces the

irreversibility due to the over-voltage phenomena (ohmic, concentration and

activation losses). The other R-field elements introduce the irreversibilities due to the

heat transfer phenomena.

The inlet and outlet valve resistances are modelled by the MR-fields described in

the previous section where subscripts mentioned in the nomenclature identify them.

The valve resistances in the MR-fields may be controlled by modifying the variables

for the stem positions. Note that although only hydrogen gas flows through the anode

side inlet valve, the information of chemical potential of water vapour ( Wμ ) inside

the anode channel is required for computing the downstream side pressure, which is

supplied by an information bond in Fig. 3.10. Similarly, the additional information of

the chemical potentials of nitrogen and oxygen in the atmosphere are required in the

anode channel outlet valve model to calculate the downstream side pressure, which is

provided by the source of efforts as shown in Fig. 3.10. The downstream side entropy

flow is the sum of the upstream side entropy flow and the entropy generated due to

the enthalpy difference between the upstream and the downstream sides (Eq. (3.75)).

In this model, the chemical potentials of the gases not only drive the electro-

chemical reaction but also, along with temperatures, determine the flow of the gases

in and out of the channels. This is because, though the mass flow through the MR-

field element is determined from the upstream and downstream side pressures (Eq.

(3.71)), the individual pressures can be written as functions of the chemical potential

and the temperature (Eqs. (3.78–3.79)). Thus, the coupling between the chemical,

thermal, mechanical and the hydraulic domains, which is encountered in a fuel cell

system, is effectively represented here in a unified manner by using true bond graphs.

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Chapter 3

68

Fig. 3.10: True Bond Graph Model of the SOFC

The transformation of power from the chemical domain into the electrical domain

is elegantly represented by means of true bonds, as shown in Fig. 3.10, instead of the

activated bonds in the preliminary models presented in Sections 3.2 and 3.3.5. This

transformation is implemented by the 1ξ junction and the transformers surrounding it.

The electrochemical relations are the same as those given in Section 3.2. The

reversible cell voltage is given by Eq. (3.24) and the relation between the reaction rate

and the current is given by Eq. (3.27). The relation between the current and the

reaction mass flow rates of the reactants and the product (in kg.s-1) is also given by Eq.

(3.28).

The transformation between the electrical and the thermal domains is due to the

RS field element shown in Fig. 3.10, which models the different polarisation losses

(activation concentration and ohmic) and the entropy generation due to them. The

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Bond Graph Model of a Solid Oxide Fuel Cell

69

constitutive relations of the RS-field element were developed in Section 3.3.5 (refer to

Eqs. (3.47–3.59)).

The constitutive relations of the convective heat transfer resistances Rcv1, Rcv2,

Rcv3 and Rcv4 are the same as those that were given in Section 3.3.5. The constitutive

relations for the convective heat transfer resistance were given in Eqs. (3.60) and

(3.63). The constitutive relations for the other heat transfer resistances are similar.

Note that the same bond numbering for the thermal part is used both in Fig. 3.6 and

Fig. 3.10. The constitutive relation of the CM element representing the thermal

capacity of the MEA solid is the same as given in Section 3.3.5 (refer to Eq. (3.62)).

The unmodelled load is represented by the flow source as shown in Fig. 3.10.

The thermal capacitance of the interconnect plates are represented by the two

capacitive elements CI1 and CI2 and their constitutive relations are given, respectively,

as

00 I1 I1

I1 I1I1 I1

exp S ST Tm c

⎛ ⎞−= ⎜ ⎟

⎝ ⎠. (3.80)

and 0

0 I2 I2I2 I2

I2 I2

exp S ST Tm c

⎛ ⎞−= ⎜ ⎟

⎝ ⎠. (3.81)

The enthalpy of the reaction is given as

H G T SΔ = Δ + Δ , (3.82)

where the part T SΔ is released as heat when the fuel cell operates reversibly. Under

irreversible operation (under all realistic circumstances), the change in the Gibbs free

energy of the reaction ( GΔ ) is not completely converted into useful electrical work.

Rather, some of it ends up as heat energy. These irreversibilities, which are called

over-voltages, give rise to entropy generation and are taken care by the RS-field

element in the model. In order to account for the entropy change of the reaction, the

following entropy flow rate is added to the MEA by means of a modulated source of

flow in Fig. 3.10:

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Chapter 3

70

( ) ( ) ( )r r rH H H O O O W W W

ran ca an

m h m h m hS

T T Tμ μ μ− − −

= + − , (3.83)

where the specific enthalpies are expressed as follows [Benson, 1977]:

( )2 3 4 51 2 3 4 5 0h R a T a T a T a T a T h= + + + + + . (3.84)

The values of the coefficients 1 6a a and 0h for the different gases are taken

from [Benson, 1977]. The source of flow MSf: rS is modulated with signals i (to

calculate rHm , r

Om and rWm according to Eq. (3.83)), Wμ , Hμ , Oμ , anT and caT (the

later five are calculable from state variables). Note that these modulating signals are

not shown in Fig. 3.10 to maintain the visual clarity of the figure.

Unlike the pseudo bond graphs, the energetic consistency of the true bond graph

presented in Fig. 3.10 is apparent. The continuity of energy flows across different

domains and across different interfaces is ensured because the effort and the flow

variables correspond to the power variables in the corresponding energy domains

throughout the bond graph model. The interface between the chemical and the

electrical domain is provided by the transformer element with en F as the modulus.

The hydraulic and the thermal domains are coupled via the entropy generation due to

the gas enthalpy difference between the cell interior and the exterior, which is called

as the entropy of mixing. These entropy generations are represented in the four MR-

field elements in Fig. 3.10. The coupling between the electrical and the thermal

energy domains is provided by the RS-field element in Fig. 3.10. The entropy

generated in the RS-field element is due to the charge transport in the electrodes,

electrolyte and interconnect plates, the charge transfer taking place at the electrode

electrolyte interface and also due to the mass diffusion through the porous electrodes.

The entropy generation due to the convective heat transfer between the gas and solid

phases are represented by the R-fields, Rcv1, Rcv2, Rcv3 and Rcv4. The expressions for

these entropy generation terms were already discussed in Sections 3.3.5 and 3.4.1.

Further, all the storage elements in the developed global model given in Fig. 3.10

are in integral causality. There is no causality violation at any place in the junction

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Bond Graph Model of a Solid Oxide Fuel Cell

71

structure. This ensures the energy consistency in the model. Moreover, this integrally

causalled model does not have algebraic or causal loops, which ensures that this

model is well computable.

3.5. Conclusions

The systematic development of a zero dimensional true bond graph model of an

SOFC system by using the concepts of network thermodynamics is discussed in this

chapter. A preliminary bond graph model of the hydrogen oxidation electrochemical

reaction is developed. This preliminary model makes use of a C-field in which the

internal energy serves as the thermodynamic potential. The preliminary model avoids

the differential causality problem present in the models proposed earlier in the

literature and gives a clear insight into the dynamics of the electrochemical hydrogen

oxidation reaction. Moreover, the relations between the enthalpies and the entropies

of the reacting gas species which are clearly represented in the bond graph model

facilitate the measurement of the reaction enthalpies. In order to enhance this model to

represent fuel cells, a C-field for representing the energy storage in a two species gas

mixture is formulated. This C-field is used in the construction of a bond graph model

of an SOFC. In this model, the two species C-fields are used to represent the

capacitance of the gas mixtures in the flow channels. Various over-voltages

encountered in an SOFC are also included. Some parts of this model are constructed

as pseudo bond graphs, which is not desirable. Also, the entropy convection processes

are not clearly represented in this model. In order to rectify these deficiencies, a true

bond graph model of the SOFC is developed by using the concepts of network

thermodynamics. For this purpose, an already available R-field formulation for single

species gas is extended to represent the convection of two component gas mixtures.

The formulations developed in this chapter can be extended to mixtures of more than

two gas species.

The true bond graph model offers significant improvements over the previous

bond graph models of the electrochemical reaction systems available in the literature.

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Chapter 3

72

They include, the use of internal energy as the thermodynamic potential of the C-field

used in modelling the chemical reaction systems, the formulation and use of a C-field

to model the energy storage in two species gas mixtures encountered in an SOFC, the

extension and the use of an R-field [Breedveld, 1984] element to represent the entropy

convection in the SOFC inlet and outlet gas streams. The couplings between the

various energy domains in a fuel cell system have been represented in a unified

manner by using the true bond graphs. The developed model ensures energy balance

at all physical process interfaces, e.g. entropy generation due to mixing, entropy

generation due to heat transfer, charge transport and diffusion phenomena. The true

bond graph model developed in this work clearly exposes the physical structure and

process dynamics of the SOFC. Causal analysis shows that model is energetically

consistent and, at the same time, it is well computable.

The developed model will be useful in designing integrated model-based control

strategies for the overall system by including the load and power conditioning

components. Moreover, various other control theoretic tools, fault detection

algorithms, and fault tolerant and robust control algorithms can be readily applied to

the bond graph model. Because this model is based on the second law of

thermodynamics of the system and the principles of network thermodynamics [Oster

et al., 1971; Perelson, 1975], it can be used for performing exergy based system

optimisation studies [Saidi et al., 2005]. Some of these possibilities are explored in

the later chapters of this thesis.

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Chapter 4

Simulation of the Open and Closed Loop

Dynamics of SOFC

4.1. Introduction

The static characteristic curves of the SOFC are useful for obtaining the voltages and

power densities for various values of current densities (j’s) and varying operating

conditions, e.g. FU’s, temperatures and pressures. They are also useful in identifying

the operating regime of the SOFC. Given the SOFC geometric and design

specifications, the static characteristic curves are extremely useful in characterizing

the stable operation of the cell under different operating conditions.

For the given values of system operating pressure, the air source and hydrogen

source pressures, the outlet pressures, the cell temperature and the inlet gas

compositions, the FU and the OU (also referred to as the air utilisation in some works)

may be interpreted in terms of the partial pressures of the gases in the anode channel

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Chapter 4

74

and the cathode channel. This knowledge is useful for obtaining the various static

characteristic curves of the SOFC.

Many emerging applications of SOFC require it to follow very fast load transients

[Mueller et al., 2006]. A detailed true bond graph model (with heat transfer,

hydraulics and the diffusion phenomena), which complies with other advanced

models available the literature [Aguiar et al., 2004], was developed in the previous

chapter. That model is used in this chapter to obtain the static characteristic curves of

the SOFC and to analyse the dynamic responses through simulations. Note that

various static characteristic curves of the SOFC given in this chapter are plotted with

current densities (j’s) rather than currents on the x-axis so as to comply with the

conventions followed in the literature. The current density is the current passing

through unit cell area, i.e. j=i/Ac.

In order to investigate the dynamic properties of a system, step response tests are

usually used. These tests can reveal key process dynamic parameters such as time

constant, overshoot, gain, etc. Therefore, in this work, the SOFC model is subjected to

step load current changes and the transient response characteristics are investigated.

Controlling average stack temperature, maintaining constant FU for all power

outputs, and ensuring that the air ratio always exceeds a minimum specified value are

some common control objectives of the SOFC. Stack temperature control is normally

provided by varying the air ratio, i.e., the supply of air for cooling [Aguiar et al.,

2005]. In [Aguiar et al., 2005], two main control loops were implemented in order to

achieve the control objectives. A master controller sets the fuel and air flow rates

proportional to the current and a typical feedback PID temperature controller which,

given the outlet fuel gas temperature, responds by changing the air flow around the

default air flow set by the master controller.

The FU is usually maintained constant by controlling the fuel flow rate into the

system [Mueller et al., 2006]. Another requirement for the operation of the SOFC,

which is often neglected by most researchers, is that the difference of the pressure

between the anode and the cathode channels has to be small [Serra et al., 2005;

Wachter et al., 2006]; the adequate value being dependent on the membrane support

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Open and Closed Loop SOFC Dynamics

75

(allowable stress) and on the age of the fuel cell (fatigue considerations). Some

authors [He, 1998; Wang et al., 2007; Zhu & Tomsovic, 2002] state that the pressure

difference between the anode and cathode channels must be kept below 8kPa under

transient conditions and below 4kPa during steady state conditions in order to prevent

damage to the electrolyte. In this chapter, a physical model-based control strategy for

the SOFC system is developed. It is based on manipulating the flows through four

inlet and outlet valves. The developed control laws satisfy all the above-mentioned

operational constraints and the control objectives.

4.2. Model Initialization and Open Loop System Simulations

In this section, the true bond graph model of the SOFC described in Section 3.4.2 of

the previous chapter is simulated to obtain the static characteristic curves and dynamic

responses to a step change in the load current. In order to simulate the steady state

operation of the SOFC, the single port C-elements in the true bond graph model

described in the previous chapter have to be initialized with the values of generalized

displacements (initial entropies in this case). Similarly, the two C-field elements have

to be initialized with the values of the initial masses of the constituent gases and their

entropies.

According to the operational requirement of the SOFC, FU must be maintained

constant. Normally, FU of 0.8–0.9 is desired [Wang et al., 2007]. In this work, a value

of 0.8 is chosen for the FU and a value of 0.125 is chosen for the OU.

The static characteristic curves of the SOFC give the behaviour of the voltage and

the current under varying operating conditions, i.e. temperatures, pressures, FU and

OU. They basically give us an idea of the contributions of the various over-voltages to

the overall voltage loss in the cell under different operating conditions. For obtaining

the static performance curves of the SOFC, the variables such as the operating

temperature, the desired current density, the desired total anode and cathode channel

pressures ( 0anp , 0

cap ), the FU ( fζ ), the OU ( oζ ), the source and the sink pressures (pHS,

pAS, pENV) and the species mass fractions at the air source ( O,ASw and N,ASw ) are

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Chapter 4

76

considered as the known variables. We need to establish relationships between the

steady state values of these known variables and the steady state partial pressures of

the individual gas species in the channels. This is necessary because it is evident from

the Nernst equation that the open circuit voltage (reversible voltage) of the cell

depends on these partial pressures. Moreover, the activation and the concentration

over-voltages also depend on the species partial pressures.

In order to establish the desired relations, the steady state mass balances for the

anode and the cathode channels are written. From those mass balance equations, the

valve coefficients (which are unknown variables) which will lead to the desired steady

state operation are obtained. In the following discussions, the superscript ‘0’ refers to

the desired steady state value of the variable which needs to be set for obtaining the

desired static characteristic curves.

The steady state mass balance in the anode channel by taking into consideration

the required value of FU is given as

i r oH H Hm m m= + , (4.1)

r oW Wm m= (4.2)

and r iH f Hm mζ= , (4.3)

where the hydrogen inlet and outlet mass flow rates and the water vapour outlet mass

flow rate are given by the linear nozzle flow equations:

( )i 0H ai HS anm K p p= − , (4.4)

( )o 0 0H ao H,an an ENVm K w p p= − , (4.5)

and ( )o 0 0W ao W,an an ENVm K w p p= − . (4.6)

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Open and Closed Loop SOFC Dynamics

77

Note that in the Eqs. (4.1–4.6), the unknown variables are aiK , aoK , 0H,anw and

0W,anw . However, the total steady state mass in the anode channel can be calculated

using the ideal gas law because the desired anode channel pressure, temperature and

the volume are known. Therefore, the three equations (Eqs. (4.1–4.3)) are solved for

the two valve coefficients ( aiK and aoK ) and for the ratio of the mass fractions, i.e.,

0 0 0W,an W,an W,an W0 0 0H,an H,an H,an H

w m n Mw m n M

= = . (4.7)

Solving the Eqs. (4.1–4.3), we get the expressions for the inlet and outlet valve

coefficients as

rH

ai 0f HS f an

mKp pζ ζ

=−

(4.8)

and rW

ao 0 0 0W,an an W,an ENV

mKw p w p

=−

, (4.9)

and the ratio of the partial pressures of the two gas species in the anode channel is

found to be

0 0H,an H f0 0W,an W f

1n pn p

ζζ−

= = . (4.10)

As we know the total pressure in the anode channel, we can calculate the

individual partial pressures of hydrogen and water vapour; thus establishing the

relationship between the FU and the partial pressures of individual species.

The mass balance in the cathode channel along with the required OU is given as

i r oO O Om m m= + , (4.11)

i oN Nm m= (4.12)

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Chapter 4

78

and r iO o Om mζ= , (4.13)

where the oxygen and nitrogen inlet and outlet mass flow rates are given by the linear

nozzle flow equations:

( )i 0O ci O,AS AS cam K w p p= − (4.14)

( )i 0N ci N,AS AS cam K w p p= − (4.15)

( )o 0 0O co O,ca ca ENVm K w p p= − (4.16)

and ( )o 0 0N co N,ca ca ENVm K w p p= − . (4.17)

Solving Eqs. (4.11–4.13) for the variables ciK , coK and 0 0N,ca O,can n (refer to the

discussion for the anode channel case), we get the expressions for the cathode channel

inlet and outlet valve coefficients as

( )

rO

ci 0o O,AS AS O,AS ca

mKw p w pζ

=−

(4.18)

and ( )( )

rO o

co 0 0 0o O,ca ca O,ca ENV

1mK

w p w pξ

ζ−

=−

, (4.19)

and also the ratio of the partial pressures of the two gas species as

( )

0 0 0N,AS N,ca N

0 0 0O,AS o O,ca O1

n n pn n pζ

= =−

. (4.20)

By assuming that N,AS O,ASn n is fixed and equal to that of the normal atmospheric

air at sea level (i.e., 3.76), the partial pressures of nitrogen and oxygen in the cathode

channel can be obtained from the known total pressure of the cathode channel.

Thus, the steady state values of the partial pressures of the gas species at the anode

and the cathode gas channels are derived as functions of the SOFC operating

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Open and Closed Loop SOFC Dynamics

79

conditions such as the temperature, the anodic and cathodic pressures, the load current,

the FU and the OU. Therefore, for any set of desired operating conditions of the

SOFC, the cell voltage can be obtained by using the Nernst equation (refer to

Appendix B) and the over-voltage relations (refer to Section 3.3.5 of Chapter 3).

Static characteristics of a fuel cell system are used to determine its operating

regime. The partial pressures of the hydrogen and water vapour in the anode channel

are set to obtain desired FU, i.e. ( )H W f f1p p ζ ζ= − . The OU is set to zero (i.e.,

there is sufficient air flow such that the rate of oxygen consumption in the reactions

can be neglected). This is achieved by setting the partial pressures of nitrogen and

oxygen in the cathode channel in such a way that N Op p is equal to the ratio

0.79 0.21, which is the same ratio as in the normally available atmospheric air.

We can also get an idea of the values of the inlet and the outlet valve coefficients

which have to be used in the dynamic model for simulating the desired steady state

operation of the SOFC. These can be obtained by substituting the relations for the

reaction mass flow rates of the gas species into Eqs. (4.8–4.9, 4.18–4.19). The

reaction mass flow rates of the gas species are (refer to Section 3.2 in Chapter 3)

r W WW

e1000i Mm

n Fν

= , (4.21)

r H HH

e1000i Mm

n Fν

= , (4.22)

and r O OO

e1000i Mm

n Fν

= . (4.23)

The initial partial pressures of the constituent gases, the initial temperatures of the

mixture and the volume of the channels are the input parameters of the C-fields. The

initial masses of the constituent gases are calculated from these inputs by using the

ideal gas law. The initial mass of hydrogen and water vapour in the anode channel and

that of oxygen and nitrogen in the cathode channel are calculated by using the ideal

gas law. For example, the initial mass of hydrogen in the anode channel is given by

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Chapter 4

80

0

0 H anH 0

H an

p VmR T

= , (4.24)

where the value of 0Hp is known from the calculations discussed earlier (refer to Eq.

(4.10)) and 0anT is the desired steady state temperature. The initial masses of other gas

species are similarly calculated.

The initial entropy of the anode channel gas mixture (hydrogen and water vapour)

is calculated as

0 0 0 0 00H v,H H H W v,W W W W WH H0 refref

0 ref ref an WHan H W ref 0 0

an H W

lnm c m R m c m R m Rm R

T ppS S ST p p

+ + +⎧ ⎫⎛ ⎞ ⎛ ⎞⎛ ⎞⎪ ⎪= + + ⎨ ⎬⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠⎪ ⎪⎩ ⎭

(4.25)

and the initial entropy of the cathode channel gas mixture (oxygen and nitrogen) is

calculated as

0 0 0 0 0 0O v,O O O N v,N N N O O N N0 ref ref

0 ref ref ca O Nca O N ref 0 0

ca O N

lnm c m R m c m R m R m R

T p pS S ST p p

+ + +⎧ ⎫⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎪ ⎪= + + ⎨ ⎬⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎪ ⎪⎩ ⎭

,(4.26)

where the values of all entropies at reference states are obtained from tables [Benson,

1977].

