Fuel Cell Modeling and Simulation: From Microscale to Macroscale 0323857620, 9780323857628

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FUEL CELL MODELING AND SIMULATION

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FUEL CELL MODELING AND SIMULATION From Microscale to Macroscale

GHOLAM REZA MOLAEIMANESH Iran University of Science and Technology (IUST) Tehran, Iran

FARSCHAD TORABI K.N. Toosi University of Technology (KNTU) Tehran, Iran

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States Copyright © 2023 Elsevier Inc. All rights reserved. MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. ISBN: 978-0-323-85762-8 For information on all Elsevier publications visit our website at https://www.elsevier.com/books-and-journals Publisher: Charlotte Cockle Acquisitions Editor: Peter Adamson Editorial Project Manager: Aera F. Gariguez Production Project Manager: Surya Narayanan Jayachandran Cover Designer: Miles Hitchen Typeset by VTeX

Contents

Preface References

ix x

1. Fuel cell fundamentals

1

1.1. Introduction 1.2. Thermodynamics 1.3. Electrochemical reaction kinetics 1.4. Charge transfer 1.5. Mass transport 1.6. Characteristic curve of a fuel cell 1.7. Summary 1.8. Problems References

2. PEMFCs 2.1. Introduction 2.2. Microscale modeling and simulation of PEMFCs 2.3. Macroscale modeling and simulation of PEMFCs 2.4. Summary 2.5. Questions and problems References

3. Solid oxide fuel cells 3.1. Introduction 3.2. Microscale modeling and simulation of SOFCs 3.3. Macroscale modeling and simulation of SOFCs 3.4. Modeling of a solid oxide electrolyzer cell (SOEC) 3.5. Summary 3.6. Problems References

2 22 34 45 50 54 55 56 56

57 57 138 190 227 228 230

237 237 247 253 264 265 265 266

4. Hydrogen storage systems

269

Introduction High-pressure tanks Hydrogen-absorbing tank Summary

269 270 272 280

4.1. 4.2. 4.3. 4.4.

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Contents

4.5. Questions and problems References

5. Fuel cell electric vehicles (FCEVs) 5.1. Introduction 5.2. Vehicle dynamics 5.3. FCEV configuration and components 5.4. Modeling and control of FCEVs 5.5. Summary 5.6. Problems and questions References

6. Fuel cell power plants

280 281

283 283 284 289 294 299 300 301

303

6.1. Applications 6.2. SOFC power plant components 6.3. Fuel 6.4. Summary 6.5. Problems References

303 310 324 335 335 335

7. Combined heat and power systems

337

7.1. CHP and fuel cells 7.2. General procedure for CHP designs 7.3. SOFC-based CHP system 7.4. PEMFC-based CHP system 7.5. Summary 7.6. Problems References

A. Lattice-Boltzmann codes A.1. A.2. A.3. A.4.

GeometryGenerator Isothermal single-phase air flow in a PEMFC channel with fiber obstacle Modeling of fluid displacement in porous media PEMFC cathode catalyst layer (CL) modeling using LBM

337 349 350 355 383 383 384

387 387 391 401 412

B. MATLAB® code for the simulation of PEMFC

443

C. Optimization methods

449

C.1. Concepts of optimization C.2. Optimization methods

449 452

Contents

C.3. Implementing optimization methods in C++ C.4. Summary References

455 472 472

D. UDFs for MH tank simulation in ANSYS Fluent

473

E. Matlab® code for calculating energy consumption of an FCEV

475

Index

479

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Preface

Global warming and the depletion of fossil fuel resources compel extensive research and development of renewable energies. In the near future, various applications of renewable energies will be observed all over the world, forecasted by climate experts. In this renewable world, green or renewable hydrogen energy (i.e., hydrogen from renewable electricity of solar plants etc.), accompanied by fuel cell systems, will have a crucial role. Evidently, in all eight scenarios proposed by the European Commission for the net zero emissions of the world economy in 2050, hydrogen energy exists [1]. To better reveal this critical role, we suggest reading the report of Bloomberg New Energy Finance in March 2020 [2]. It states that the clean hydrogen can address the most challenging third of global greenhouse gas emissions by 2050. This clean energy can play several roles; e.g., it can be used to store the generated heat in a solar power plant by employing an electrolyzer and producing hydrogen fuel; the produced hydrogen can be used in a fuel cell system to deliver electricity to the grid for peak power shaving. Hydrogen energy and fuel cell engineering is an active field in both the academic and industrial sectors. Most universities in the world teach courses such as fuel cell systems in different departments, including chemical engineering, energy systems engineering, mechanical engineering, etc. The present book deals with all aspects of fuel cell modeling and simulation, from microscale to macroscale, and can be a precious source for such courses. For the numerical simulation of fuel cells, the book is of special interest. More specifically, for higher education students, the topic is of more interest. In the industrial sector, many companies in the world are working on the installation of fuel cell systems. These companies use different software like GT-SUITE, AVL, ANSYS, COMSOL Multiphysics, etc., for fuel cell simulations, which all use a numerical model or several numerical models. A key feature of this book is the classified insight provided for the reader. This can effectively help the reader to decide which numerical model or simulation technique is appropriate for him/her according to the required accuracy and available computational capacity. This book includes seven chapters and five appendixes. Chapter 1 talks about fuel cell fundamentals concisely as a prestep for fuel cell modeling. However, those familiar with fuel cell fundamentals can directly study the other applied six chapters. These six chapters are about the modeling and simulation of several independent applied topics, from PEMFCs, SOFCs, and hydrogen storage, to FCEVs, fuel cell power plants, and fuel cell-based CHPs. Chapter 2, the largest and most important, discusses PEMFCs. After a brief introduction about PEMFCs and the transport phenomena in different parts of them, the microscale simulation techniques such as LBM, PNM, and VOF are presented. After that, single- and multiphase macroscale CFD models and 1D