The initial entropy of the element CM representing the thermal capacity of the

MEA is given as

00 ref MM M M M ref

M

ln TS S m cT⎛ ⎞

= + ⎜ ⎟⎝ ⎠

(4.27)

and the initial entropies of the capacitance elements CI1 and CI2 are, respectively,

given as

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Open and Closed Loop SOFC Dynamics

81

00 ref I1I1 I1 I1 I1 ref

I1

ln TS S m cT⎛ ⎞

= + ⎜ ⎟⎝ ⎠

(4.28)

and 0

0 ref I2I2 I2 I2 I2 ref

I2

ln TS S m cT⎛ ⎞

= + ⎜ ⎟⎝ ⎠

. (4.29)

Use of the values of the valve coefficients calculated from Eqs. (4.8–4.9, 4.18–

4.19) in the dynamic model will not result in the exact desired steady state conditions

at the beginning of the simulation due to the influence of unaccounted thermal

dynamics. Therefore, a first phase simulation is run until the transients phase is over,

and the final values of the states are taken as the initial values for the next phase of the

simulation. This next phase simulation is run at a steady state operating mode with the

desired values of the FU and the OU. A step change in the load current is then

imposed in order to study the dynamic response.

4.2.1. Static Characteristics

The simulations were performed using the software SYMBOLS Shakti [Samantaray

& Mukherjee, 2006]. The parameter values used in the simulation are given in Table

4.1. In Fig. 4.1, the polarization and current density curves obtained from the model

are compared with the data from [Aguiar et al., 2004] in which the fuel considered

was CH4 and the fuel composition for obtaining the static characteristic curves was

fully reformed steam and methane mixture. It can be seen that that the difference

between the results is small because the principal gaseous species in the anode

channel are still H2 and H2O, which is a valid assumption used in [Aguiar et al., 2004].

A part of the small difference between the results can also be attributed to the

difference in the calculations of the activation over-voltage between this model and

[Aguiar et al., 2004].

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Chapter 4

82

Fig. 4.1: Polarisation and Power Density Curves of the SOFC

Fig. 4.2 shows the reversible cell voltage as a function of the FU with the system

pressure as the parameter. From these curves, it is evident that the reversible cell

voltage decreases with the increase in the FU and also that increasing system pressure

results in increased Nernst voltage. However, this increase is quite small. Moreover,

high-pressure operation may lead to other complications. Therefore, the cell pressure

is kept slightly above the atmospheric pressure. It can also be seen that the reversible

cell voltage drops significantly for FU’s near the value of unity. That is why; a FU of

more than 0.9 is normally not desired [Wang et al., 2007]. On the other hand, a low

FU is economically unviable. These curves agree with similar curves given in

[Singhal & Kendall, 2003].

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Open and Closed Loop SOFC Dynamics

83

Table 4.1: Parameters Used in the Simulations Data Description Value Unit Van Volume of the anode chamber 0.0001 m3 Vca Volume of the cathode chamber 0.0001 m3

0anT Initial temperature of anode chamber gases 1073 K 0

caT Initial temperature of cathode chamber gases 1073 K 0Hp Initial partial pressure of hydrogen 22000 Pa 0Wp Initial partial pressure of water vapour 88000 Pa

0Np Initial partial pressure of nitrogen 89242.62 Pa 0Op Initial partial pressure of oxygen 20757.38 Pa

ENVp Pressure of environment 100000 Pa

ASp Pressure at air source 120000 Pa

HSp Pressure at hydrogen source 120000 Pa

ENVT Temperature of environment 300 K

AST Temperature of air source 745 K

HST Temperature of hydrogen source 745 K Ean Activation energy per mole of the anode 110000 J mol-1

Eca Activation energy per mole of the cathode 120000 J mol-1

anψ Pre-exponential co-efficient for anode 7x109 A m-2

caψ Pre-exponential co-efficient for cathode 7x109 A m-2

Rohm Ohmic resistance per unit area 0.2 Ωcm2 R Universal gas constant 8.314 J mol-1 K-1 ne Number of electrons involved in the reaction 4 - F Faraday’s constant 96493 C mol-1 Ac Cell effective area 0.01 m2 iL Limiting current 1000 A

anm Mass of anode solid 0.02950 kg

cam Mass of cathode solid 0.00295 kg

elm Mass of electrolyte solid 0.00118 kg

inm Mass of interconnect solid 0.04 kg can, cca, cel , cI1 & cI2

Common specific heat capacity of anode, cathode, electrolyte and interconnect 500 J kg-1 K-1

aλ Convection heat transfer coefficient in anode channel 2987 J m-2 s-1 K-1

cλ Convection heat transfer coefficient in cathode channel 1322.8 J m-2 s-1 K-1

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Chapter 4

84

Fig. 4.2: Characteristic Curves Showing the Variation of Nernst Voltage as a Function of FU

The power density and the polarisation curves for a cell operating at 1073K and

1bar with undepleted air (zero OU) and FU of 0.03 are shown in Fig. 4.3, where the

various internal cell voltage losses are also indicated. For the cell under consideration,

it can be seen from Fig. 4.3 that the ohmic and the activation losses are the major

losses while the concentration voltage loss is minimum. Concentration losses cause

the cell potential to drop to zero sharply with a concave curvature at a current density

called the limiting current density [Aguiar et al., 2004]. For the cell and the operating

conditions chosen in this work, no concave curvature is observed as high ohmic and

activation losses cause the cell voltage to drop to zero much before the limiting

current density ( 2L 10 A/cmj = ) is reached.

Fig. 4.3: The Polarisation Curve Showing the Contribution of Various Voltage Losses

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Open and Closed Loop SOFC Dynamics

85

The power density and polarisation curves for a cell operating at 1bar with

undepleted air and FU of 0.03 and with the operating temperature as a parameter are

shown in Fig. 4.4a. At very low current densities, the cell voltage is almost equal to

the reversible Nernst voltage. Thus at zero current density, the cell potential is higher

for lower operating temperatures. At about 0.05 Acm-2, the curves cross and the trend

is reversed (refer to Fig. 4.4b). This trend is also reported in [Petruzzi et al., 2003].

This is because of the fact that at higher current densities, the polarisation losses are

significant and they decrease with increase in the cell operating temperature. Thus,

higher operating temperatures for the cell are preferred. It can also be seen that the

power densities increase with the increase in the cell operating temperature.

(a) (b) Fig. 4.4: (a) Polarisation and Power Density Curves for Different Cell Operating Temperatures and (b)

an Enlarged Version Showing the Voltage Curves at Lower Current Densities

The power density and polarisation curves for a cell operating at 1bar and 1073K,

with undepleted air and with the FU as a parameter, are shown in Fig. 4.5. It can be

concluded from Fig. 4.5 that increase in the FU results in decrease in the cell voltage

and power density. This is primarily due to the fall in the Nernst voltage caused by the

decrease in the hydrogen partial pressure. These curves (Figs. 4.3, 4.4 and 4.5) are in

agreement with those given in [Aguiar et al., 2004].

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Chapter 4

86

Fig. 4.5: Polarisation and Power Density Curves for Different values of FU

4.2.2. Selection of the Operating Regime

Fuel cells are normally designed to operate at a cell voltage between 0.6 V and 0.7 V

because this operating range gives a good compromise between the cell efficiency, the

power density, the operating cost (FU) and stable operation. It further avoids possible

anode oxidation at low cell voltage [Chan et al., 2001]. It may not be beneficial to

operate the fuel cell at the operating point corresponding to maximum power density

as the cell voltage becomes low (about 0.5V). With reference to Fig. 4.3, the ideal

operating range of the current densities is from 1.2 Acm-2 (at 0.7V) to 1.6 Acm-2 (at

0.6V) and the corresponding range of power densities is from 0.8 Wcm-2 to 0.92

Wcm-2.

4.2.3. Dynamic Response

The dynamic response of the fuel cell to a step change in the load current is shown in

Fig. 4.6. The dynamic response study helps us in understanding the various physical

processes involved in the functioning of the fuel cell and ultimately guides us in

developing efficient control strategies to the improve SOFC system’s load following

capability.

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Open and Closed Loop SOFC Dynamics

87

The valve coefficients for the four valves are fixed at the values given by Eqs.

(4.8–4.9, 4.18–4.19) to obtain the open-loop dynamic response of the SOFC. Step

changes are made in the load current from 100A to 80A at 500s and from 80A to 90A

at 2000s. The dynamic responses of the cell voltage, current, FU and OU to the step

changes of load current, the data being normalised with respect to the initial steady

state conditions (Voltage = 0.609036 V, Current = 100 A, FU = 0.8 and OU = 0.125),

are shown in Fig. 4.6a.

When the current is decreased, the combined effect of the changes in the partial

pressures and the polarization losses results in the increase of the cell voltage. It is

observed that the cell voltage initially overshoots before settling to a steady state

value. This overshoot in the cell voltage was also reported in [Achenbach, 1995; Qi et

al., 2006]. The FU and the OU, which are proportional to the current, also decrease.

The reverse phenomena are observed with the increase in the external load current.

The sudden decrease in the load current results in the decrease in the rate of

hydrogen and oxygen consumption and the rate of water vapour formation. In other

words, the reaction rate decreases. This results in the accumulation of hydrogen and

oxygen in the chambers and hence their partial pressures increase almost abruptly. At

the same time, the partial pressure of water vapour falls, as shown in Fig. 4.6b. The

decrease in the current also results in the decrease in the polarization losses and the

decrease in the reaction entropy flow rate ( rS ) (due to reduced mass flow rates) and

thereby results in the fall of the system temperature. The reverse phenomena are

observed (at t = 2000 s in Fig. 4.6) when the load current density is increased.

It can be seen from Fig. 4.6b that the hydraulic (pressure) dynamics is much faster

than the thermal (temperature) dynamics. This was also observed in the results of

[Murshed et al., 2007; Qi et al., 2006]. Although the entropy flow due to the ohmic

losses and the reaction are directly proportional to the current, the entropy flow due to

the activation and concentration losses depend upon the gas species partial pressures.

Therefore, the change in the heat production does not happen instantaneously with the

change in the current. Further, the solid components of the SOFC have high thermal

capacities. Due to these reasons, it usually takes a long time (in the order of several

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Chapter 4

88

minutes [Achenbach, 1995; Mueller et al., 2006; Murshed et al., 2007; Qi et al.,

2006]) for the cell temperature to settle down to a steady state value after a

disturbance. On the other hand, the pressure dynamics is faster due to the reason that

the changes in the inlet and outlet mass flow rates are fast (only the small time delay

associated with valve actuation, which is not modelled in this work, can affect it).

(a) (b)

Fig. 4.6: Dynamic Response Curves for (a) Voltage, Current, FU and OU Normalised to their Initial Values and (b) Species Partial Pressures and Cell Temperature

Note that the dynamic characteristics are useful in the feedback control of the fuel

cell. The static and dynamic characteristics presented in this section show that the

model produces comparable results to various published researches [Achenbach,

1995; Aguiar et al., 2004; Chan et al., 2001; Mueller et al., 2006; Petruzzi et al.,

2003; Qi et al., 2006; Singhal & Kendall, 2003] and it captures all the essential

dynamics of the SOFC system.

4.3. Detailed Bond Graph Model for Dynamic Simulation

The true bond graph model discussed in Chapter 3 is further improved in this section

to account for certain un-modelled dynamics. These improvements shown in Fig. 4.7

include more detailed representation of the concentration over-voltage, inclusion of

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Open and Closed Loop SOFC Dynamics

89

the conduction and radiation heat transfer effects in the cell and representation of the

flow resistance by isentropic nozzle flow equation rather than a linear flow relation.

Fig. 4.7: True Bond Graph Model of the Closed-Loop SOFC System

The MEA solid is represented by three control volumes; one each for the anode,

the cathode and the electrolyte. The temperatures of the solid anode, cathode and

electrolyte control volumes are represented by the junctions ‘0an,s’, ‘0ca,s’ and ‘0el’,

respectively.

It is assumed that the entropy generated due to the ohmic resistance is added to the

solid electrolyte. The ohmic resistance is modelled by the resistive field RSO between

the 1i junction and the 0el-junction in Fig. 4.7. The inputs to this field are the current

( i ) and the electrolyte temperature ( elT ). The outputs are the over-voltage ( ohmη ) and

the entropy flow rate ( elS ), which are calculated as

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Chapter 4

90

ohm ohmiRη = (4.30)

and 2

ohmel

el

i RST

= . (4.31)

The concentration over-voltage is represented in more detail in the improved

model as follows. The concentration over-voltage is given in terms of the partial

pressures of the gas species at the flow bulk and at the TPB as (refer to Section 3.3.5

in Chapter 3)

W W

O OH H

W,b W,TPBconc

H,b O,b H,TPB O,TPB

R ln R lnp p

T Tp p p p

ν ν

ν νν νη⎛ ⎞ ⎛ ⎞

= − +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

, (4.32)

which can be rearranged to separate out the anodic and cathodic contributions as

O WH

O W H

O,b H,b W,TPBconc conc,an conc,ca

O,TPB W,b H,TPB

R ln R lnp p p

T Tp p p

ν νν

ν ν νη η η⎛ ⎞ ⎛ ⎞

= + = +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

. (4.33)

The values of the stoichiometric coefficients and ne are substituted in Eq. (4.33) to

yield [Aguiar et al., 2004],

W,TPB H,banconc,an

W,b H,TPB

R ln2

p pTF p p

η⎛ ⎞

= ⎜ ⎟⎜ ⎟⎝ ⎠

, (4.34)

and O,bcaconc,ca

O,TPB

R ln4

pTF p

η⎛ ⎞

= ⎜ ⎟⎜ ⎟⎝ ⎠

. (4.35)

The partial pressures of the gas species at the TPB are calculated as follows [Chan

et al., 2001; Kim et al., 1999; Virkar et al., 2000; Zhu & Kee, 2003]. At steady state,

the fluxes (J) of the reactants (hydrogen and oxygen) and the product (water vapour)

through the electrodes are related to the current (refer to Section 3.2 of Chapter 3) as

follows.

H O W22iJ J JF

− = = − = . (4.36)

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Open and Closed Loop SOFC Dynamics

91

Equimolar counter current diffusion of hydrogen and water vapour takes place in

the anode. For one dimensional diffusion, the hydrogen flux is given by [Chan et al.,

2001]

HH eff,an

ddcJ Dx

= − . (4.37)

As H H and dp Rc T= , Eq. (4.37) becomes

Heff,an

an

dp2 R di DF T x= . (4.38)

Integration of Eq. (4.38) yields

an anH,TPB H,b

eff,an

R2

T ip pFD

τ= − . (4.39)

Therefore the partial pressure of hydrogen at the TPB can be calculated by using

Eq. (4.39). The second term on the right hand side of Eq. (4.39) represents the

pressure loss due to the limitations imposed by the diffusion process and is

responsible for the loss in the cell voltage. Similarly, the partial pressure of water

vapour at the TPB can be obtained as

an anW,TPB W,b

eff,an

R2

T ip pFD

τ= − . (4.40)

On the cathode side, the nitrogen flux is zero. The oxygen flux is given by [Chan

et al., 2000]

O,TPB O OOO eff,ca

ca

dd

p JcJ Dx p

δ= − + . (4.41)

As O O cad d Rc p T= , Eq. (4.41) becomes

O,TPB OOeff,ca

ca ca

d4 R d 4

p ipi DF T x Fp

δ= − + . (4.42)

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Chapter 4

92

Integration of Eq. (4.42) and substitution of O 1δ = gives the expression for the

partial pressure of oxygen at the TPB as

( ) ca caO,TPB ca ca O,b

eff,ca ca

Rexp4

T ip p p pFD p

τ⎛ ⎞= − − ⎜ ⎟⎜ ⎟

⎝ ⎠. (4.43)

The activation and the concentration over-voltages at the anode are modelled by

the resistive field element anA,CRS between the 1i-junction and the 0an,s-junction in Fig.

4.7. The inputs to the resistive field are the current and the anode temperature. In

addition, this element is modulated with the signals of gas species partial pressures.

These signal bonds are not shown in the figure for maintaining the visual clarity. The

outputs of the field are the total anodic over-voltage ( anη ) and the entropy generated

( act,conc,anS ) due to the anodic over-voltages. The anodic over-voltage is given as

an act,an conc,anη η η= + . (4.44)

The entropy flow rate is obtained from the energy balance by assuming that all the

energy dissipated by the activation and the concentration losses is converted into heat.

From the energy balance, we get

act,conc,an an,s act,an conc,anS T i iη η= + . (4.45)

Substitution of the expressions for act,anη and conc,anη from Eq. (3.48) and Eq.

(4.34), respectively, into Eq. (4.45) yields

W,TPB H,b-1 anact,conc,an

e 0,a an,s W,b H,TPB

R2 R sinh ln2 2

p pi Ti iSn F i FT p p

⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠. (4.46)

Similarly, the activation and the concentration over-voltages at the cathode are

modelled by the resistive field element between the 1i-junction and the 0ca,s-junction

in Fig. 4.7, where the 0ca,s-junction represents the common temperature of the cathode

solid. The constitutive relations of this field are given as

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Open and Closed Loop SOFC Dynamics

93

ca act,ca conc,caη η η= + , (4.47)

and O,b-1 caact,conc,ca

e 0,c ca,s O,TPB

R2 R sinh ln2 4

pi Ti iSn F i FT p

⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠. (4.48)

The conduction heat transfers inside the MEA are included in the improved model

and are represented by the two-port field elements Rcd1 and Rcd2 as shown in Fig. 4.7.

The constitutive relations of the R-field element Rcd1 are given as

( )M c el an,scd1,el

M el

k A T TS

Tτ−

= (4.49)

and ( )M c el an,s

cd1,an,sM an,s

k A T TS

Tτ−

= , (4.50)

where cd1,elS and cd1,an,sS are the entropy flow rates at the electrolyte and the solid

anode sides, respectively. The constitutive relations of the R-field element Rcd2 are

similar.

As the SOFC operates at high temperatures, thermal radiation is a significant

mode of heat transfer [Murshed et al., 2007]. The radiation heat transfer between the

solid anode and the interconnect is modelled by the field element Rrd1 and that

between the solid cathode and the interconnect is modelled in Fig. 4.7 by the field

element Rrd2. The constitutive relations of the field element Rrd1 are given as

( )( )( ) ( )( )

4 4c an,s an,s I1

rd1,an,san in1/ 1/ 1

A T T TS

σ

ε ε

−=

+ − (4.51)

and ( )( )( ) ( )( )

4 4c I1 an,s I1

rd1,I1an in1/ 1/ 1

A T T TS

σ

ε ε

−=

+ −, (4.52)

where rd1,an,sS and rd1,I1S are the entropy flow rates at the solid anode and the

interconnect sides, respectively. The constitutive relations of the R-field element Rrd2

are similar.

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Chapter 4

94

The mass flow rates through the valve resistances are given by the isentropic

nozzle flow equations. The overall mass flow rate of the gas mixture is given by the

formulae for the mass flow through an isentropic nozzle [Karnopp et al., 2006] as

( )2 ( 1)

u d d

u uu

2( 1)

p p pm A xR p pT

γ γ γγγ

+⎛ ⎞ ⎛ ⎞

= −⎜ ⎟ ⎜ ⎟− ⎝ ⎠ ⎝ ⎠. (4.53)

where the valve areas are given by ( )A x Ax= by assuming linear valve characteristic,

i.e. the coefficient of discharge varies linearly with the valve stem displacement. Such

a relation has been used for modelling flow through valves in a fuel cell system and in

other process engineering systems in the literature [Caux et al., 2005; Pukrushpan et

al., 2002; Thomas, 1999; Yu et al., 2005].

4.4. Model Validation and Control Strategy Formulation

In this section, the true bond graph model of the SOFC, to which the changes detailed

in Section 4.3 are incorporated, is validated by comparing its characteristic curves

with those of [Aguiar et al., 2004]. Also, a physical model-based control strategy is

formulated for improving the dynamic performance of the SOFC during a load change.

The parameters used for simulating the detailed model shown in Fig. 4.7, other than

those given in Table 4.1, are given in Table 4.2.

Table 4.2: Parameters Used in the Simulations of the Detailed Model Data Description Value Unit σ Boltzman constant 5.6704x10-8 J m-2 s-1K-4

anε , caε , inε Emissivity constants of anode, cathode and interconnect 0.9 -

eff,anD Anode diffusion coefficient 3.66x10-5 m2 s-1

eff,caD Cathode diffusion coefficient 1.37x10-5 m2 s-1

PK PI controller proportional gain -3.44e-7 K-1

IK PI controller integral gain -1e-8 K-1 s-1

Mk MEA thermal conductivity 2 J m-1 s-1 K-1

anτ Anode thickness 500 x10-6 m

caτ Cathode thickness 50 x10-6 m

elτ Electrolyte thickness 20 x10-6 m

inτ Interconnect thickness 500 x10-6 m

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Open and Closed Loop SOFC Dynamics

95

The comparison between the polarization and the power density curves from the

developed model and from the literature [Aguiar et al., 2004] is shown in Fig. 4.8. It

is evident from the Fig. 4.8 that there is good agreement between the characteristic

curves obtained from the improved bond graph model and those from [Aguiar et al.,

2004]. This agreement is better than the one shown in Fig. 4.1, which can be

attributed to the more detailed representation of the concentration over-voltage in the

improved model.

Fig. 4.8: Comparison of characteristic Curves of the Improved Model with Data from Literature

The compliance elements and the compliance fields in the improved model of the

SOFC are also initialized in the same manner as was discussed in Section 4.2. The

mass balances for individual gas species are written so as to obtain the values of the

valve displacements which will give desired steady state operation of the SOFC. The

steady state mass balances in the anode channel are given by the Eqs. (4.1–4.3). The

gas species inlet and outlet mass flow rates are given by the isentropic nozzle flow

equations as

H H H2 ( 1)0 0i ai ai HS an anHH

H HS HSHS

2( 1)

x A p p pmR p pT

γ γ γγγ

+⎛ ⎞ ⎛ ⎞

= −⎜ ⎟ ⎜ ⎟− ⎝ ⎠ ⎝ ⎠, (4.54)

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Chapter 4

96

an an an2 γ ( 1)0 0H,an ao ao an0 an ENV ENV

H 0 00an an anan

2( 1)

w A x p p pmR p pT

γ γγγ

+⎛ ⎞ ⎛ ⎞

= −⎜ ⎟ ⎜ ⎟− ⎝ ⎠ ⎝ ⎠, (4.55)

and an an an2 γ ( 1)0 0

W,an ao ao an0 an ENV ENVW 0 00

an an anan

2( 1)

w A x p p pmR p pT

γ γγγ

+⎛ ⎞ ⎛ ⎞

= −⎜ ⎟ ⎜ ⎟− ⎝ ⎠ ⎝ ⎠. (4.56)

In a procedure similar to that followed in Section 4.2, the anode chamber inlet and

outlet valve displacements required for obtaining a steady state operation with the

desired FU and OU are obtained from the mass balance equations, respectively, as

H H H

rH

ai 2 ( 1)0 0f ai HS an anH

H HS HSHS

2( 1)

mxA p p p

R p pT

γ γ γζ γ

γ

+=

⎛ ⎞ ⎛ ⎞−⎜ ⎟ ⎜ ⎟− ⎝ ⎠ ⎝ ⎠

(4.57)

and an an an

rW

ao 2 γ ( 1)0 0W,an ao an an ENV ENV

0 00an an anan

2( 1)

mxw A p p p

R p pT

γ γγγ

+=

⎛ ⎞ ⎛ ⎞−⎜ ⎟ ⎜ ⎟− ⎝ ⎠ ⎝ ⎠

, (4.58)

subject to the condition that

0 0H,an H f0 0W,an W f

1n pn p

ζζ−

= = . (4.59)

The steady state mass balances in the cathode channel are given by the Eqs. (4.11–

4.13). The gas species inlet and outlet mass flow rates are given by the isentropic

nozzle flow equations as

air air air2 ( 1)0 0O,AS ci ci ASi air ca ca

Oair AS ASAS

2( 1)

w A x p p pmR p pT

γ γ γγγ

+⎛ ⎞ ⎛ ⎞

= −⎜ ⎟ ⎜ ⎟− ⎝ ⎠ ⎝ ⎠, (4.60)

air air air2 ( 1)0 0N,AS ci ci ASi air ca ca

Nair AS ASAS

2( 1)

w A x p p pmR p pT

γ γ γγγ

+⎛ ⎞ ⎛ ⎞

= −⎜ ⎟ ⎜ ⎟− ⎝ ⎠ ⎝ ⎠, (4.61)

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Open and Closed Loop SOFC Dynamics

97

air air air2 ( 1)0 0O,ca co co cao air ENV ENV

O 0 00air ca caca

2( 1)

w A x p p pmR p pT

γ γ γγγ

+⎛ ⎞ ⎛ ⎞

= −⎜ ⎟ ⎜ ⎟− ⎝ ⎠ ⎝ ⎠, (4.62)

and air air air2 ( 1)0 0

N,ca co co cao air ENV ENVN 0 00

air ca caca

2( 1)

w A x p p pmR p pT

γ γ γγγ

+⎛ ⎞ ⎛ ⎞

= −⎜ ⎟ ⎜ ⎟− ⎝ ⎠ ⎝ ⎠. (4.63)

The cathode chamber inlet and outlet valve displacements required for obtaining a

steady state operation with the desired FU and OU are obtained from the cathode

channel mass balance equations as

air air air

rO

ci 2 ( 1)0 0o O,AS ci AS air ca ca

air AS ASAS

2( 1)

mxw A p p p

R p pT

γ γ γζ γ

γ

+=

⎛ ⎞ ⎛ ⎞−⎜ ⎟ ⎜ ⎟− ⎝ ⎠ ⎝ ⎠

(4.64)

and ( )air air air

ro O

co 2 ( 1)0 0o O,ca co ca air ENV ENV

0 00air ca caca

1

2( 1)

mx

w A p p pR p pT

γ γ γ

ζ

ζ γγ

+

− ×=

⎛ ⎞ ⎛ ⎞−⎜ ⎟ ⎜ ⎟− ⎝ ⎠ ⎝ ⎠

, (4.65)

subject to the condition that

( )

0 0 0N,AS N,ca N

0 0 0O,AS o O,ca O1

n n pn n pζ

= =−

. (4.66)

The valve displacements of the two inlet and the two outlet valves are set

according to the Eqs. (4.57–4.58, 4.64–4.65) and the C-fields are initialized by using

the partial pressures of the gas species in the anode and the cathode channel according

to Eqs. (4.59) and (4.66) so as to obtain a steady state operation of the SOFC with the

desired FU and OU. Also, the compliance elements representing the capacities of the

solids are initialized properly. Even with these initializations, the system response will

exhibit small transients due to the thermal dynamics. A first phase simulation is run

until the transients phase is over, and the final values of the states are taken as the

initial values for the next phase of the simulation. This next phase simulation is run at

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Chapter 4

98

a steady state operating mode with the desired values of the FU and the OU. A step

change in the load current is then imposed in order to study the dynamic response.

The motivation for the proposed control strategy in order to achieve the control

objectives comes from Eqs. (4.57–4.58) and Eqs. (4.64–4.65). If the temperature is

assumed to be constant, it is evident that the valve displacements given in these

equations are only functions of the current density. This means that if all the valve

displacements are varied proportionally to the current (in the case of a change in load

current) then the steady state operation with the desired values of OU, FU and channel

pressures will be maintained. Note that the reaction mass flow rates are functions of

current only. But the temperature of the gases in the channels will not remain constant,

i.e., they would vary from their initial steady state values when the load current

changes. Hence, it is necessary to measure both the chamber gas temperatures. The

control strategy proposed in this work consists of a primary controller, which

simultaneously controls the four valves by varying their valve displacements in order

to maintain constant FU and OU.

However, the above-mentioned control action does not ensure constant cell

temperature. The usual method followed in literature [Aguiar et al., 2005] for

controlling the SOFC temperature is the manipulation of the excess air supplied to the

cell. This method is also adopted in this work to maintain a constant temperature of

the SOFC despite load changes. For this purpose, a secondary PI temperature

controller, which manipulates the air ratio around the value set by the primary

controller, is added. According to [Aguiar et al., 2005], the maximum and minimum

limits of the OU are set as 0.5 and 0.07, respectively. The cathode chamber gas

temperature is compared with the set point value of the temperature and the objective

of the PI controller is to reduce the temperature error signal by manipulating the flow

through cathode chamber inlet and outlet valves by means of varying their valve

displacements. It is assumed that the economic cost of increased airflow is

insignificant with respect to other operational costs.

The outputs of the primary controller are the four valve displacements, which are

given as

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Open and Closed Loop SOFC Dynamics

99

H H H

H Hai 2 ( 1)0 0

f ai e HS an anH

HS HSHS

2( 1)H

i MxA n Fp p p

R p pT

γ γ γ

ν

ζ γγ

+=

⎛ ⎞ ⎛ ⎞−⎜ ⎟ ⎜ ⎟− ⎝ ⎠ ⎝ ⎠

, (4.67)

an an an

W Wao 2 ( 1)0 0

W,an ao e an an ENV ENV0 0

an an anan

1000 2( 1)

i Mxw A n Fp p p

R p pT

γ γ γ

ν

γγ

+=

⎛ ⎞ ⎛ ⎞−⎜ ⎟ ⎜ ⎟− ⎝ ⎠ ⎝ ⎠

, (4.68)

air air air

1 O Oci 2 ( 1)0 0

o O,AS ci e AS air ca ca

air AS ASAS

1000 2( 1)

i Mxw A n Fp p p

R p pT

γ γ γ

ν

ζ γγ

+=

⎛ ⎞ ⎛ ⎞−⎜ ⎟ ⎜ ⎟− ⎝ ⎠ ⎝ ⎠

(4.69)

and ( )air air air

o O O1co 2 ( 1)0 0

O,ca o co e ca air ENV ENV0 0

air ca caca

1

1000 2( 1)

i Mx

w A n Fp p pR p pT

γ γ γ

ζ ν

ζ γγ

+

−=

⎛ ⎞ ⎛ ⎞−⎜ ⎟ ⎜ ⎟− ⎝ ⎠ ⎝ ⎠

. (4.70)

The secondary controller acts only on the cathode inlet and outlet valves by means

of manipulating their valve displacements ( cix and cox ) in order to vary the OU. The

modified expression for the cathode inlet valve displacement is given by the sum of

the displacements due to the primary controller ( 1cix given by Eq. (4.69)) and the

output of the PI controller. Note that the PI controller proportional and the integral

gains are negative values in Table 4.2. This is because the error input of the PI

controller is defined as the difference between the reference or set point temperature

and the cathode channel gas temperature (refer to Fig. 4.7). Therefore, a positive

value for the error means that the cell temperature is less than the controller set point

temperature. In order to increase the temperature of the cell, the air flow rate should

be decreased which requires that the cathode inlet valve displacement should also be

reduced. The negative values of the proportional and the integral gains make the

output of the PI controller denoted by 2cix (refer to Fig. 4.7) negative thereby

decreasing the overall cathode channel inlet valve displacement ( cix ). The controller

works in the opposite way when the cell temperature becomes more than the set point

temperature.

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Chapter 4

100

Once the cathode inlet valve displacement has been changed, the cathode outlet

valve displacement given by Eq. (4.70) is no longer valid. It becomes necessary to

obtain a modified expression for the cathode outlet valve displacement, which along

with the modified cathode inlet valve displacement will satisfy the mass balance

equations given in Eqs. (4.1–4.3) and (4.11–4.13), thereby giving the desired pressure

and OU. The relation between cix and cox , subject to the constraints of the mass

balance equations, can be obtained by equating the oζ ’s in Eq. (4.64) and Eq. (4.65).

From this relation, the modified formulae to calculate the cathode outlet valve

displacement is given by

air air air

air air air

2 ( 1)0 0rci AS air ca ca O O

ci O,AS Oair AS AS eAS

co 2 ( 1)00 co ca air ENV ENVO,ca 0 0

air ca caca

2( 1) 1000

2( 1)

A p p p i Mx w mR p p n FT

xA p p pw

R p pT

γ γ γ

γ γ γ

γ νγ

γγ

+

+

⎛ ⎞ ⎛ ⎞− −⎜ ⎟ ⎜ ⎟− ⎝ ⎠ ⎝ ⎠=

⎛ ⎞ ⎛ ⎞−⎜ ⎟ ⎜ ⎟− ⎝ ⎠ ⎝ ⎠

.(4.71)

Note that the measured value of cix must be used in the above expression (i.e., the

total valve displacement of the primary and the secondary controllers). The block

diagram of the above control strategy, with both the primary and secondary

controllers, is shown in the model given earlier in Fig. 4.7, where the outputs of the

blocks FC1, FC2, FC3 and FC4 are given by Eqs. (4.67), (4.68), (4.69) and (4.71),

respectively.

4.5. Open and Closed Loop Dynamic Responses

The dynamic response of the fuel cell to a step change in the load current with and

without the control system is discussed in this section. The open loop dynamic

responses of the gas species partial pressures, the pressure difference between the

cathode and the anode gas chambers and the temperature to the step changes in load

current are shown in Fig. 4.9a. In order to show the details of the transients, the same

curves are shown with a smaller time scale in Fig. 4.9b. The dynamic responses of the

cell voltage, current, the FU and the OU to the step changes of load current are shown

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Open and Closed Loop SOFC Dynamics

101

in Fig. 4.10, the data being normalized with respect to the initial steady state

conditions (Voltage = 0.602663 V, Current = 100 A, FU = 0.8 and OU = 0.125).

All the dynamic system responses are very similar to the ones given in Section

4.2.3 with small changes in the steady state values. The transient thermal response is

similar for both the cases with nearly the same time constants. The voltage responses

are also similar with both cases exhibiting the characteristic over-shoot. However, the

dynamic pressure responses of the improved model shown in Fig. 4.9 has a larger

time constant as compared to that shown in Fig. 4.6b. This is due to the nonlinear

relation used to model the valve flow resistance in the improved model. Because of

this pressure dynamics, the FU and OU responses also exhibit increased response

times as shown in Fig. 4.10. The simulation time requirement (computational cost) for

the improved model is much larger than that of the model described in Section 3.4.2

of Chapter 3.

(a) (b)

Fig. 4.9: Dynamic Pressure and Temperature Responses of the Open-Loop SOFC System

Comparing the temperature response in Fig. 4.6b and Fig. 4.9, it can be concluded

that there is negligible change in both the steady state values and the response times.

The inclusion of the radiation heat transfer effects has only resulted in many fold

increase in the simulation time (computational load) without significantly changing

the dynamic response. Therefore, in future analyses, the radiation heat transfer is not

considered.

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Chapter 4

102

Fig. 4.10: Dynamic Voltage, OU and FU Responses of the Open-Loop SOFC System

The dynamic responses of the fuel cell with only the primary controller (i.e.,

without the PI temperature controller) to the same step changes in the load current are

shown Fig. 4.11. As the load current changes from 100A to 80A, the increase in the

cell voltage is less than that observed in the previous case. The FU and the OU are

maintained at their steady state values as the fuel flow and the airflow are manipulated

by the controller. There is a very little change in the partial pressures of the reactants

and the product as shown in Fig. 4.11b. The temperature of the cathode chamber

gases decreases from 1073K to 1056K. The voltage increase in this case can be

attributed partly to the temperature change (however, this part is very small) and

partly to the decrease in the cell losses due to the lower current density. It can be seen

that the pressure difference between the anode and the cathode chambers is

maintained at zero. The reverse phenomena are observed with an increase in the

external load current.

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Open and Closed Loop SOFC Dynamics

103

(a) (b)

Fig. 4.11: Dynamic Closed Loop Responses of the SOFC with Only the Primary Controller

The dynamic response of the fully controlled fuel cell (i.e. with the primary and

the secondary controllers) to the same step changes in the load is shown in Fig. 4.12.

As the load current changes from 100A to 80A, there is an increase in the cell voltage.

The FU is maintained at its initial steady state value. The OU is varied by the

controller so as to bring back the cell temperature to the initial steady state value.

There is a very little change in the partial pressures of the reactants and the product, as

can be seen from Fig. 4.12b. The voltage increase in this case can be attributed to the

decrease in the cell losses due to the lower current density. However, the pressure

difference between the anode and the cathode chambers is not zero (62 Pa). The

reason for this increase can be understood from Eq. (4.66). The term 0 0N,AS O,ASn n is

constant and N Op p varies with the variation in the OU ( oζ ). The reverse

phenomena are observed with an increase in the external load current.

From these results, it is evident that the temperature and the pressure control

requirements are conflicting and some trade off between them may be required. The

maximum allowable pressure difference value depends on the membrane support and

the age of the fuel cell [Serra et al., 2005]. In the considered design, the pressure

difference, obtained from the simulations (62 Pa) is small. If, in some case, the

pressure difference turns out to be large enough then a third controller which resets

the temperature set point of the PI controller when the chamber pressure crosses a

certain limit may be added in order to retain the pressure difference within the

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Chapter 4

104

allowable limits. Additionally, the temperature control can also be accomplished by

varying the temperature of the input fuel and the air by means of external heat

exchangers. However, these aspects are not studied in this work.

(a) (b)

Fig. 4.12. Dynamic Closed Loop Responses of the SOFC with both the Primary and the Secondary Controllers

The computer simulation results show that the proposed control strategy satisfies

the control objectives and can be treated as a promising candidate for online

implementation.

4.6. Conclusions

In this chapter, the true bond graph model presented in the last chapter is properly

initialized and simulations are performed to obtain the static and the dynamic

characteristics of the SOFC. For obtaining the static characteristic curves of the

SOFC, the FU and the OU have been interpreted in terms of the partial pressures of

the gas species in the channels, for a given set of known and input parameters. In

order to study the transient response characteristics, the model is subjected to step

changes in the load current. A number of static characteristic curves for the SOFC

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Open and Closed Loop SOFC Dynamics

105

have been obtained. These curves are used to determine the operating regime of the

SOFC. The polarisation and power density curves obtained from the model are found

to agree with the data from the literature. From the results, it is found that the

developed model satisfactorily captures all the essential dynamics of the SOFC. The

physical explanations for the behaviour of the various dynamic responses are given.

Certain further improvements relating to the conduction and radiation effects in

the cell, the flow of the reactants and products, and the diffusion of the reactants and

the products through the electrodes are incorporated into the model. This revised

model is then used for control system synthesis. A physical model-based control

strategy for the SOFC which requires simultaneous control of all the four valves (two

inlet and two exhaust valves) is developed. From the simulations it is concluded that

all the control objectives involving the constraints on the FU, the OU, the cell

operating temperature and the pressure difference between the anode and the cathode

channels are achieved by the proposed control system. It is also shown that the

temperature and the pressure control objectives are conflicting requirements and some

trade off may be required between them in practice.

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Chapter 4

106

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Chapter 5

Optimisation of Operational Efficiency

Under Varying Loads

5.1. Introduction

One of the main advantages of a fuel cell system is its high-energy conversion

efficiency as compared to other conventional power sources. The efficiency of the

fuel cell system depends upon the operating conditions of the cell, which is

determined by the inputs to the system. The models of the SOFC system available in

the literature are either too simple or too detailed and hence cannot be satisfactorily

used for optimization purposes. There is a lack of a comprehensive static model of the

fuel cell system that can calculate the states of the system given the inputs so that it

can be used for system optimization studies.

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Chapter 5

108

An SOFC system is expected to operate under varying load conditions. This raises

a question as to how to control the operating conditions of the cell to achieve

maximum efficiency for all load current densities. This work aims to investigate the

optimal operating conditions of an SOFC system that will result in the maximum

energy and exergy efficiencies. For this purpose, a comprehensive steady state model

of the fuel cell is derived in this chapter from the true bond graph model of the SOFC

presented in Section 3.4.2 of Chapter 3. This steady state model can be used to

calculate the steady state values of variables such as voltage, partial pressures of the

gas species in the anode and cathode channels, temperatures of the solids and the

gases, the reaction and the outlet mass flow rates of the gas species, the fuel and the

oxygen utilizations, power and the energy and exergy efficiencies from the inputs

specified in terms of the cathode and anode inlet mass flow rates and the current

drawn by the load. The cell stack, which has the highest irreversibility as compared to

other system components, makes the prime contribution to the overall exergetic cost

in a fuel cell system. Moreover, unlike other system components, the irreversibilities

of the stack are highly dependent on the current density because the activation and the

ohmic over-voltages increase with the current density. As the true bond graph model

given in Chapter 3 (from which the steady state model of the SOFC is derived) is

developed from the second law analysis, it includes the contributions of the cell

irreversibilities to the system’s dynamics.