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Preface

models for the simulation at both cell and stack levels are presented. In all these simulation techniques, the governing equations and the solution procedures are presented. This chapter consists of several examples and problem as well as several Fortran codes for the pore-scale simulation of PEMFC electrodes by LBM in Appendix A. Chapter 3 talks about SOFCs in a similar structure. It introduces transport phenomena, microand macroscale models. In Chapter 4, different methods of hydrogen storage and their simulation and modeling are presented. Special focus of this chapter is on the hydrogen storage in modern absorbers such as metal hydride and advanced carbon materials; a UDF for the simulation of an MH tank in ANSYS Fluent is provided in Appendix D. Chapter 5 is quite interesting for vehicle engineers; it explains vehicle dynamics concisely and then a 1D model of a vehicular powertrain, which can be used for the design and analysis of FCEVs. The MATLAB® code of the model is also provided in Appendix E. The topics of the last two chapters are about the modeling and simulation of power plants and CHP systems based on fuel cells. Since optimization is an important factor in design phase, its concept is explained in Appendix C, and different codes are provided to implement different optimization methods. Finally, we sincerely hope that the goals of this book are achieved and welcome any feedback, such as comments, suggestions, queries, finding a typo, providing better explanations and discussing other important undiscussed concepts. Please help us to improve the next edition of the book with your precious feedback via sending emails to [email protected] or [email protected].

References [1] European Commission, A Clean Planet for all. A European long-term strategic vision for a prosperous, modern, competitive and climate neutral economy, Depth Analysis in Support of the Commission; Communication COM 773 (2018) 2018. [2] BloombergNEF, Hydrogen Economy Outlook: Key Messages, New York, USA, 2020.

CHAPTER 1

Fuel cell fundamentals Contents 1.1. Introduction 1.1.1 Fuel cell perspective 1.1.1.1 Roadmap of Japan 1.1.1.2 Roadmap of EU 1.1.1.3 Roadmap of the United States 1.1.1.4 Roadmap conclusions 1.1.2 Fuel cell operation 1.1.3 Fuel cell types 1.1.3.1 Proton-exchange membrane fuel cell (PEMFC) 1.1.3.2 Direct methanol fuel cells (DMFCs) 1.1.3.3 Alkaline fuel cells (AFCs) 1.1.3.4 Phosphoric acid fuel cells (PAFCs) 1.1.3.5 Solid-oxide fuel cell (SOFCs) 1.1.3.6 Molten-carbonate fuel cell (MCFC) 1.1.3.7 Other technologies 1.2. Thermodynamics 1.2.1 Gibbs free energy 1.2.2 Second law of thermodynamics and fuel cells 1.2.3 Fuel cell efficiency 1.2.4 Role of effective factors 1.2.4.1 Effect of temperature 1.2.4.2 Effect of pressure 1.3. Electrochemical reaction kinetics 1.3.1 Exchange current density 1.3.2 Butler–Volmer equation 1.3.3 Role of effective factors 1.4. Charge transfer 1.4.1 Electronic resistance 1.4.2 Ionic resistance 1.4.3 Role of effective factors 1.5. Mass transport 1.5.1 Convective mass transfer from flow channel to GDL 1.5.2 Diffusive mass transfer 1.5.3 Role of effective factors 1.6. Characteristic curve of a fuel cell 1.7. Summary 1.8. Problems References Fuel Cell Modeling and Simulation https://doi.org/10.1016/B978-0-32-385762-8.00005-1

2 3 3 6 7 7 7 11 12 13 15 17 18 19 21 22 23 25 26 28 28 33 34 36 38 44 45 46 47 49 50 52 52 53 54 55 56 56 Copyright © 2023 Elsevier Inc. All rights reserved.