In a hydrogen-fed SOFC system, the anode and the cathode outlet gases go to an

after-burner, where all the remaining hydrogen is combusted, from where they pass

through two pre-heaters which are used for heating the inlet hydrogen and air streams.

Thus, the temperatures of the anode and the cathode inlet streams are not independent

variables in a fuel cell system; they depend upon the fuel cell outlet gas temperatures

and mass flow rates. In order to introduce this dependency, the after-burner and the

pre-heater models have to be included in the efficiency analysis.

A schematic representation of the SOFC system considered for this study is shown

in Fig. 5.1. The exhaust gases from the anode and the cathode sides are fed to an after-

burner where all the remaining hydrogen is combusted. The exhaust gases from the

after-burner first pass through a hydrogen heat exchanger, HX1, where they lose some

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Optimisation of Operational Efficiency Under Varying Loads

109

of their heat energy to the fuel cell inlet hydrogen stream and then they enter an air

heat exchanger, HX2, where they heat up the fuel cell inlet air and are then released

into the atmosphere.

Fig. 5.1: A Schematic of the SOFC System Considered in This Work

Commercializing the SOFC requires efficient control systems for maximizing the

fuel efficiency of the system. Maximizing the operating efficiency of the SOFC under

varying load conditions and achieving smooth operation during load changes are

important issues in SOFC system control. Research on such issues is being actively

pursued ( [Aguiar et al., 2005; Golbert & Lewin, 2007; Hasikos et al., 2009; Mueller

et al., 2006; Stiller et al., 2006]). The former is achieved through complex control

methods like model predictive control whereas the latter is usually achieved by

maintaining a constant fuel concentration within the cell. However, the implications

of maintaining constant FU on the efficiency of the SOFC system have not been

investigated so far.

In this work, an algorithm, which uses the comprehensive steady state model of

the SOFC system in order to determine the cell operating conditions and control laws

corresponding to the maximum system efficiency, is developed. In the literature

[Aguiar et al., 2005; Mueller et al., 2006; Singhal & Kendall, 2003; Stiller et al.,

2006], maintaining constant FU and constant cell temperature is advocated as the

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Chapter 5

110

desirable operating condition for an SOFC. Constant FU operation of the fuel cell is

recommended because it minimises the dynamics during load changes [Mueller et al.,

2006] and also avoids uneven distribution of voltage and temperature within the cell

[Stiller et al., 2006]. The constant temperature operation is recommended in order to

avoid thermal cracking [Stiller et al., 2006]. However, the implications of maintaining

these operating conditions on the efficiency of the SOFC system have not been

investigated in any work. This analysis is carried out in this chapter by comparing the

dynamic responses of the SOFC system operated with three different control

strategies: (1) maximum efficiency operation, (2) a suitably chosen constant FU

operation, which approximates the maximum efficiency operation and (3) constant

FU and constant cell-temperature operation.

5.2. Steady State Model of the SOFC System

In this section, the steady state model of the SOFC is obtained from the previously

developed true bond graph model. The differential equations representing the SOFC

are written down from the bond graph. The steady state model which is derived from

the dynamic model is discussed in detail. The formulations of the steady state models

of the after-burner and the heat exchangers are also given.

5.2.1. The Dynamic Equations of the SOFC Bond Graph

Model

The differential equations obtained from the previously developed true bond graph

model, which represent the dynamics of the SOFC system, will be used in this section.

The improvements incorporated into the model in chapter 4 produce only small

changes in the system steady states and in the transient responses, but they also make

the simulation very slow. Therefore, those modifications are not considered in the

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111

present analysis. However, some modifications to the SOFC model, which are

described below, are required so as to incorporate it into the circuit shown in Fig. 5.1.

For the study detailed in this chapter, the anode and the cathode inlet mass flow

rates are considered as the input variables which are to be imposed on the system. On

the other hand, the anode and the cathode channel outlet mass flow rates are decided

by the pressure differences between the cell and the after-burner and are given by

linear nozzle flow equations. This is why the inlet flow resistances are represented by

MR-field elements whereas the outlet flow resistances are represented by R-field

elements in Fig. 5.2. Also, the intensive variables at the channel inlets and outlets are

not constants but are variables corresponding to the heat exchanger outlets and the

after-burner inlet, respectively. Therefore, these are represented by modulated effort

sources in Fig. 5.2.

Fig. 5.2: True Bond Graph Model of the SOFC

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Chapter 5

112

The mass balances for the two gas species at the anode channel gases control

volume, which are given by the zero junctions H

0μ and W

0μ in the Fig. 5.2, are

i r oH H H Hm m m m= − − (5.1)

and r oW W Wm m m= − , (5.2)

where the inlet mass flow rate of hydrogen is a controlled and measured input to the

system.

The entropy balance at the anode channel gases control volume, which is

modelled by the an

0T junction in Fig. 5.2, is

i oan an an 1S S S S= − − . (5.3)

The expansions for the various terms in the Eqs. (5.1–5.3) are discussed in the

following.

The total outlet mass flow rate from the anode channel that is calculated in the

aoR -field in Fig. 5.2 is given by the linearised nozzle flow equation as

( )oan ao an ABm K p p= − , (5.4)

from which, the outlet mass flow rates of the two constituent gases can be obtained as

( )oH H,an ao an ABm w K p p= − (5.5)

and ( )oW W,an ao an ABm w K p p= − . (5.6)

The total pressure of the gases at the anode channel is represented as a function of

the temperatures and the chemical potentials of the two gas species (refer to Eqs.

(3.80–3.81) in Section 3.4.1 of Chapter 3) as

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113

H Wv v

H W00

0 0an W W anH Han H W0 0 0 0

an H an H an an W an W an

exp exp

c cR RT Tp p p

T R T R T T R T R Tμ μμ μ⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞

= − + −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠

. (5.7)

The mass flow rates of the hydrogen consumed in the reaction and that of the

water vapour produced due to the reaction are functions of the current drawn from the

cell, which is a system input (i.e., measured), and are given (in kg/s) by (refer to the

1n junction and the transformer structure in Fig. 5.2) the charge balance equations

(refer to Section 3.2 of Chapter 3):

r H HH

e1000i Mm

n Fν

= (5.8)

and r W WW

e1000i Mm

n Fν

= . (5.9)

Note that the molar masses are given in grams per mole and thus the factor 1000

appears in the denominator of Eqs. (5.8–5.9) for conversion of mass flow rates from

grams per second to kilograms per second. The same factor appears in some other

equations (e.g., Eq. (5.27) and Eq. (5.48)) in this work precisely due to the same

reason.

The entropy flow rate into the anode channel control volume, ianS , which is

calculated in the MRai field shown in Fig. 5.2, is given by the sum of the convective

entropy flow due to the inlet mass flow and the entropy generated due to the enthalpy

difference between the inlet and the anode channel gases as (refer to Section 3.4.1 of

Chapter 3)

i ian H ai gen,aiS m s S= + , (5.10)

where the entropy generated at the anode inlet is given by

( )( )iH H,ai H,an ai ai an

gen,aian

m s T TS

Tμ μ− + −

= (5.11)

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Chapter 5

114

and the specific entropy of the anode side upstream gas is

Hp H,ai

aiai

cs

Tμ−

= . (5.12)

The entropy flow out of the anode channel volume, which is calculated in the Rao

field shown in Fig. 5.2, is

oo o an anan an an

H W

m SS m sm m

= =+

. (5.13)

From the effort balance at the an

0T junction in Fig. 5.2, it can be seen that the

entropy flow rate due to the convective heat transfer with the solids (MEA and the

interconnect) is

1 2 3S S S= + , (5.14)

where 2S and 3S are given by the constitutive relations of Rcv2 and Rcv1, respectively,

in Fig. 5.2 as (refer to Section 3.3.5 of Chapter 3)

( )an an M2

an

T TS

Tλ −

= (5.15)

and ( )an an I13

an

T TS

Tλ −

= . (5.16)

The temperature and the chemical potentials of the anode channel gases are given

by the constitutive relations for the anode channel C-field in terms of the state

variables as (refer to Section 3.3.4 of Chapter 3)

W WH HH WH W

H WH W v vv v0 WHan an 1 0 0

H W

m Rm Rm c m cm c m c mmT T X

m m

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ ++ ⎝ ⎠⎝ ⎠ ⎛ ⎞⎛ ⎞

= ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

, (5.17)

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115

where

00an W WH H

H W 0 H 0 W 0 H 0 WH v W H H H W H W W Wv v v v v

1

S m Sm Sm c m c m m c m m c m m c m m c

X e

⎛ ⎞⎜ ⎟− −⎜ ⎟+ + +⎝ ⎠= ,

( ) H,an0 0H,an H an H,an H an 0

H,an

, lnp

T p R Tp

μ μ⎛ ⎞

= + ⎜ ⎟⎜ ⎟⎝ ⎠

(5.18)

and ( ) W,an0 0W,an W an W,an W an 0

W,an

, lnp

T p R Tp

μ μ⎛ ⎞

= + ⎜ ⎟⎜ ⎟⎝ ⎠

. (5.19)

Similarly, the mass balances at the cathode channel gases control volume, given

by the zero junctions O

0μ and N

0μ in the Fig. 5.2, are

i r oO O O Om m m m= − − (5.20)

and i oN N Nm m m= − , (5.21)

where the inlet mass flow rate of the air is an controlled and measured input to the

system from which, the inlet mass flow rates of oxygen and nitrogen are obtained as i iO O,AS cam w m= and i i

N N,AS cam w m= .

The entropy balance at the cathode channel gases control volume, which is

represented by the ca

0T junction in Fig. 5.2, is

i oca ca ca 8S S S S= − − . (5.22)

The expansions for the various terms in the Eqs. (5.20–5.22) are discussed in the

following.

The total outlet mass flow rate from the cathode channel which is calculated in

the Rco-field element in Fig. 5.2 is given by the linearized nozzle flow equation as

( )oca co ca ABm K p p= − , (5.23)

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Chapter 5

116

from which, the outlet mass flow rate of the two constituent gases can be obtained as

( )oO O,ca co ca ABm w K p p= − (5.24)

and ( )oN N,ca co ca ABm w K p p= − . (5.25)

The total pressure of the gases at the cathode is represented as a function of the

temperatures and the chemical potentials of the two cathode gas species (refer to

Section 3.4.1 of Chapter 3) as

O Nv v

O N0 0

0 0O O ca N N caca O N0 0 0 0

ca O ca O ca ca N ca N ca

exp exp

c cR RT Tp p p

T R T R T T R T R Tμ μ μ μ⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞

= − + −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠

. (5.26)

The mass flow rate of oxygen consumed in the reaction (refer to the 1n junction

and the transformer structure in Fig. 5.2) is given (in kg/s) by (refer to Section 3.2 of

Chapter 3)

r O OO

e1000i Mm

n Fν

= . (5.27)

The entropy flow rate into the cathode channel control volume, which is given by

the MRci field element in Fig. 5.2, is (refer to Section 3.4.1 in Chapter 3)

i ica ca ci gen,ciS m s S= + , (5.28)

where the entropy generated at the cathode inlet is

( ) ( ) ( )( )ica O,ci O,ci O,ca N,ci N,ci N,ca ci ci ca

gen,cica

m w w s T TS

T

μ μ μ μ− + − + −= , (5.29)

and the specific entropy of the cathode side upstream gas mixture is

O,ci O,ci N,ci N,ciO Nci p O,ci p N,ci

ci

w ws c w c w

Tμ μ+

= + − . (5.30)

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117

The entropy flow out of the cathode channel volume, which is calculated in the

Rco field in Fig. 5.2, is

oo o ca caca ca ca

O N

m SS m sm m

= =+

. (5.31)

From the effort balance at the ca

0T junction in Fig. 5.2, the entropy flow rate due

to the convective heat transfer with the solids (the MEA and the interconnect) is

8 9 7S S S= − , (5.32)

where 9S and 7S are given by the constitutive relations of Rcv3 and Rcv4, respectively,

in Fig. 5.2 as (refer to Section 3.3.5 of Chapter 3)

( )ca ca I29

ca

T TS

Tλ −

= (5.33)

and ( )ca ca M7

ca

T TS

Tλ −

= . (5.34)

The temperature and the chemical potentials of the cathode channel gases are

given in terms of the state variables by the constitutive relations for the cathode

channel C-field as (refer to Section 3.3.4 of Chapter 3)

( ) ( )O O N N

O N O NO N O Nv v v v

0 0 0ca ca 2 O O N N

m R m Rm c m c m c m cT T X m m m m

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟+ +⎝ ⎠ ⎝ ⎠= , (5.35)

where

0 0ca O O N N

O N 0 O 0 N 0 O 0 NO v N O O O N O N N Nv v v v v

2

S m S m Sm c m c m m c m m c m m c m m c

X e

⎛ ⎞⎜ ⎟− −⎜ ⎟+ + +⎝ ⎠= ,

( ) O,ca0 0O,ca O ca O,ca O ca 0

O,ca

, lnp

T p R Tp

μ μ⎛ ⎞

= + ⎜ ⎟⎜ ⎟⎝ ⎠

(5.36)

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Chapter 5

118

and ( ) N,ca0 0N,ca N ca N,ca N ca 0

N,ca

, lnp

T p R Tp

μ μ⎛ ⎞

= + ⎜ ⎟⎜ ⎟⎝ ⎠

. (5.37)

The entropy balance for the solid (MEA and the interconnects) control volumes

gives rise to three more differential equations as follows. The flow balance at the I1

0T

and I2

0T junctions (refer to Fig. 5.2) give (refer to Section 3.3.5 of Chapter 3)

( )an an I1I1 4

I1

T TS S

Tλ −

= = (5.38)

and ( )ca ca I2I2 10

I2

T TS S

Tλ −

= = . (5.39)

The flow balance at the M

0T junction (refer to Fig. 5.2) gives

M PL 5 6 rS S S S S= + − + , (5.40)

where 5S and 6S are the entropy flow rates due to the convection heat transfer with

the gases and are given by the constitutive relations of the Rcv2 and Rcv4 fields,

respectively, as (refer to Section 3.3.5 of Chapter 3)

( )an an M5

M

T TS

Tλ −

= (5.41)

and ( )ca ca M6

M

T TS

Tλ −

= . (5.42)

The expansions for the various terms in the Eq. (5.40) are discussed in the

following. The entropy flow rate due to the reaction, which is applied to the M

0T

junction in Fig. 5.2, is (refer to Section 3.4.2 of Chapter 3)

( ) ( ) ( )r r rW W W H H H O O O

ran an ca

m h m h m hS

T T Tμ μ μ− − −

= − − . (5.43)

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119

The temperature of solids are given in terms of the state variables by the

constitutive relations of the compliance elements MC , I1C and I2C , respectively, as

00 M M

M MM M

exp S ST Tm c

⎛ ⎞−= ⎜ ⎟

⎝ ⎠, (5.44)

00 I1 I1

I1 I1I1 I1

exp S ST Tm c

⎛ ⎞−= ⎜ ⎟

⎝ ⎠, (5.45)

and 0

0 I2 I2I2 I2

I2 I2

exp S ST Tm c

⎛ ⎞−= ⎜ ⎟

⎝ ⎠. (5.46)

The reversible cell voltage is given by the Nernst equation [Singhal & Kendall,

2003] as

re

GVn FΔ

= − , (5.47)

where GΔ is the change in the Gibbs free energy [Singhal & Kendall, 2003]:

W W W H H H O O O

1000M M MG ν μ ν μ ν μ− −

Δ = . (5.48)

The calculation of GΔ is realised through the 1n junction in Fig. 5.2.

The actual cell voltage is less than the reversible voltage due to the irreversibilities

in the system. The irreversibilities cause voltage losses called over-voltages (refer to

Section 3.3.5 of Chapter 3). The actual cell voltage is obtained by subtracting the

over-voltages from the reversible voltage (refer to 1i junction in Fig. 5.2) as

rV V η= − , (5.49)

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Chapter 5

120

where, conc o act,a act,cη η η η η= + + + . The sum of all the over-voltages, η , is calculated

in the RS field in Fig. 5.2, where the various types of over-voltages are calculated as

described in the following.

The ohmic over-voltage is given by the Ohm’s law: o ohmiRη = . Activation over-

voltage, which refers to the over potential required to exceed the activation energy

barrier so that the electrode reactions proceed at the desired rate are governed by the

Butler-Volmer equation [Qi et al., 2005], which in its general form is given as

( ) e acte act0

1exp exp

R Rn Fn Fi i

T Tβ ηβ η⎧ ⎫⎛ ⎞− −⎪ ⎪⎛ ⎞= −⎨ ⎜ ⎟⎬⎜ ⎟

⎝ ⎠⎪ ⎪⎝ ⎠⎩ ⎭. (5.50)

If the transfer coefficient ( β ) is 0.5, which is normally the case [Qi et al., 2005],

then the anodic and cathodic activation over-voltages can be obtained from Eq. (5.50)

as

1Mact,a

e 0,a

2R 0.5sinhT in F i

η − ⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠ (5.51)

and 1Mact,c

e 0,c

2R 0.5sinhT in F i

η − ⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠, (5.52)

where, W anH0,a an c

amb amb M

expR

p Epi Ap p T

ψ⎛ ⎞⎛ ⎞ ⎛ ⎞−

= ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠⎝ ⎠

and 0.25

O ca0,c ca c

amb M

expR

p Ei Ap T

ψ⎛ ⎞ ⎛ ⎞−

= ⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

.

Concentration over-voltage results due to the difference between the concentration

of the gases at the electrode-electrolyte interface, where the reaction occurs, and those

in the bulk. The physical processes that contribute to its occurrence include gas

species molecular transport in the electrode pores, solution of reactants into the

electrolyte, dissolution of the products out of the electrolyte and diffusion of the

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121

reactants/products through the electrolyte to/from the reaction sites. The concentration

over-voltage is given as [Mueller et al., 2006]

Mconc

e L

R ln 1T in F i

η⎛ ⎞

= − −⎜ ⎟⎝ ⎠

. (5.53)

The entropy flow rate due to the polarisation losses (over-voltage losses), which is

calculated from energy balance in the RS-field in Fig. 5.2, is given as

21 1 ohm

PLe 0,a 0,c L M

R 0.5 0.52sinh 2sinh ln 1 i Ri i i iSn F i i i T

− −⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞

= + − − +⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠. (5.54)

The nine differential equations (Eqs. (5.1–5.3, 5.20–5.22, 5.38–5.40)), after

relevant substitutions, represent the state equations of the SOFC system consisting of

nine state variables: Om , Hm , Wm , Nm , anS , caS , I1S , I2S and MS . This model is

based upon the second law of thermodynamics analysis of the fuel cell system and

therefore, it is suitable for exergy calculations.

5.2.2. The Steady State Model of the SOFC

In this work, the optimum operating conditions of the SOFC are investigated. In order

to perform the optimisation process we need a model that should be comprehensive

(i.e., able to predict all the required variables of the system) and at the same time

should also be computationally efficient. The use of the dynamic model in the

optimisation algorithm is computationally expensive. For this purpose, it is desired to

formulate a comprehensive steady state model of the fuel cell system which can be

solved to find the steady state values of variables such as temperatures, pressures,

outlet mass flow rates, voltage, the fuel and the oxygen utilizations, power and

efficiency, given the inputs such as the anode and the cathode inlet mass flow rates

and the current drawn from the fuel cell. This model could then be used to study the

optimum operating conditions of the SOFC system.

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Chapter 5

122

The steady state model of the SOFC system can be obtained from the dynamic

model by means of omitting the dynamic (derivative) terms from its system equations.

It results in a set of algebraic equations, which can be solved to obtain the steady state

values of the state variables, using which, the steady state values of other variables of

interest can be calculated. In a bond graph model, this procedure translates to

assignment of derivative causalities to storage elements (C and I-elements) and

replacement of the differentially causalled storage elements by appropriate null

sources. The resulting model has a junction structure determined solely by the

resistive elements and sources. However, this procedure sets up a set of algebraic

loops between the resistive elements in the model. The objective is to assign

causalities in such a way that the junction structure becomes explicitly solvable.

However, it is impossible to assign such causalities in our case due to the complex

non-linearities involved in the SOFC model. The other approach is to directly write

down the constraints and solve them implicitly. When this procedure is applied to the

fuel cell dynamic model described earlier, the resulting algebraic equations became

too complicated. Moreover, they could not be solved by sophisticated numerical

methods such as those available in Matlab software. This is the reason why a steady-

state model of a fuel cell unit has not been developed so far. It is found in the

literature that either a full model is recursively simulated or an over-simplified steady

state model is used in all fuel cell optimisation problems [Chen et al., 2006; Cownden

et al., 2001; Douvartzides et al., 2003; Frangopoulos & Nakos, 2006; Hussain et al.,

2005; Na & Gou, 2007; Palazzi et al., 2007; Saidi et al., 2005; Subramanyan &

Diwekar, 2007; Yeh & Chen, 2008].

This is the motivation to develop an alternative simpler formulation of a steady

state SOFC model. In this work, first, the partial pressures are calculated and the

chemical potentials and various specific quantities are expressed as empirical

functions of the known partial pressures and the unknown temperatures. Then the

resulting implicit equations for unknown temperatures are solved. Later, it is

shown/validated that the steady state model obtained through this simplification

approach indeed gives good predictions.

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123

It is possible to calculate the anode and the cathode channel mass fractions and the

partial pressures directly from the system inputs and known parameters as follows.

At steady state, Eq. (5.1) becomes

i r oH H H 0m m m− − = . (5.55)

Use of the expression for the outlet mass flow rate of hydrogen in the above

equation gives

( )o i rH H H H,an ao an ABm m m w K p p= − = − . (5.56)

From Eq. (5.2) and the expression for the water vapour outlet mass flow rate, we

get

( )oW W,an ao an ABm w K p p= − . (5.57)

Because of the fact that H,an W,an 1w w+ = , Eq. (5.57) may be written as

( ) ( )rW H,an ao an AB1m w K p p= − − . (5.58)

By eliminating H,anw from Eqs. (5.56) and (5.58), we obtain an expression for anp

as

r i rao AB W H H

anao

K p m m mpK

+ + −= . (5.59)

By substituting Eq. (5.59) into Eq. (5.56), and noting that H,an W,an 1w w+ = , we

arrive at

i rH H

H,an r i rW H H

m mwm m m

−=

+ − (5.60)

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Chapter 5

124

and rW

W,an r i rW H H

mwm m m

=+ −

. (5.61)

The partial pressures of hydrogen and water vapour inside the anode channel are

then calculated as

H,an W anH

W H,an H W,an

w M pp

M w M w=

+ (5.62)

and W,an H anW

W H,an H W,an

w M pp

M w M w=

+. (5.63)

Similarly, at steady state, Eq. (5.20) becomes

i r oO O O 0m m m− − = . (5.64)

Using the expression for the outlet mass flow rate of oxygen in the above equation

gives

( )o i rO O O O,ca co ca ABm m m w K p p= − = − . (5.65)

From Eq. (5.21) and the expression for the nitrogen outlet mass flow rate, we get

( )oN N,ca co ca ABm w K p p= − . (5.66)

From H,an W,an 1w w+ = , Eq. (5.66) may be written as

( ) ( )oN O,ca co ca AB1m w K p p= − − . (5.67)

Eliminating O,caw from Eqs. (5.65) and (5.67), we obtain an expression for cap as

i i rco AB N O O

caco

K p m m mpK

+ + −= . (5.68)

By substituting Eq. (5.68) into Eq. (5.67), and noting that O,ca N,ca 1w w+ = , we get

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125

i rO O

O,ca i i rN O O

m mwm m m

−=

− − (5.69)

and iN

N,ca i i rN O O

mwm m m

=− −

. (5.70)

The partial pressures of the oxygen and the nitrogen inside the cathode channel

are then calculated as

O,ca N caO

N O,ca O N,ca

w M pp

M w M w=

+ (5.71)

and N,ca O caN

N O,ca O N,ca

w M pp

M w M w=

+. (5.72)

Now that the steady state values of the partial pressures of the gas species are

known, we just need to calculate the steady state temperatures so that the values of all

the variables of interest can be computed. To find the temperatures, anT , caT and MT ,

the steady state forms of the state equations (Eqs. (5.3, 5.22 and 5.40)) are used.

The entropy balance equations (Eqs. (5.3, 5.22 and 5.40)) are written in terms of

the temperatures and the pressures instead of the dynamic model’s state variables. The

chemical potentials and the specific enthalpies and entropies encountered in the

expressions are represented as polynomial functions of the temperatures and the

pressures [Benson, 1977], whose coefficients are calculated from the spectroscopic

data.

The chemical potential is calculated from

( )00

ln pT RTp

μ μ⎛ ⎞

= + ⎜ ⎟⎝ ⎠

, (5.73)

where ( )0 Tμ is purely a function of the temperature. The values of ( )0 Tμ for the

individual gas species are calculated as:

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Chapter 5

126

( ) ( )2 43

3 540 1 1 2 6R ln

2 3 4 oa T a Ta TT T a a T a T aμ μ

⎡ ⎤⎛ ⎞= − − + + + − +⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦. (5.74)

The specific enthalpies of the gases are calculated as

( )2 3 4 51 2 3 4 5 0Rh a T a T a T a T a T h= + + + + + (5.75)

and the specific entropies of all the gas species are calculated as

2 3 41 2 3 4 5 1 6 0

0

3 4 5R 2 2 ln ln2 3 4

ps a a T a T a T a T a T a sp

⎛ ⎞⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + + + + − − − +⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠,(5.76)

where the coefficients a1,…,a6 are taken from [Benson, 1977].

Note that the temperatures I1T and I2T become equal to anT and caT , respectively,

at steady state and hence the unknowns in Eqs. (5.3, 5.22 and 5.40) are the three

temperatures anT , caT and MT . Thus, we arrive at a set of three algebraic equations in

three unknown variables anT , caT and MT , which is lot simpler to solve than the

original set of nine equations (Eqs. (5.1–5.3, 5.20–5.22, 5.38–5.40)) given in terms of

nine state variables. The new set of equations can now be solved numerically in order

to find the fuel cell temperatures. Using these, the other variables of interest such as

the voltage, power, OU, FU, system efficiencies etc. can be calculated.

The cell voltage is calculated by using the Eqs. (5.49–5.53) given in the last

section. In the literature, the FU and the OU are considered to be the most important

variables that have to be controlled during the operation of the fuel cell [Aguiar et al.,

2005]. The FU may be given as

rH H H

f i iH e H1000

m i Mm n Fm

νζ = = . (5.77)

The OU of the fuel cell may be quantitatively given as

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127

rO O O

o i iO e O1000

m i Mm n Fm

νζ = = . (5.78)

5.2.3. Validation of the Steady State Model of the SOFC

In this section, the developed steady state model of the SOFC is validated by

comparing the steady state model results with the simulation results from the dynamic

model. The dynamic model of the SOFC, which was described in Section 5.2.1, is

used to perform the simulations. The relevant parameter values are given in Table 5.1.

The input variables have been arbitrarily selected as follows: ianm = 1.30781 610−×

kg/s; icam = 2.84754 410−× kg/s; aiT = 700 K; ciT = 700 K; 80i = A and 100 A (two

cases). Corresponding to these input variables, the steady-state model predicts the

following steady-state values for case 1 ( 80i = A): an 926 622T .= K, ca 929 202T .= K,

M 928 015T .= K, O 21247 5p .= Pa, N 88812 4p .= Pa, H 39085 5p .= Pa,

W 69188 7p .= Pa and 0 63888V .= V. For case 2 ( 100i = A), the predicted steady-

state values are an 1017 251T .= K, ca 1020 733T .= K, M 1019 019T .= K,

O 20757 4p .= Pa, N 89242 6p .= Pa, H 22136 1p .= Pa, W 87866 1p .= Pa and

0 59161V .= V.

The objective of the validation step is to verify whether the dynamic responses

converge to the predicted steady-state values for any arbitrarily chosen initial

conditions. Two different initial condition sets were tried out to check the

independence of the steady state response to the initial values. In the first scenario, the

following arbitrary initial conditions were chosen: 0an 0 272576S .= J/K,

0ca 0 285695S .= J/K, 0

M 20S = J/K, 0I1 20S = J/K, 0

I2 20S = J/K, 0 5N 2 73347 10m . −= × kg,

0 6O 7 2627 10m . −= × kg, 0 7

H 4 84941 10m . −= × kg and 0 5W 1 73338 10m . −= × kg. These

values correspond to the following initial partial pressures and temperatures: 0O 20757 38p .= Pa, 0

N 89242 62p .= Pa, 0H 22000p = Pa, 0

W 88000p = Pa, 0an 1100T = K,

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128

0ca 1100T = K and 0

M 1100T = K. The second set of initial conditions was chosen as

follows: 0an 0 321541S .= J/K, 0

ca 0 339347S .= J/K, 0M 20S = J/K, 0

I1 20S = J/K,

0I2 20S = J/K, 0 5

N 3 34091 10m . −= × kg, 0 6O 8 87664 10m . −= × kg, 0 7

H 5 92706 10m . −= × kg

and 0 5W 2 11858 10m . −= × kg, which correspond to the following initial partial

pressures and temperatures: 0O 20757 38p .= Pa, 0

N 89242 62p .= Pa, 0H 22000p = Pa,

0W 88000p = Pa, 0

an 900T = K, 0ca 900T = K and 0

M 900T = K.

Comparison between the results of the dynamic and the steady state models are

shown in Fig. 5.3. In the Fig. 5.3a, the cathode temperatures obtained from the steady

state and the dynamic model are plotted for the two different values of the current.

The corresponding voltage responses of the dynamic and the steady state models are

shown in Fig. 5.3b. It can be seen from Fig. 5.3 that the results of the steady state

model agree with that of the dynamic model with very small differences and that the

steady state values are independent of the initial conditions. The small differences

may be attributed to the approximations we have used to derive the steady state model,

namely the use of the polynomial functions of the temperatures and the pressures

[Benson, 1977] (as obtained from the spectroscopic data) to calculate the chemical

potentials, the specific enthalpies and entropies.

(a) (b)

Fig. 5.3: Comparison of the (a) Cathode Temperature and the (b) Voltage of the SOFC Between the Dynamic and the Steady state Models

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129

5.2.4. Steady State Models of the After-burner and the Heat

Exchangers

A simple steady state model of the after-burner is used in this work. This model

assumes complete combustion of hydrogen at constant pressure [Murshed et al.,

2007]. Moreover, as the pressure loss in the after-burner volume is very small, it has

negligible effect on the after-burner outlet temperature [Kandepu, 2007]. Hence the

pressure at the after-burner (pAB) is assumed to be constant and equal to 1 bar. Also,

ideal gas mixing inside the after-burner channel is assumed so that the exit gas

temperature is same as the inside temperature. By assuming that the after-burner is

well insulated so that the energy losses to the environment are negligible, the energy

balance for the after-burner control volume is

( )

abi fco abi fco abi fcoH H O O N N

abi fco fco abo abo abo abo abo aboW W H O O N N W WH

LHV

h m h m h m

h m m h m h m h m

+ + +

+ = + +. (5.79)

In Eq. (5.79), ( )HLHV means the Lower Heating Value of the reaction, which is a

synonym for the enthalpy of the combustion reactions when all the water in the

products is in the gaseous state. The lower heating values for various reactions at

atmospheric pressure and at various temperatures are taken from the tables of

experimental data [Benson, 1977].

Further assuming that all the hydrogen remaining in the fuel cell exhaust is

combusted in the after-burner, the after-burner outlet mass flow rates of oxygen

( aboOm ), nitrogen ( abo

Nm ) and water vapour ( aboWm ) can be calculated by using the

principle of mass conservation as

fcoabo fco O HO O

H2M mm m

M= − , (5.80)

fcoabo fco W HW W

H

M mm mM

= + (5.81)

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Chapter 5

130

and abo fcoN Nm m= . (5.82)

The inlet specific enthalpies of the gas species in Eq. (5.79) can be calculated by

using the previously discussed polynomial equation [Benson, 1977], where the

temperature is equal to the fuel cell outlet temperature ( FCOT ), which is assumed to be

equal to the cathode gas temperature ( caT ). The after-burner outlet specific enthalpies

are expressed in terms of the polynomial equations given in Eq. (5.75) and then Eq.

(5.79) is solved numerically in order to find the after-burner outlet temperature ( ABOT ).

The SOFC system consists of two counter-current heat exchangers, HX1 and HX2,

as shown in Fig. 5.1. Because the outlet temperatures of the cold and the hot fluids of

the heat exchangers need to be calculated from the inlet temperatures and mass flow

rates, the effectiveness-number of transfer units ( -NTUε ) method [Incropera &

Dewitt, 1996] is used to model the heat exchangers. The heat capacity rates of the

cold and hot fluids are calculated as

c c p,cC m c=∑ and h h p,hC m c=∑ . (5.83)

By comparing their values, the lower and higher values are assigned as minC and

maxC , respectively. The ratio of the heat capacity rates is calculated as r min maxC C C= .

The effectiveness of the counter-current heat exchangers can be calculated as

( )( )( )( )

r

r r

1 exp NTU 11 exp NTU 1

CC C

ε− − −

=− − −

for r 1C < (5.84)

or NTU1 NTU

ε =+

for r 1C = , (5.85)

where, minNTU UA C= , U is the overall heat transfer coefficient and A is the

effective heat transfer area of the heat exchanger.

The heat exchange rate between the hot and the cold gas streams is maxq qε= ,

where the theoretical maximum heat transfer rate is

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Optimisation of Operational Efficiency Under Varying Loads

131

( )max min h,i c,iq C T T= − , (5.86)

h,iT refers to the inlet temperature of the hot fluid, which is equal to the after-burner

outlet temperature in the case of HX1 and the hot fluid outlet temperature of HX1

( HX1h,oT ) in the case of HX2, and c,iT refers to the inlet temperature of the cold fluid to

the heat exchangers, which is 298K for both the heat exchangers HX1 and HX2.

From the energy balance, the exit temperatures of the hot and the cold fluids can

be calculated as

h,o h,ih

qT TC

= − and c,o c,ic

qT TC

= − . (5.87)

Given the anode and cathode inlet mass flow rates, the temperatures and the

current drawn by the load, the steady state values of the fuel cell outlet mass flow

rates and the temperatures of the gases can be obtained from the steady state fuel cell

model described earlier. By using the fuel cell outlet mass flow rates and the

temperatures as the inputs to the models of the after-burner and the heat exchangers

described by Eqs. (5.79–5.87), the temperatures of the cathode and the anode inlet

streams can be calculated, which will further be used for the calculation of the fuel

cell outlet stream temperatures. This forms an algebraic loop, which is why the

calculations are repeated iteratively until the difference between the fuel cell inlet

stream temperatures and the heat exchanger cold fluid outlet temperatures converges

to within a specified small value. Such recursive methods have been adopted in

various studies on exergy analysis and optimisation of fuel cell systems [Akkaya et al.,

2007; Calise et al., 2006; Chan et al., 2002; Douvartzides et al., 2003; Saidi et al.,

2005].

5.3. Optimisation of the SOFC Operating Conditions

The aim of this work is to find the optimal working conditions of the SOFC system in

terms of the first and the second law efficiencies, and to investigate the optimal

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Chapter 5

132

operating strategies of the SOFC system. A design problem is not considered here, i.e.,

it will be assumed that the fuel cell system components are available and their

parameters are fixed. The inputs to the fuel cell system model are the anode and the

cathode inlet mass flow rates and the current drawn by the load. The question that

motivated this study is, “For a given value of current drawn from the fuel cell by the

load, what values of anodic and cathodic inlet mass flow rates will result in maximum

system efficiency?”

For the SOFC system under consideration, the operating conditions solely depend

upon the current drawn by the load and the anodic and the cathodic inlet mass flow

rates. In this investigation, the current drawn from the fuel cell is considered as a

disturbance (i.e., it is not a variable that can be manipulated) as the load usually

decides it. The anode and the cathode inlet mass flow rates are considered as the

design variables, which can be manipulated in order to optimise the objective function.

The objective of this optimization study is to maximise the system efficiency, which

is defined as [Rosen, 1990]

( )en iH H

LHVVi

mη = . (5.88)

Because our model accounts for the various irreversibilities in the fuel cell, the

exergetic efficiency of the system can also be calculated as [Cownden et al., 2001;

Hussain et al., 2005; Rosen, 1990; Rosen & Scott, 1988]

ex i 0H T

Vim G

η =Δ

. (5.89)

where 0TGΔ is the standard free enthalpy of reaction at temperature T and pressure p0

= 1bar (also known as the standard Gibbs function change).

Exergy efficiency (also known as the second-law efficiency or rational efficiency)

computes the efficiency of a process by taking the second law of thermodynamics into

account. According to the second law of thermodynamics, no system can be 100%

energy efficient. Therefore, the energy efficiency does not give any indication of how

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Optimisation of Operational Efficiency Under Varying Loads

133

the system compares to a thermodynamically perfect one operating under the same

conditions. In comparison, the exergy efficiency or the rational efficiency of a system

can reach 100% because the work output is compared to the potential of the input to

do work. As for all fuels, o oG HΔ < Δ , the exergy efficiency must always be greater

than the energy efficiency.

The destruction of exergy is closely related to the creation of entropy and any

system containing highly irreversible processes will have low exergy efficiency.

Modelling the thermodynamics of the system by using the second law analysis of the

system helps in identifying the causes of entropy generation in the system. This

knowledge could be utilised to modify the system (operating conditions, geometry,

material properties, etc.) so that the irreversibilities in the system are minimised and it

operates at maximum exergy efficiency. Note that, in recent years, the second law

based modelling methods for thermodynamic optimisation known as entropy

generation minimisation (also called thermodynamic optimisation or finite time

thermodynamics) is being widely used for engineering design of thermal systems

[Balaji et al., 2007; Bejan, 1996; Bejan, 2002; Krane, 1987; Maximov, 2006; Ogulata

& Doba, 1998]. In this method, the models incorporate basic principles of

thermodynamics and heat transfer, and the optimization is subjected to finite-size and

finite-time constraints. The parameters required for the system optimization and the

simulations, excluding the ones given in the Table 4.1 in Chapter 4, are given in Table

5.1.

Table 5.1: Parameters used in the Efficiency Optimisation and Simulation Parameter Description Value Unit

h,HX1C Hot fluid thermal capacitance of HX1 0.04224 J K-1

c,HX1C Cold fluid thermal capacitance of HX1 0.41919 J K-1

h,HX2C Hot fluid thermal capacitance of HX2 0.62083 J K-1

c,HX2C Cold fluid thermal capacitance of HX2 0.73702 J K-1

ABC Thermal capacitance of after-burner gases 0.1919 J K-1 UHX1 Overall heat transfer coefficient of HX1 10 J m-2 s-1 K-1 UHX2 Overall heat transfer coefficient of HX2 10 J m-2 s-1 K-1 AHX1 Area of HX1 0.0038525 m2 AHX2 Area of HX2 0.0452986 m2

AST , HST Temperatures of air and hydrogen sources 298 K

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Chapter 5

134

5.3.1. Optimisation Algorithm

A program is written in Matlab for maximizing the system efficiency at a given value

of the load current by taking ianm and i

cam as the design variables. This program also

calculates the corresponding steady state values of the variables of interest such as the

temperatures, pressures, mass flow rates, voltage, FU, OU and power. The Matlab

function ‘fminsearch’, which finds the minimum of a scalar function of several

variables, starting at an initial estimate and using the Nelder-Mead Simplex algorithm

(refer to Appendix C), is used in this routine. In this program a cascaded optimization

is used wherein another optimization loop runs inside the main optimization loop.

The main optimization loop of the algorithm searches for the optimum values of ianm and i

cam which maximize the system efficiency for a given value of the load

current. The initial guess values of the design variables ( ianm and i

cam ) are passed on

to the function ‘fminsearch’ along with the function ‘Func 1’ which describes the

negative value of system efficiency (the objective function to be minimized). The

algorithm for the function ‘Func1’ is given in the form of a flow chart in Fig. 5.4. This

algorithm is executed every time the objective function evaluation is required for the

test points generated by the main loop (according to the Nelder-Mead Simplex

algorithm).

There are two more instances of the usage of the unconstrained multivariable

optimization function ‘fminserch’ within the function ‘Func 1’. In the first instance,

shown as the predefined process ‘Func 2’ in Fig. 5.4, it is used to solve the nonlinear

algebraic equations obtained from the steady state forms of the equations (Eqs. (5.3,

5.22 & 5.40)) in order obtain the steady state fuel cell temperatures anT , caT and MT .

The objective function to be minimized therein is given as

( ) ( ) ( )2 2 2i o i o2 an an 1 ca ca 8 PL 5 6 rF S S S S S S S S S S= − − + − − + + − + . (5.90)

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135

Fig. 5.4: Flow Chart of the Function ‘Func 1’ Used in the Optimization Routine

The second instance of its use is shown as the predefined process ‘Func 3’ in Fig.

5.4, where it is used for obtaining the after-burner outlet stream temperature ( ABOT ) by

Calculate oHm , o

Wm , oOm , o

Nm , Hp , Wp , Op , Np , H,anw , W,anw ,

O,anw & N,anw using Eqs. (5.55–5.72)

Start

Assign initial guess values to aiT , ciT , HX1c,oT & HX2

c,oT

Is HX1ai c,o 0.1T T− <

and HX2ci c,o 0.1T T− <

No

Set HX1ai c,oT T= and HX2

ci c,oT T=

Yes

Assign initial guess values to anT , caT & MT

Func 2

Set FCO caT T=

Calculate aboOm , abo

Wm & aboNm using Eqs. (5.80–5.82)

Assign initial guess values to ABOT

Func 3

Calculate V , fζ , oζ & exη using Eqs. (5.47–5.53, 5.77–

5.78, 5.89)

Define ( )( )i1 H H

F Vi m LHV= −

Stop

Calculate HX1c,oT & HX2

c,oT using Eqs. (5.83–5.87)

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Chapter 5

136

solving Eq. (5.79). The objective function to be minimized therein is given in least

squares form as

( ) 2fco abi fco abi fco abi fcoH H H O O N NH

3 abi fco abo abo abo abo abo aboW W O O N N W W

LHVm h m h m h mF

h m h m h m h m

⎛ ⎞+ + += ⎜ ⎟⎜ ⎟+ − − −⎝ ⎠

. (5.91)

For a specific value of the current density (e.g., 1 A/cm2), the initial estimates for

the optimisation routine were obtained through trial and error. In this trial and error

approach, both the above-mentioned optimisation routines and a dynamic simulation

model described later were used. Once an optimal solution was obtained for that

specific value of the current density, in the next step, i.e., for the next value of the

current density (e.g., 0.99 A/cm2 and 1.01 A/cm2), the initial guess values were taken

to be the solutions obtained from the previous step. This way of successive

approximation of initial guess values reduced the convergence time of further

optimisations by many folds.

5.3.2. Discussions on the Results of the Efficiency

Optimization Study

The results obtained from the efficiency optimization study are summarised in the

following. The cathode and the anode inlet mass flow rates that give the maximum

system efficiency for various current densities are shown in Fig. 5.5a. At low current

densities, lower inlet mass flow rates give maximum efficiency; whereas, at higher

current densities, higher inlet mass flow rates are required. This is expected because at

higher current densities, the reaction rate is higher and hence more amounts of

reactants are consumed, which have to be replenished by the inlet flow. Several

publications in the literature [Aguiar et al., 2005; Mueller et al., 2006] recommend

constant FU operating strategy for the SOFC in order to minimise the transients

during the load changes. An interesting observation from Fig. 5.5a is that the anode

inlet mass flow rate that gives maximum system efficiency has an almost linear

relationship with the current density. This observation served as a motivation for

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137

investigating whether the constant FU operation of the SOFC could closely

approximate the operation with maximum efficiency operating conditions. This is

because the FU of an SOFC can be maintained constant if the anode inlet mass flow

rate is varied proportionally to the current (this can be seen from Eq. (5.77)).

Fuel cells are normally designed to operate at a cell voltage between 0.6V and

0.7V, as this range of operation is found to be a good compromise between the cell

efficiency, the power density, capital cost, stability of the operation and avoids

(a) (b)

(c) (d)

Fig. 5.5: SOFC System Operating Conditions for Maximum Efficiency and Constant FU Operations

Showing (a) the Anode and the Cathode Inlet Mass Flow Rates, (b) the FU and the OU, (c) Voltages and

Power Densities and (d) the Pressures and the Temperatures

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138

possible anode oxidation at low cell voltage [Aguiar et al., 2005]. This operating

voltage range also offers a compromise between the cell efficiency and the power

density for the case of maximum efficiency operation also(this can be verified from

Fig. 5.5c and Fig. 5.6). From Fig. 5.5c, it can be seen that the above-mentioned

voltage range corresponds to the current densities between 0.45 to 0.88 A/cm2.

In order to compare the maximum efficiency operation and constant FU operation

of the SOFC, it is desired to obtain the linear relation between the anode inlet mass

flow rate and the current density, which will closely approximate the anode inlet mass

flow rate curve corresponding to the maximum efficiency operation within the desired

operating range (0.45 to 0.88 A/cm2). This is done by performing a least squares fit of

the data ( ianm and i ) corresponding to maximum efficiency in Fig. 5.5a within the

operating range (0.45 to 0.88 A/cm2) to a straight line passing through the origin

( ianm ci= ). Thus, the constant FU value which is expected to approximate the

maximum efficiency operation can be calculated as

H Hf

e1000Mn Fc

νζ = . (5.92)

It is found to be equal to 0.891 for the considered system. Note that the FU value

0.891 is specific for the data considered here and this value would change for

different cell and heat exchanger parameter values. The cathode inlet mass flow rate

corresponding to the maximum efficiency in Fig. 5.5a within the operating range is

fitted (least squares fit of icam vs. i ) to a cubic polynomial ( i 3 2

cam ai bi ci d= + + + ).

By using these linear and the cubic polynomial relations for the inlet mass flow rates

as inputs to the steady state SOFC system model, the operating conditions

corresponding to the constant FU operation are calculated for various current densities.

The constant FU operation imposed by the polynomials for the inlet mass flow rates is

expected to be a close enough approximation to the maximum efficiency operation in

terms of the system efficiencies and the operating conditions.

The FU and the OU for constant FU operation and the maximum efficiency

operation are compared in Fig. 5.5b. The value of the constant FU (for the system

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139

considered) at which the efficiencies of both the strategies are nearly equal is equal to

0.891, whereas the FU for the maximum efficiency operation reduces gradually with

the increase in the current density. Note that the FU value 0.891 is specific for the

data considered here and this value would change for different cell and heat

exchanger parameter values. The OU curves for both the operating strategies coincide

within the operating current density range as shown in Fig. 5.5b.

The cell voltage and the power density curves for the maximum efficiency

operation and the constant FU operation are shown in Fig. 5.5c. At low current

densities, the over-voltages are less and hence the cell voltage is large. This is also the

reason for high efficiencies observed in the low current density range. It can be seen

from Fig. 5.5c that the difference in the cell voltages and the power densities between

the maximum efficiency operation and the constant FU operation is small, more so in

the operating range. As per the definition of efficiency in Eq. (5.88), the maximum

possible power density at a particular value of the current density will result in the

maximum efficiency. At low current densities, the maximum power density is less

and at high current densities, the power density is more, which is mainly due to the

fact that the power density depends more on the current density than the voltage.

The anode pressures for both the operating strategies and the cell temperatures for

both the operating strategies are shown in Fig. 5.5d. Although the pressures are nearly

the same for both the operating strategies, the temperatures show slight deviations in

the operating range. The energy and the exergy efficiencies for the maximum

efficiency and the constant FU operating strategies are plotted in Fig. 5.6.

The exergy efficiency is higher than the energy efficiency as can be seen in Fig.

5.6, which is also reported in other literature [Chan et al., 2002; Cownden et al., 2001;

Hussain et al., 2005] for hydrogen fed SOFC systems. Within the operating range of

the fuel cell, the energy and exergy efficiencies for both the cases almost coincide.

This result is important because it establishes that if the SOFC is operated at a

particular value of FU (which depends upon the system), the efficiency of the system

remains very near to the maximum despite the changes in the load current density.

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However, it is to be noted that this applies only when the cathode inlet mass flow rate

is equal to that required for the maximum efficiency operation.

Fig. 5.6: The Energy and Exergy Efficiencies of the SOFC System for Maximum Efficiency and

Constant FU Operations

5.4. Simulation of the Dynamic Responses

Comparison of the closed loop dynamic response of the SOFC system, which is

operated by using different control strategies during a step change in the load current,

is given in this section. The true bond graph model of the SOFC shown in Fig. 5.2

along with the pseudo bond graph models of the after-burner and the heat exchangers

are used to perform the dynamic simulations of the SOFC system. The after-burner

and the heat exchangers are modelled by using the pseudo bond graph approach

[Karnopp, 1978; Thoma & Ould Bouamama, 2000] (the effort and flow variables are

T and H , respectively) as shown in Fig. 5.7. As a usual assumption, the hydraulic

storage is neglected in heat-exchanger design [Incropera & Dewitt, 1996]. It may be

assumed that the hydraulic resistance has been already modelled at the fuel cell exit.

As the after burner and the heat exchanger models are obtained from the first law

analysis, it is convenient to represent their dynamics in terms of pseudo bond graphs.

Note that there is no convenient way of deriving the after-burner model using second

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Optimisation of Operational Efficiency Under Varying Loads

141

law of thermodynamics. Therefore, the energy balance equations used in the after-

burner modelling are represented in a pseudo-bond graph. To maintain compatibility,

the heat exchanger is also modelled as a pseudo-bond graph.

The CAB element in the after-burner section marked in Fig. 5.7 represents the

thermal capacitance of the after-burner gases. The two modulated sources of flows to

the left of the CAB element give the after-burner inlet enthalpy flow rate ( abiH ) and

the enthalpy flow rate due to the hydrogen combustion reaction in the after-burner

( rH ). These flow sources are modulated by the signals of the fuel cell outlet gas

species mass flow rates and the cathode gas temperature (which come from the true

bond graph model of the fuel cell shown in Fig. 5.2). All the hydrogen gas coming out

of the fuel cell is consumed in the instantaneous combustion reaction taking place at

the after-burner, which is justified from the fact that the combustion reaction time

constant is sufficiently small as compared to the other dynamics in the system.

Fig. 5.7: Pseudo Bond Graph Models of the After-burner and the Heat Exchangers

The coupling element for thermo-fluid (CETF) systems, described in [Bouamama

et al., 2006], is extended here for multi-component gas mixture as shown in Fig. 5.8.

The after-burner outlet enthalpy flow rate, which is also the hot fluid inlet enthalpy

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Chapter 5

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flow rate of the hydrogen heat exchanger (HX1), is given by the CETF element. The

CETF element is modulated by the after-burner outlet species mass flow rates given

by Eqs. (5.80–5.82). The heat exchanger is assumed to be an ideal one in which the

pressure and the mass flow rates do not change between the inlet and the outlet sides.

This is justified from the point that SOFC systems operate at a little above the

atmospheric pressure and thus the pressure difference across the heat exchangers is

negligible. The thermal capacitances of the hot and the cold fluids of the heat

exchanger, HX1, are represented by Ch and Cc elements, respectively, in Fig. 5.7. The

capacitance of the solid wall separating the two fluids is neglected [Chantre et al.,

1994] because we are not interested in the start-up dynamics. The cold fluid inlet

enthalpy flow rate is given by the CETF element which is modulated by the anode

inlet mass flow rate signal ( ianm ), whereas the cold fluid outlet mass flow rate is given

by the MSf element which is modulated by the signals of the anode inlet mass flow

rate ( ianm ) and the temperature of the cold fluid. The heat transfer rate between the hot

and the cold fluids of the hydrogen heat exchanger (HX1) is modelled by the MR

element (modulated with mass flow rate signals) using the NTU formulation given in

Eqs. (5.83–5.86) as the constitutive relations. The air heat exchanger (HX2) is also

modelled in a similar fashion. The temperatures defined by the cold fluid thermal

capacitances (Cc) of the heat exchangers HX1 and HX2 are given as those of the

anode and the cathode inlet temperatures (Tai and Tci, respectively) to the SOFC bond

graph model shown in Fig. 5.2. The chemical potentials of the gas species at the

channel inlets ( H,aiμ , N,ciμ and O,ciμ ) are also represented as functions of these

temperatures by using Eqs. (5.73 and 5.74).

The simulations are performed with the same parameters (given in Table 5.1),

which were used in the optimisation algorithm. The aim of the simulations is to

compare the dynamic responses of the SOFC system, operated by using three

different control strategies, subjected to a step change in the load current. The first

control strategy, referred to as ‘Strategy 1’, consists of operating the fuel cell system

under the maximum efficiency conditions. This is achieved by manipulating the anode

and the cathode inlet mass flow rates according to the polynomial relations in terms of

the current, which are obtained by fitting the data (within the operating current density

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Optimisation of Operational Efficiency Under Varying Loads

143

range) resulting from the optimization program. The second control strategy, referred

to as ‘Strategy 2’, consists of operating the fuel cell under constant FU and variable

OU conditions. This is achieved by the manipulation of the inlet mass flow rates

according to the polynomial relations discussed in the Section 5.3.2.

Fig. 5.8: Expanded Form of CETF Element for Multi-component Gas Mixtures

In addition to the constant FU requirement, another requirement for the fuel cell

operation is to maintain constant cell temperature despite the changes in the load

[Stiller et al., 2006]. This is usually achieved by manipulating the cathode inlet mass

flow rate [Aguiar et al., 2005; Mueller et al., 2006; Stiller et al., 2006] (i.e.,

manipulating the OU). In this work, the constant cell temperature requirement is

achieved by means of introducing a PI controller, which manipulates the cathode inlet

mass flow rate around the value, set by using the cubic polynomial relation mentioned

earlier. Doing so will cause the cathode inlet mass flow rate to deviate from the value

corresponding to that of the maximum efficiency operation. Naturally, this will also

result in decrease of the system efficiency. The absolute values of the proportional

and the integral gains used for the PI controller are 1e-4 and 1e-10, respectively. The

constant FU and the constant temperature operation of the SOFC system, referred to

as ‘Strategy 3’, is simulated in order to investigate the effect of maintaining constant

cell temperature on the system efficiency. The parameter values used in the

simulation are the same as those given in Table 5.1.

In the simulation, the load current density is changed from 0.8 A/cm2 to 0.6 A/cm2

at 500s. The closed loop dynamic responses of the cell voltage, the temperature and

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Chapter 5

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the energy and the exergy efficiencies of the SOFC system operated by using the three

above mentioned three operating strategies are given in Figs. 5.9 and 5.10,

respectively.

The dynamic responses of the cell voltage due to the step change in the current

density for the SOFC system operated by using the three operating strategies are

given in Fig. 5.9a. It can be seen from the Fig. 5.9a that the difference in the cell

voltage between the three operating strategies is very less. The initial steady state

voltages of the cell operating using ‘Strategy 2’ and ‘Strategy 3’ coincide because the

temperature set-point for the PI controller is given as the initial steady state

temperature. The cell operation by using ‘Strategy2’ gives the highest voltage and is

closely followed by ‘Strategy 3’. Maximum efficiency operation of the fuel cell gives

the minimum voltage. Overshoot in the voltage response is observed in all three cases.

The highest overshoot is observed in the case of ‘Strategy 1’ followed by ‘Strategy 2’.

Operating the cell using ‘Strategy 3’ gives rise to very little overshoot, which means a

good transient response. From the dynamic voltage response, it can be concluded that

operating the fuel cell using ‘Strategy 3’ is better than the other two strategies as it

results in high voltage and a good transient response.

(a) (b)

Fig. 5.9: Dynamic Response Curves of (a) Voltage and (b) Temperature of the SOFC System Operated

by Using Three Different Control Strategies

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Optimisation of Operational Efficiency Under Varying Loads

145

The temperature dynamic response is given in Fig. 5.9b. Here too it is found that

‘Strategy 3’ is the best operating strategy for the fuel cell, as the temperature is

maintained constant, while ‘Strategy 1’ is the worst as it gives rise to a large change

in cell temperature.

The energy and exergy efficiencies of the fuel cell system operated using the three

operating strategies are shown in Fig. 5.10 for comparison. It can be seen that the

difference in the system efficiencies (energy and exergy) between the three operating

strategies is very less. Operating the fuel cell by using ‘Strategy 1’ gives the

maximum system efficiencies (both energy and exergy), whereas ‘Strategy 3’ gives

the lowest system efficiencies (both energy and exergy). From the results, we

conclude that for obtaining a constant FU operation, some system efficiency has to be

sacrificed. A further sacrifice of system efficiency is necessary if the cell temperature

is to be maintained constant.

Fig. 5.10: Energy and Exergy Efficiencies of the SOFC System Operated by Using Three Different

Control Strategies

‘Strategy 3’ gives the most favourable operating conditions (constant FU and

temperature) although it results in slightly decreased system efficiency (both energy

and exergy). However, this decrease in the system efficiency is very small. Thus, it

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Chapter 5

146

can be concluded that ‘Strategy 3’ closely approximates the maximum efficiency

operation of the fuel cell in terms of the system efficiencies, in addition to providing

the most favourable operating conditions, i.e. constant FU and cell temperature.

5.5. Conclusions

An algorithm has been developed in this chapter to find the optimal operating

conditions of a hydrogen fed SOFC system. This algorithm makes use of a steady

state model of the SOFC and also the steady state models of the after-burner and the

heat exchangers. For this purpose, a comprehensive steady state model of the SOFC

has been developed for the first time from the previously developed true bond graph

model. This steady state model can be solved to find the steady state values of the

variables of interest such as the temperatures, pressures, outlet mass flow rates,

voltage, fuel and oxygen utilizations, power and efficiency from the given inputs: the

anode and the cathode inlet mass flow rates and the current drawn from the fuel cell.

For a particular value of the current density, the developed algorithm maximises the

system efficiency by considering the anodic and the cathodic inlet mass flow rates as

the design variables and it also gives the values of the corresponding cell operating

conditions.

It is observed from the results of the optimization study that the anode inlet mass

flow rate corresponding to the maximum system efficiency has an approximately

linear relationship with the current density. This implies that the constant FU

operation of the fuel cell at a particular value of the FU (which depends upon the

system) can closely approximate the maximum efficiency operation of the fuel cell in

terms of the cell operating conditions as well as the energy and the exergy efficiencies.

The comparison between constant FU operation and the maximum efficiency

operation curves shows that the operating conditions of the cell are nearly the same

for both the cases. These results indicate that within the operating range of the SOFC

system, the sensitivity of the system efficiency to the variation in the FU is small.

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Optimisation of Operational Efficiency Under Varying Loads

147

The closed loop responses of the SOFC system operated by using three different

control strategies (maximum efficiency, constant FU, constant FU and temperature;

referred to as ‘Strategy 1’, ‘Strategy 2’ and ‘Strategy 3’, respectively), during a step

change in the load current are compared. It is found from the simulations that the

differences in the operating conditions between the fuel cell operated using the three

control strategies is small. It is also found that some system efficiency has to be

sacrificed to operate the fuel cell under constant FU and constant temperature

conditions. However, this decrease in the system efficiency is very small to be of any

significance. Thus, it can be concluded that ‘Strategy 3’ operation of the SOFC

closely approximates the maximum efficiency operation in terms of the system

efficiencies, in addition to providing the most favourable operating conditions, i.e.

constant FU and cell temperature. However, the constant FU value for the specific

fuel cell system has to be designed from the optimization studies and it may change

due to gradual changes to the system, e.g., degradation of the heat transfer coefficients

of the heat exchangers over long periods of operation.

Moreover, it is of interest to note that if an MEA material that has good capacity

to endure cyclic temperature stresses were developed in the future then ‘Strategy 2’

would be the preferred control law because it would give near maximum efficiency

with a very simple control law.

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Chapter 6

Conclusions

This dissertation deals with modelling the dynamics of the multi disciplinary SOFC

system through bond graphs. Control strategies for improving the SOFC’s dynamic

performance and for optimising its operational efficiency under varying loads have

also been investigated.

A zero-dimensional true bond graph model of the SOFC is developed in Chapter 3

by using the concepts of network thermodynamics. This model incorporates several

corrective modifications over the previous bond graph models of the electrochemical

reaction systems available in the literature. They include the use of internal energy as

the thermodynamic potential of the C-field used in modelling the chemical reaction

systems, the formulation and use of a C-field to model the energy storage in two

species gas mixtures encountered in an SOFC, and the extension and use of an R-field

to represent the entropy convection in the SOFC inlet and outlet gas streams. The

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Chapter 6

150

problem of differential causality, which was present in the earlier models given in the

literature, is eliminated by using the internal energy as the thermodynamic potential

for the C-field in the developed bond graph model.

This is the first time a completely true bond graph model of a fuel cell has been

developed. This model ensures the thermodynamic consistency of the system and it

clearly illustrates the physical structure of the system. The couplings between the

various energy domains in a fuel cell system have been represented in a unified

manner by using true bond graphs. The developed model is useful for designing

integrated model-based control strategies for the SOFC system.

The true bond graph model is simulated after proper initialisation and the

simulation results are given in Chapter 4. For a given set of input parameters, the

partial pressures of the gas species in the channels are related to the FU and the OU of

the SOFC by solving the steady state mass balance equations. Using these relations,

the static characteristic curves of the SOFC have been obtained for various operating

conditions. SOFC’s transient response to step changes in the load current have been

studied through simulations. It is concluded from the transient response results that

the true bond graph model developed in this work captures all the essential dynamics

of the SOFC.

Thereafter, the radiation heat transfer effects within the cell and an improved

representation of the concentration polarisation (considering diffusion of gases

through the electrodes) are included in the bond graph model. Then the improved

model is used to test a physical model-based control strategy developed in this work

to achieve the common SOFC control objectives. It is concluded from the simulations

that all the control objectives involving the constraints on the FU, the OU, the cell

operating temperature and the pressure difference between the anode and the cathode

channels are achieved by the proposed control system. It is also shown that the

temperature and the pressure control objectives are conflicting requirements and some

trade off may be required between them in practice.

In the last part of the work, a comprehensive steady state model of the SOFC has

been developed from the true bond graph model. An algorithm has been developed to

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Conclusions

151

optimise the operational efficiency of the SOFC system (consisting of the cell, the

after-burner and two recuperators) under varying load conditions. This algorithm

utilises the developed steady state model of the SOFC and also the steady state

models of the after-burner and the heat exchangers. From the optimisation results, it is

found that the constant FU operation of the fuel cell at a particular value of FU (which

depends upon the system) can closely approximate the maximum efficiency (both the

energy and the exergy efficiencies) operation of the fuel cell and at the same time it

can maintain favourable cell operating conditions, i.e. satisfy the constraints on the

temperatures and the difference between the anode and the cathode channel pressures.

Dynamic responses of the SOFC system which was operated using three different

control strategies have been compared. It is concluded that the constant FU and

constant temperature operating strategy is best for the SOFC system in terms of both

the system efficiency and the cell life. However, if an MEA material that has good

capacity to endure cyclic temperature stresses is developed in the future then the

constant FU operation would be the preferred control strategy because it would give

the near maximum efficiency with a very simple control law.

Based on this work, the following areas are suggested for further exploration.

• A one or two dimensional model of the SOFC may be constructed by

considering more lumps of the cell control volume, each lump being

represented by the model developed in this work. Such a higher dimensional

model may be used for studying the spatial distribution of the intensive

variables such as the temperature, pressure, voltage etc. under different

operating conditions.

• In this work, the electrical load part is represented by a flow source. In future

work, the electrical domain model may be extended by including the models

of the power conditioning components and various kinds of loads. Such an

extended model may be used for designing integrated model based control

strategies for the overall system.

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152

• The non-linear bond graph model of the SOFC developed in this work may

be used for state estimation so as to implement advanced control methods such

as the model predictive control for the SOFC system.

• Pure hydrogen has been considered in this work as the fuel for the SOFC.

However, the SOFC is capable of operating with other hydrocarbon fuels such

as methane. In the future, the model may be extended to include such fuels.

This will involve modelling the reforming and the shift reactions (reactions

through which the hydrocarbons are broken down to hydrogen). The

reforming reactions are endothermic reactions and when they occur inside the

SOFC, they take up the heat released due to the exothermic hydrogen

oxidation reaction. Sometimes, these reforming and shift reactions are

designed to occur at a reformer so that pure hydrogen is supplied to the cell.

Models of the reforming and the shift reactions may be included, either in the

cell or in an upstream side reformer so as to study their effect on the SOFC

system’s dynamics.

• SOFCs are sometimes coupled with gas turbines to improve the efficiency of

the system by utilising the energy in the exhaust gases. The model of a gas

turbine may be integrated with the SOFC model developed in this work to

study the dynamics of such hybrid power generation systems.

• In this work, the activation and the concentration polarisations are modelled

as voltage losses and they are represented in the electrical side of the system.

However, the activation polarisation losses can be modelled by including the

models of the charge transfer phenomena at the electrode electrolyte

interfaces, the charge transport through the electrolyte and the micro kinetics

of the electrode surface reactions. The time scales of such processes are very

small when compared to the thermal and hydraulic dynamics of the SOFC.

Therefore, coupling them with the model developed in this work and

simulating them together is a challenging task. Such multi-scale simulations

may be performed in the future. Also, detailed models of the diffusion

processes leading to the concentration over-voltage may be incorporated into

the model developed in this work.

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Appendices

Appendix A

Bond graph Notations Revisited

A.1. Introduction

In this appendix, a concise introduction to the bond graph modelling method is given.

Bond graph is a graphical tool based on the power exchanges between different

dynamical parts in a system. A bond graph model is capable of providing physical

insight into the dynamic behaviour of a system. Moreover, the notation of causality

provides a tool not only for formulation of system equations but also for qualitative

analysis of the system behaviour, viz. controllability, observability, fault diagnosis, etc.

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In 1959, Paynter [Paynter, 1961] gave the revolutionary idea of portraying

systems in terms of power bonds, connecting the elements of the physical system to

the so called junction structures which were manifestations of the constraints. Later

on, bond graph theory has been developed further by many researchers [Breedveld &

Dauphin-Tanguy, 1992; Brown, 2007; Cellier, 1991; Gawthrop & Smith, 1996;

Karnopp et al., 1990; Karnopp & Rosenberg, 1983; Mukherjee et al., 2006; Thoma,

1990] who have worked on extending this modelling technique to power hydraulics,

mechatronics, general thermodynamic systems and recently to electronics and non-

energetic systems like economics and queuing theory.

Through bond graph approach, a physical system can be represented by symbols

and lines, identifying the power flow paths. The lumped parameter elements of

resistance, capacitance and inertance are interconnected in an energy conserving way

by bonds and junctions resulting in a network structure. From the pictorial

representation of the bond graph, the derivation of system equations is so systematic

that a computer can be programmed to do it. The whole procedure of modelling and

simulation of the system may be performed by some of the existing software, e.g.

ENPORT, Camp-G, SYMBOLS, 20Sim, Dymola, etc.

The factors of power, i.e. Effort and Flow, have different interpretations in

different physical domains. Yet, power can always be used as a generalized co-

ordinate to model coupled systems residing in several energy domains. Table A.1

gives the interpretations of the effort and the flow variables in different energy

domains. In bond graphs, one needs to recognize only four groups of basic symbols,

i.e., three basic one port passive elements: inertance (I), capacitance (C), and

resistance (R); two basic active source elements: source of effort (Se or SE) and

source of flow (Sf or SF); two basic two port elements: gyrator (GY) and transformer

(TF); and two basic junctions: constant effort junction (0) and constant flow junction

(1). The basic variables are the effort (e), the flow (f), the time integral of effort (p)

and the time integral of flow (Q). In a bond graph, the assignment of power directions

may be as arbitrary as fixing the co-ordinate systems in classical analysis. The

reference power direction is shown by a half-arrow at one end of the bond. The

assignment of bond numbers fixes the names of the elements or junctions. This is a

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bookkeeping technique adopted by most of the existing bond graph modelling

software. Table A.2 gives definition of Bond graph elements with integral causality.

Note that in Table. A.2, the representation of the transformer elements are different

from that of the ones used in some other works. This notation (followed in

[Mukherjee et al., 2006]) has the advantage that it removes the dependence of the

transformer constitutive relation on the power direction as was the case in other

notations [Karnopp et al., 1990]. Rather, the constitutive relations of the transformer

only depend on the arrow placed over it but not on the power directions. When

causalities are assigned to the model, the causal forms of the two-port element’s

equations are derived from the acausal constitutive relations. The two-port’s relations

in the two possible causalled forms are given in Table A.2. This is the notation that

has been followed throughout this work.

Table A.1: Effort and Flow Variables in Different Energy Domains Domain Effort Flow

Electrical Voltage (V) Current (A)

Translational mechanics Force (N) Velocity (m s-1)

Rotational mechanics Torque (N m) Angular velocity (rad s-1)

Hydraulics Pressure (N m-2) Volume flow (m3 s-1)

Thermal Temperature (K) Entropy flow (W K-1)

Chemical Chemical potential (N m mol-1) Molar flow (mol s-1)

A.2. Causality

Causality establishes the cause and effect relationships between the factors of power.

In bond graphs, the inputs and the outputs are characterized by the causal stroke

which is shown by a short line perpendicular to the bond at one of its ends. The causal

stroke indicates the direction in which the effort signal is directed. The end of the

bond that does not have a causal stroke is the end towards which the flow signal is

directed. The proper causality for a storage element (I or C) is called integral

causality, where the cause is integrated to generate the effect. Sometimes the causal

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stroke for a storage element has to be inverted, which means the constitutive

relationship for the corresponding element is to be written in a differential form. At an

1 junction, only one bond should bring the information of flow. This uniquely

causalled bond at a junction is termed as the strong bond. Similarly, at a 0 junction,

only one bond can be stroked at the junction side; this strong bond determines the

effort at the junction. The weak bonds are the bonds other than the strong bond.

Table A.2: Definition of Bond Graph Elements

Definition Type Name Symbol Linear Nonlinear

1 df e tI

= ∫ ( )I df e tψ= ∫ Inertance

ddfe It

= Iddfet

φ ⎛ ⎞= ⎜ ⎟⎝ ⎠

1 de f tC

= ∫ ( )C de f tφ= ∫ Storages

Capacitance

ddef Ct

= Cddeft

ψ ⎛ ⎞= ⎜ ⎟⎝ ⎠

Ref /= ( )Rf eψ= Dissipation Resistance

e R f= ( )Re fφ=

Gyrator I

12 fre =

21 fre = 12 )( fxre =

21 )( fxre =

Gyrator II

12 )/1( erf =

21 )/1( erf =

( )2 11/ ( )f r x e=

( )1 21/ ( )f r x e=

Transformer I

12 ff μ=

21 ee μ= 12 )( fxf μ=

21 )( exe μ=

Transducers (ideal)

Transformer II

12 )/1( ee μ=

21 )/1( ff μ=

( )2 11/ ( )e x eμ=

( )1 21/ ( )f x fμ=

Effort (Source of effort) )(tee =

Sources Flow (Source of flow) ( )f f t=

Zero (0)

1 2 3e e e= =

1 1 2 2 3 3 0f e f e f e+ − =

2 3 1f f f= − Junctions

One (1)

2 3 1f f f= =

1 1 2 2 3 3 0f e f e f e+ − =

1 3 2e e e= −

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A.3. Activation

Some bonds in a bond graph may be only information carriers. These bonds are not

power bonds. Such bonds, where one of the factors of the power is masked are called

activated bonds. For example, in a bond representing the velocity pick-up, the

information of force must be masked and in a bond representing a force sensor, the

information of the flow must be masked. A full arrow somewhere on the bond shows

that some information is allowed to pass and some information is masked. The

information which is allowed to pass may be written near that full arrow (refer to

Table A.3). The concept of activation is very significant to depict feedback control

systems.

The term activation initially seems a misnomer. However, Paynter's idea was

based on the fact that though the information of a factor of power is masked on one

end, an activated bond on the other end can impart infinite power which is derived

from a tank circuit used for both the measurement or actuation device (for instance,

the pick-up, the amplifier and the exciter, all have external power sources).

A.4. Sensors, Actuators and Instrumentation Circuits

Additional pseudo-states can be added for measurement of any factor of power on a

bond graph model by using pseudo-storage elements. A flow activated integrally

causalled C-element would observe the flow (and consequently time integral of flow),

whereas an effort activated integrally causalled I-element would observe the effort

(and consequently generalized momentum). Activated storage elements are perceived

as conceptual instrumentation on a model. They do not interfere in the dynamics of

the system.

The flow activated integrally causalled C-element and the effort activated

integrally causalled I-element are also referred to as detector of flow (Df element) and

detector of effort (De element), respectively, in some bond graph literature. The

symbols for the effort and the flow detectors are shown in Table. A.3. Detectors are

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usually weak bonds at the connected junctions. Therefore, the flow detector is usually

connected to an 1-junction whereas the effort detector is usually connected to a 0-

junction.

Table A.3: Definition of Activated Bonds and Sensors

Name Type Symbol Definition

Effort ( )

0e e tf=

=

Activated Bond

Flow

( ) 0f f te=

=

Effort ( )

0e e tf=

= Sensors

(Detectors) flow

( ) 0f f te=

=

Activated bonds are used to implement modulated sources, and non-linear

constitutive relations of two-port and one-port elements. These are called modulated

sources, elements or two-ports and their bond graph notations are given by prefixing

‘M’ to their respective normal symbolic names. The modulating signal(s) may come

from a real sensor (De or Df, if the modulation is due to control action) or through an

activated bond emanating from a junction (if the modulation is a natural physical

phenomena, e.g., the modulated source of effort representing the torque due to gravity

in a vertical pendulum is a function of the pendulum’s angle with the vertical axis).

In order to illustrate the use of activated bonds and detectors in instrumentation

circuits, an example consisting of two articulated vehicles shown in Fig. A.1 is

considered. The vehicles are connected by means of a spring. The motions of the cars

are sustained by applying force on the second car. It is proposed to control the

velocity of the two cars by picking up the velocity of the first car and applying an

effort which is proportional to the error in velocity to the second car. The bond graph

for the system using the conventions followed in this work is shown in the Fig. A.2a.

Note that the transformer element used in between the two flow activated bonds is not

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power conserving. In fact it is just a gain that is to be multiplied with the input signal

to give the output signal. The corresponding bond graph of the same system using the

alternative convention being used by other researchers (the so-called international

convention) is shown in Fig. A.2b. The later model, which is a hybrid bond graph and

block diagram representation, explicitly shows the sensors (Df) and actuators (MSe)

in the model. The bond graph models which were given in the Figs. 3.2, 3.6 and 3.10

in Chapter 3 are redrawn using the international conventions in Figs. A.3, A.4 and A.5,

respectively.

Fig. A.1: Example System Considered for Illustrating the Use of Activated Bonds

(a) (b)

Fig. A.2: Bond Graph of the Example System in (a) the Convention Used in this Work and (b) the International Convention Being Followed Now-a-days

A.5. Systems with Differential Causality

The situation of differential causality arises for system models in which some storage

elements are not dynamically independent. This, however, does not mean that the

parameters of these storage elements do not appear in the equations. In such cases,

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these parameters get associated with the parameters of other storage elements which

have achieved integral causality. The occurrence of differential causalities in a system

may indicate serious violations of principles of conservation of energy. Any storage

element with differential causality does not contribute state variables. The

constitutive relation of a differentially causalled element is written in the derivative

form. Therefore, initial conditions for such storage elements cannot be independently

specified. For example, the constitutive relation for a differentially causalled C-

element is given as

ddef Ct

= . (A.1)

Fig. A.3: Preliminary Bond Graph Model of the Hydrogen Oxidation Electrochemical Reaction (Original Fig. 3.2) in International Convention

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Fig. A.4: Bond Graph Model of the SOFC (Original Fig. 3.6) Using the 4 Port C-fields in International Convention

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Fig. A.5: True Bond Graph Model of the SOFC (Original Fig. 3.10) in International Convention

A.6. Field Elements

The single port C, I and R elements can also be generalized for multiple ports. Such

multi-port storage and dissipative elements are called field elements. Let us consider

the C-field for example. The efforts in a set of bonds are determined by displacements

in the bonds of the same set as

1

,n

i ij jj

e K Q=

= ∑ (A.2)

where, 1, 2,...,i n= and Kij’s are the stiffness influence coefficients and

00

t

j jQ f dt Q= +∫ .

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Such a relation may be represented by a multiport C element called C-field. C-

fields are a common feature in modelling of thermodynamic systems. For instance, a

collapsible chamber in an engine or a compressor chamber can store energy through

interaction of three modes, viz. the mechanical port associated with the piston motion,

the thermal port for the heat transfer and the chemical work done by mass transfer and

combustion.

The constitutive relations for an integrally causalled nonlinear C-field (with n

bonds, all of which have causal strokes placed away from the C-field) are given by

( )

( )

( )

1 C1 1 2 C1 1 2

2 C2 1 2 C2 1 2

Cn 1 2 Cn 1 2

, , , , , , , , ,

, , , , , , , , ,

, , , , , , , , ,

t t t

n n

t t t

n n

t t t

n n n

e Q Q Q f dt f dt f dt

e Q Q Q f dt f dt f dt

e Q Q Q f dt f dt f dt

φ θ φ θ

φ θ φ θ

φ θ φ θ

−∞ −∞ −∞

−∞ −∞ −∞

−∞ −∞ −∞

⎛ ⎞= = ⎜ ⎟

⎝ ⎠⎛ ⎞

= = ⎜ ⎟⎝ ⎠

⎛ ⎞= = ⎜ ⎟

⎝ ⎠

∫ ∫ ∫

∫ ∫ ∫

∫ ∫ ∫

(A.2)

where C1φ , C2φ , , Cnφ are scalar valued non-linear functions written in terms of the

state variables and a set of parameters (θ ) associated with the field element.

Similar relationships can be established for linear and non-linear I and R field

elements. The parameters associated with linear fields are always referred enclosed

within square braces ([K], [I] and [R]) as matrices.

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174

Appendix B

Derivations of Some Important Relations

B.1. Chemical Potential of a Constituent in an Ideal Gas

Mixture

A brief derivation of the expression for the chemical potential of a constituent in an

ideal gas mixture [Liebhafsky & Cairns, 1968] is given in this section.

The Gibbs function G is defined as

G U pV TS= + − . (B.1)

Differentiating Eq. (B.1) and substituting the expression for dU , which is given

as

d d d dU T S p V mμ= − + , (B.2)

we get

d d d dG V p S T mμ= − + , (B.3)

which implies

,T n

G Vp

⎛ ⎞∂=⎜ ⎟∂⎝ ⎠

. (B.4)

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175

Substituting the equation of state for ideal gas in Eq. (B.4), we get

,T n

G mRTp p

⎛ ⎞∂=⎜ ⎟∂⎝ ⎠

, (B.5)

integration of which yields

00ln pG G G mRT

p⎛ ⎞

Δ = − = ⎜ ⎟⎝ ⎠

, (B.6)

which further leads to

00ln pRT

pμ μ

⎛ ⎞− = ⎜ ⎟

⎝ ⎠. (B.7)

Since any ideal gas in a mixture of ideal gases behaves as though it were present

alone at its partial pressure, Eq. (B.7) also applies to every constituent in the mixture

of ideal gases, provided that the partial pressure of that constituent is used in Eq. (B.7).

Equation (B.7) can also be written in terms of the mole fraction (xi) as

( )lnoi i iRT xμ μ− = , (B.8)

where oiμ is the chemical potential of the i-th constituent at xi =1.

The chemical potential for a solute in a dilute solution is generally given in terms

of the concentrations, which can be obtained as follows. Dilute solutions can be

considered like ideal gases with regard to their state properties, i.e., the solute

molecules may be considered to occupy the whole volume with negligible interaction

with the solvent molecules. Therefore, Eq. (B.8) is valid for dilute solutions. Also, as

solventin n ,

solvent solvent

i ii

i

n nxn n n

=+

. (B.9)

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176

where ni is the number of moles of the i-th solute and nsolvent is the number of moles of

the solvent.

Substituting solvent solvent solvent solvent1000n V Mρ= into Eq. (B.9) and noting that

solvent totalV V , we get

solvent

total solvent1000i

in Mx

V ρ= . (B.10)

As ( )totali in V c= and ( )solvent solvent solvent1000 M cρ = ,

solvent

ii

cxc

= . (B.11)

Therefore, the chemical potential for a solvent in a dilute solution is given as

( )lnoi i iRT cμ μ− = , (B.12)

where the concentration of the solvent has been taken as unity.

B.2. Gibbs-Duhem Equation

The Gibbs-Duhem equation [Callen, 1985; Denbigh, 1955] gives the relationship

between the intensive thermodynamic variables, the pressure, the temperature and the

chemical potential. The derivation of the Gibbs-Duhem equation from other standard

thermodynamic relations is given in the following.

The Gibbs function is defined as

G U PV TS= + − , (B.13)

which may be written as

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177

U n PV TSμ= − + . (B.14)

In differential form, Eq. (B.14) can be written as

d d d d d d dU n n P V V P T S S Tμ μ= + − − + + . (B.15)

The change in the internal energy may be given as

d d d dU U UU V S nV S n∂ ∂ ∂

= + +∂ ∂ ∂

. (B.16)

From well-known thermodynamic relations ,U V p∂ ∂ = − U S T∂ ∂ =

and U m μ∂ ∂ = , Eq. (B.16) can be expressed as

dU TdS PdV dnμ= − + . (B.17)

Subtracting Eq. (B.17) from Eq. (B.15), we arrive at the Gibbs-Duhem equation as

0nd VdP SdTμ − + = . (B.18)

This equation can also be generalized for multi-species mixtures as

0i in d VdP SdTμ − + =∑ . (B.19)

B.3. Isentropic Nozzle Flow Equation

The derivation of the mass flow relation for an isentropic nozzle [Sabersky et al.,

1971] is presented in this section. The energy equation for the nozzle may be written

as

2 2u d

u d2 2u uh h+ = + , (B.20)

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178

where h is the specific enthalpy and u is the velocity. The subscripts ‘u’ and ‘d’

represent the upstream and the downstream sides, respectively. The outlet mass flow

rate from the nozzle may be given as

dd

d

Aum Auv

ρ= = , (B.21)

where ρ is the density of the fluid, A is the throat area of the nozzle and vd is the

specific volume of the downstream side fluid. Substituting the expression for ud

obtained from Eq. (B.20) into Eq. (B.21), we obtain

( ) 2u d u

d

2A h h um

v− +

= . (B.22)

In most of the practical applications, the velocity at the inlet of the nozzle (i.e., the

upstream side) is negligibly small when compared to the outlet velocity (i.e., the

velocity at the throat of the nozzle). Therefore, Eq. (B.22) becomes

( )u d

d

2A h hm

v−

= . (B.23)

For an ideal gas, the specific enthalpy is given as ph c T= and the specific volume

is given according to the ideal gas law as d d dv RT p= . Therefore, considering

constant cp, Eq. (B.23) becomes

( )d p u d

d

2p A c T Tm

RT−

= . (B.24)

If the flow through the nozzle is assumed to be adiabatic and reversible then the

process is isentropic and is governed by the following relation:

( )1

d d

u u

T pT p

γ γ−⎛ ⎞

= ⎜ ⎟⎝ ⎠

, (B.25)

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179

where p vc cγ = .

Substituting Eq. (B.25) into Eq. (B.24), we obtain the relation for the mass flow

rate through an isentropic nozzle as

( )2 ( 1)

u d d

u uu

2( 1)

p p pm A xR p pT

γ γ γγγ

+⎛ ⎞ ⎛ ⎞

= −⎜ ⎟ ⎜ ⎟− ⎝ ⎠ ⎝ ⎠. (B.26)

B.4. Nernst Equation

The reversible voltage of an electrochemical cell is given by the Nernst equation

[Liebhafsky & Cairns, 1968]. A brief derivation of the Nernst equation is given in this

section.

The change in Gibbs free energy per mole for an electrochemical reaction is given

as

P Ri i i i

i iG μν μν

∈ ∈

Δ = −∑ ∑ , (B.32)

where i refers to a particular reactant or product species and P and R are the set of

products and reactants participating in the reaction, respectively.

For the reversible operation of the cell, this change in the Gibbs free energy is

completely converted into electrical energy. Therefore,

e rG n FVΔ = − . (B.33)

The reversible voltage rV can be interpreted as being equal and opposite to the

electromotive force needed to establish virtual chemical equilibrium in an ideal

electrochemical cell; no matter how far removed the state of the reactants and

products are from the actual chemical equilibrium.

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180

For ideal gas mixture, the chemical potentials of the constituents are given by

(refer to Section B.1)

( ),0 R lni i iT pμ μ= + , (B.34)

where the reference pressure is taken as unity.

Substituting the chemical potentials of the reactants and the products in the form

given in Eq. (B.34) into Eq. (B.32) gives

P,0 ,0

P RR

R lni

ii i i i

i i ii

pG T

pμ ν μ ν ∈

∈ ∈∈

⎛ ⎞⎜ ⎟Δ = − + ⎜ ⎟⎜ ⎟⎝ ⎠

∏∑ ∑ ∏

, (B.35)

where 0,0 ,0

P Ri i i i

i iGμ ν μ ν

∈ ∈

− = Δ∑ ∑ , which is the Gibbs free energy of the reaction at the

standard state.

From Eqs. (B.33) and (B.35), we get

0 Pe r

R

R lni

i

ii

pn FV G T

p∈

⎛ ⎞⎜ ⎟− = Δ + ⎜ ⎟⎜ ⎟⎝ ⎠

∏∏

. (B.36)

This equation is called the Nernst equation. The Nernst equation is widely used in

electrochemistry to calculate the effect of changes in the partial pressure and the

concentration of the substances, which participate in the cell reaction, on the cell

voltage.

At equilibrium, 0GΔ = and then Eq. (B.35) becomes

,eq0 P

,eqR

R lni

i

ii

pG T

p∈

⎛ ⎞⎜ ⎟Δ = − ⎜ ⎟⎜ ⎟⎝ ⎠

∏∏

. (B.37)

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181

Defining a constant eq ,eq ,eqP R

i ii i

K p p∈ ∈

=∏ ∏ , known as the equilibrium constant

for the reaction, Eq. (B.37) may be written as

0

eq expRGKT

⎛ ⎞Δ= −⎜ ⎟

⎝ ⎠. (B.38)

This equilibrium constant is a constant for a particular reaction at a particular

temperature. It depends on the equilibrium values of the partial pressures of the

reactants and the products of the reaction at that temperature and it denotes the

equilibrium composition of the mixture. However, it varies with change in the

temperature.

Using Eq. (B.38), the Nernst equation given in Eq. (B.36) can also be written as

Pe r eq

R

R ln R lni

i

ii

pn FV T K T

p∈

⎛ ⎞⎜ ⎟− = − + ⎜ ⎟⎜ ⎟⎝ ⎠

∏∏

. (B.39)

This equation may also be written in terms of the concentrations of the reacting

species by using the following form for the chemical potentials instead of Eq. (B.34).

( )i i,0 iR lnT cμ μ= + . (B.40)

Note that in Eq. (B.40), the reference concentration has been taken as unity. Using

Eq. (B.40) and following the same procedure, the Nernst equation may be derived as

Pe r eq

R

R ln R lni

i

ii

cn FV T K T

c∈

⎛ ⎞⎜ ⎟− = − + ⎜ ⎟⎜ ⎟⎝ ⎠

∏∏

. (B.41)

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182

B.5. Butler-Volmer Equation

The Butler-Volmer equation [Bockris et al., 1998] describes how the electrical current

on an electrode depends on the electrode potential. A charge double layer usually

forms at the interface between the electrode and the electrolyte. Many models have

been put forward to explain the double layer phenomenon. The simplest one is the

Helmholtz model according to which, the charge accumulation at the two sides of the

interface forms an ideal electrical capacitor. The charge densities on the two sheets

are equal in magnitude but opposite in sign. Therefore, the drop in potential between

these two layers of charge is linear. Charge transfer across this double layer is

essential for the electrochemical reaction to proceed. This process of charge transfer

across the potential difference created due to the double layer is described by the

Butler-Volmer equation. This charge transfer can be considered as a rate process.

In any rate process such as the charge transfer process, the initial condition of the

system is characterised by a certain configuration of involved entities (ions, atoms,

sites etc.). Similarly, the final state is characterised by a different configuration of

those entities. During the transfer of the ion from the initial to the final site, the

configuration of the system keeps on changing. There is a critical configuration

beyond which the configuration of the ion and the sites more closely resembles the

final state. This critical configuration is known as the activated complex. Each

configuration during the process of charge transfer corresponds to a particular energy

for the system. Hence, this phenomenon can be represented by the motion of a point

along the potential energy curve shown in Fig. B.1. The peak of the potential energy

curve corresponds to the configuration of the system known as the activated complex.

As the system has to climb a potential energy barrier, it has to be activated with

critical activation energy H ≠Δ for the process to occur. If the entropy change S ≠Δ in

going from the initial to this activated state is taken into account then the rate process

needs a critical free energy of activation G H T S≠ ≠ ≠Δ = Δ − Δ .

The system in the activated state (i.e., the activated complex) has a property that it

will break down to the final state if there are specific vibrations of the configurations.

The vibration leading to the breakdown of the activated complex, given by the

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183

Planck’s law, takes place with a certain vibration frequency ν which has an energy E

associated with it:

E hv= , (B.42)

where h is the Planck’s constant.

According to the kinetic theory, the translation of the entities is associated with a

translational energy, which is given as 2kT per entity and kT for a pair of entities,

where k is the Boltzmann constant. Hence, if the vibrations of the configurations are

viewed as translations, the associated energy is given by

E kT= . (B.43)

Fig. B.1: The Free Energy of Activation and the Electrical Potential Across the Double Layer

Equating Eq. (B.42) and Eq. (B.43), we get

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184

kTh

ν = . (B.44)

The important assumption in the rate process theory is that the activated

complexes are in equilibrium with the reactants. Therefore, the mass action law can

be used to give

RACeq

R

oG Tc K ec

≠−Δ= = , (B.45)

where ACc and Rc are the concentrations of the activated complex and the reactants,

respectively, eqK is the equilibrium constant and o

G≠

Δ is the standard free energy of

activation.

The number of times the rate process occurs per second ( eν ) is equal to the

concentration of the activated complexes ( ACc ) times the frequency (ν ). Therefore,

Re R

oG TkT c eh

ν≠−Δ= , (B.46)

where o

G≠

Δ is the free energy of activation of the reaction. This represents the

chemical energy barrier that has to be overcome by the charged particles. This

chemical energy barrier has a peak at some distance across the interface which also

corresponds to a particular configuration of the reaction system known as activated

complex, i.e. the critical configuration that must be attained before the charge transfer

process can occur.

The above formulations do not account for the electric field created by the double

layer. The double layer results in a linearly varying potential difference across the

interface as shown in Fig. B.1. Let eφΔ be the equilibrium potential difference across

the interface. We define a factor β , which is called as the symmetry factor or the

transfer coefficient, such that

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185

Distance across the double layer to the summitDistance across the whole double layer

β = . (B.47)

This electrical field lowers the energy barrier by the amount eFβ φΔ for one mole

of charged particles to move in one direction and raises the energy barrier by the

amount ( ) e1 Fβ φ− Δ for one mole of charged particles to move in the opposite

direction. Let us consider here that the movement of the charged particle is in such a

direction that it is aided by the electrical field. The oG ≠Δ in Eq. (B.46) can be

considered to consist of two parts, a chemical energy part and an electrical energy part.

chem eo oG G Fβ φ≠ ≠Δ = Δ + Δ . (B.48)

Substituting Eq. (B.48) into Eq. (B.46), the rate of charge transfer is given as

chem eR Re i

oG T F TkT c e eh

β φν≠−Δ − Δ= . (B.49)

The current is obtained by multiplying Eq. (B.49) by the Faraday’s constant as

chem eR Ri

oG T F TkTi F c e eh

β φ≠−Δ − Δ= . (B.50)

According to the principle of microscopic reversibility, there must also be a

current in the opposite direction at equilibrium. In this case, the potential difference

across the interface raises the energy barrier by the amount ( ) e1 Fβ φ− Δ and hence

( )chem e1o oG G Fβ φ≠ ≠Δ = Δ − − Δ . (B.51)

Therefore, the current in the opposite direction is given as

( ) echem 1 RRi

o F TG TkTi F c e eh

β φ≠ − Δ−Δ= . (B.52)

At equilibrium, the currents in both the directions must be equal and this

equilibrium current is known as the exchange current:

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186

( ) ee 1 RR0 chem i chem i

F TF Ti k c e k c e β φβ φ − Δ− Δ= = , (B.53)

where chemk is the chemical part given by

chem Rchem

oG TkTk eh

≠−Δ= . (B.54)

Let φΔ be the interface potential of the system when it is driven out of

equilibrium. The deviation of the interface potential from its equilibrium value is

termed as the over-potential (η ). In a fuel cell, this over-potential is actually the

voltage loss as the cell is taken out of equilibrium by means of drawing current from it.

Thus, the interface potential is given as

eφ φ ηΔ = Δ − . (B.55)

Away from equilibrium, the net current at the electrode-electrolyte interface is

non-zero and it is given by

( )1 RRchem i chem i

F TF Ti k c e k c e β φβ φ − − ΔΔ= − . (B.56)

Substituting Eq. (B.55) into Eq. (B.56), we get

( ) ( ) ( )e eR 1 Rchem i chem i

F T F Ti k c e k c eβ φ η β φ ηΔ − − − Δ −= − , (B.57)

which can be written as

( )( )1 RR0

F TF Ti i e e β ηβ η − −= − . (B.58)

This equation is called the Butler-Volmer equation. This formula holds good for

both the electrode electrolyte interfaces as the same charge transfer phenomena occurs

at both the places.

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187

Appendix C

The Nelder-Mead Simplex Algorithm

The Nelder and Mead simplex method [Lagarias, 1998; Nelder and Mead, 1965] is

implemented for solving n-dimensional unconstrained minimization problem to find

( )1 2 n, ,..., x x x=X which minimizes ( )f X . The geometric figure formed by a set of

n+1 points in an n-dimensional space is called a simplex. The basic idea in the

simplex method is to compare the value of the objective function ( )f X at the n+1

vertices Xj of a simplex and move the simplex gradually toward the optimum point

during the iterative process. Starting from an initial simplex with n+1 known vertices

Xj, a new vertex will be computed to define a new simplex using reflection,

expansion, or contraction.

One iteration of the Nelder-Mead method consists of the following three steps.

1. Ordering: Determine the worst (Xh), second worst (Xsh) and the best

vertex (Xl), in the current working simplex by evaluating the objective

function ( )jf X . These evaluations are called trials and denoted by jT ,

i.e., ( )h hT f= X , ( )sh shT f= X and ( )l lT f= X .

2. Centroid: Calculate the centroid 0X of the best side, i.e., the one opposite

the worst vertex hX as 1

01, h

1 n

jj jn

+

= ≠

= ∑X X .

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188

3. Transformation: Compute the new working simplex from the current one.

First, try to replace only the worst vertex Xh with a better point by using

reflection, expansion or contraction with respect to the best side. All test

points lie on the line defined by Xh and 0X , and at the most two of them

are computed in one iteration. If this succeeds, the accepted point becomes

the new vertex of the working simplex. If this fails, reduce the simplex

towards the best vertex Xl. In this case, n new vertices are computed. The

different transformations are illustrated in Fig. C.1.

Simplex transformations in the Nelder-Mead method are controlled by four

parameters: α for reflection, β for contraction, γ for expansion and δ for

reduction. They should satisfy the constraints: 0α > , 0 1β< < , 1γ > , γ α> and

0 1δ< < . The standard values used in most implementations are 1α = , 1/ 2β = ,

2γ = and 1/ 2δ = .

Fig. C.1: Different Transformations of a Two-dimensional Simplex

The flow chart of the Simplex algorithm is shown in Fig. C.2. The working of

simplex transformations in step 3 is described in the following:

• Reflect: Compute the reflection point ( )r o h oα= − +X X X X and ( )r rT f= X .

If l r shT T T≤ < then accept rX and terminate the iteration.

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Appendices

189

• Expand: If r lT T< then compute the expansion point ( )e r o oγ= − +X X X X

and ( )e eT f= X . If e rT T< then accept eX and terminate the iteration.

Otherwise (if e rT T≥ ), accept rX and terminate the iteration.

• Contract: If r shT T≥ then compute the contraction point cX by using the better

of the two points hX and rX .

o Outside: If sh r hT T T≤ < then compute ( )c+ o r oβ= + −X X X X and

( )c+ c+T f= X . If c+ rT T≤ then accept c+X and terminate the iteration.

Otherwise, perform a reduction transformation.

o Inside: If r hT T≥ then compute ( )c- o h oβ= + −X X X X and

( )c- c-T f= X . If c- hT T< then accept c-X and terminate the iteration.

Otherwise, perform a reduction transformation.

• Reduce: Compute n new vertices ( )l lj jδ= + −X X X X and ( )j jT f= X , for

0,..., j n= , with lj ≠ .

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190

Fig. C.2: Flow Chart of the Nelder-Mead Simplex Algorithm

Calculate first simplex vertices

Rank trials in order: (Tl, …, Tsh, Th)

Make reflection to get Xr

Tr<Tl Tr<Tsh

Make expansion to get Xe

Te<Tr

Xh=Xe Xh=Xr

Convergence criteria

Stop Optimization

Start Optimization

No

Yes

Make +ve contraction to get Xc+

Tr<Th

Tc+≤Tr Xh=Xc+

Make reduction to get ‘n’ new

vertices Xj

Make -ve contraction to get Xc-

Xh=Xc- Tc-<Th

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Appendices

191

Appendix D

List of Files and Programs Given in the

Compact Disc

S. No. File Name Location/

Directory Description Compatibility

(OS & Software)

Remarks

1 Thesis.pdf Root\Thesis\ Full thesis Adobe Acrobat 5.0+

2 Basic_model.bgp, Detailed_model_no_control.bgp, Detailed_model_with_control.bgp, SOFC_system .bgp

Root\Models\Bondpad files

Bondpad (*.bgp) files of the SOFC models described in the thesis

Windows XP, Symbols Shakti

Use Runge-Kutta Gill method with 1000 steps, 16384 records and an error limit of 5e-5 for simulation

3 ANODE.cap, CATHODE.cap, convres.cap, convresmulti.cap, CR.cap, HEEX.cap, MCR.cap, OCR.cap, OMCR.cap, TC.cap, TR.cap, TRad.cap

Root\Models\Capsule files

Symbols capsules used in the bond graph models

Windows XP, Symbols Shakti

4 Optimisation_code.m, ConstFU_operation.m,

Root\Matlab\ Matlab code for the optimisation routine and for finding operating conditions corresponding to constant FU operation

Windows XP, Matlab 7.0

To be run for different values of current density within the cell operating range

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192

Page 213: Modelling, Simulation and Control of a Solid Oxide Fuel Cell System: A Bond Graph Approach

Curriculum Vitae

P. Vijay did his graduation, Bachelor of Mechanical Engineering, from K.S.

Rangasamy College of Technology, Madras University, Tamil Nadu in the year 2000.

He obtained his Master of Engineering (M. E) degree from Government College of

Engineering, Salem in the year 2002. After that he served as faculty in the mechanical

engineering department of Sethu Institute of Technology, Madurai and in the school

of mechanical engineering of SASTRA deemed university, Tanjavur. In the year

2006, he joined in the Ph.D. Programme at IIT, Kharagpur as a regular scholar to

pursue his research work in the area of fuel cell modelling and control. Following are

the list of papers published and communicated, from the present research work.

In Journals

1. Vijay, P., Samantaray, A. K., and Mukherjee, A. Bond graph model of a solid oxide fuel cell with a C-field for mixture of two gas species. Proceedings of the Institution of Mechanical Engineers, Part I: J. Systems and Control Engineering, 2008, 222 (4), 247–259.

2. Vijay, P., Samantaray, A. K., and Mukherjee, A. On the rationale behind constant fuel utilization control of solid oxide fuel cells. Proceedings of the Institution of Mechanical Engineers, Part I: J. Systems and Control Engineering, 2009, 223 (2), 229–252.

3. Vijay, P., Samantaray, A. K., and Mukherjee, A. A Bond Graph Model-based Evaluation of a Control Scheme to Improve the Dynamic Performance of a Solid Oxide Fuel Cell. Mechatronics, 2009, 19 (4), 489–502.

4. Vijay, P., Samantaray, A. K., and Mukherjee, A. Constant Fuel Utilization Operation of a SOFC System: An Efficiency Viewpoint. Accepted for publication in ASME Journal of Fuel Cell Science and Technology.

In Proceedings of Conferences

1. Vijay, P., Mukherjee, A., and Samantaray, A. K. Bond Graph Model of a Solid Oxide Fuel Cell. Millennium Energy Summit, Central Glass and Ceramic Research Institute, Kolkata, September 27–29, 2007.

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