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1.1. Introduction A fuel cell is an electrochemical device in which a fuel is oxidized and generates electricity, heat, water, and in some cases, other materials such as carbon dioxide. For most cases, the fuel is hydrogen, and the oxidant is the oxygen available in atmosphere. In this view, fuel cells are nothing than the conventional electrochemical batteries. The only difference is that the available energy in batteries is limited to the amount of active mass that is confined in the battery, but a fuel cell does not store any energy. In fact, as long as the fuel and oxygen are fed into the device, they produce electricity. In other words, a fuel cell is just a reactor whose reactants (fuel and oxygen) are fed from external storage devices. Although fuel cells are recently considered as a promising source of electricity generation, its invention dates back to the 19th century when the renown British physicist, Sir William Grove found out that steam can be dissociated into hydrogen and oxygen in a reversible reaction. In other words, the hydrogen and oxygen can be combined through an electrochemical reaction to produce water. After the invention, fuel cells did not get much attention until the 20th century where alkaline fuel cells were used in space programs. However, at that time, the cost of cells was too high to become interesting for residential applications. One of the main reasons that contributed to their high cost was the usage of platinum (Pt), which is too expensive. Many efforts were carried on to reduce the amount of Pt to reduce the cost. The result was a significant reduction in cost, which in turn resulted in more economic cells. Nowadays, fuel cells are well known all over the world, and many factories and manufacturers produce different cell types. However, they are still costly devices, and their usage is still under debate. Although many efforts have been done to reduce the amount of Pt, the cost of devices is not still as low as it should be. One of the reasons is that the other parts of the cell are also expensive. For example, in many fuel cells the separator is made of Nafion, and in some others, it is made of a solid ceramic material. Both materials are still expensive, which increases the cost of the cells. Recent researches show that although the fuel cells are still expensive, if we consider the whole life cycle assessment, they may be much better devices than the present engines. Note that the fuel cells have no moving part, are noiseless, and have a long service life and low maintenance. These factors make them quite attractive in comparison with the present generators such as gas engines or turbomachines. In addition to the above, fuel cells are a part of hydrogen economy, which many scientists believe would be the future of human energy source. Without doubt, the fossil fuels will come to an end, and human has no choice but to rely on natural sources of energy such as solar energy, wind, biomass, ocean energy, and so on. These sources of energy have their own benefits and drawbacks. The good news about them is that they are renewable and will not come to an end. The bad news is that they may not be available in places where they are needed. For example, the solar or wind energy may

Fuel cell fundamentals

not be available while we need to drive a car. Therefore we have to store their energy for our needs such as transportation and so. There are different ways of energy storage, one of which is storing the energy in chemical compounds. For example, we may use the renewable energies to dissociate water into hydrogen and store hydrogen as a fuel in specific tanks. These tanks may be used wherever we need including transportation, home applications, power plants, etc. The stored hydrogen can be burnt in conventional burners and engines, but as we know, the thermodynamic efficiency of these burners is very low comparing to fuel cells. In practice the efficiency of fuel cells may reach more than twice the efficiency of a conventional Otto engine. Therefore, from the energy point of view, it would be much better to use fuel cells instead of the present engines and burners.

1.1.1 Fuel cell perspective To have a better understanding about the perspective of the fuel cells, a good idea is to study the programs and roadmaps of the pioneers in this field. In this section, we study the roadmaps of Japan, the European Union (EU), and the United States to see the targets of their roadmaps and their plans to achieve the hydrogen economy.

1.1.1.1 Roadmap of Japan The leading countries such as Japan have solid programs for hydrogen economy or what is called the hydrogen world. In Japan, there are two main targets on FC economy, one for residential sector and the other for fuel cell vehicles (FCVs). The residential program focuses on the usage of solid oxide fuel cells (SOFCs), because these devices produce both electricity and heat. The working temperature of a SOFC is something between 500 and 1000 ◦ C, which is quite suitable for combined heat and power (CHP) installation. By this architecture an SOFC produces both the required heat and electric power for a residential building. The benefit of this design is that the overall efficiency reaches above 80%. Fig. 1.1 shows Japan’s roadmap for hydrogen technology in residential sector. This roadmap started around 2005 and, as is shown in the figure, will continue till 2030. The program has three distinct phases: Large-scale validation, in which the concept was going to be validated till the production of a commercializing product. Market creation with policy support was the commercialization phase in which SOFC packages were to penetrate the market. Since at the time the price was so high, the penetration was supported by support. Establishment of self-sustained market is the current stage in which the price of the packages is affordable by individuals, and the market is becoming more an more sustained.

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Figure 1.1 Japan’s program for residential sector.

The roadmap predicts that by the end of the present decade, 5.3 million units will be sold, and the price of each unit reaches from less than 500 to 600 JPY. Japan’s program on FCVs is based on proton exchange membrane fuel cells (PEMFCs). The roadmap is shown in Fig. 1.2 and started from around 2005 and will continue up to 2030. This program is demonstrated in four different phases: Technology Demonstration, in which the technology was to be developed. Technology and Market Demonstration is the continuation of the first phase with market consideration. At the end of the phase the first commercialized cars including Toyota Mirai (2014), Honda Clarity (2016), and Nissan X-TRAIL FCV (2017) were introduced. Early Commercialization is the current stage. Early FEVs are produced and sent to market. However, the infrastructures such as H2 production, hydrogen filling stations, and other facilities should be expanded. Full Commercialization is a continuation of the previous phase in which the prices should be reduced and the number of FEVs in the market should be increased. Japan’s program on hydrogen and FC revised and extended for beyond 2040 as shown in Fig. 1.3. The revised roadmap focuses on three major phases: Installation of Fuel Cell focuses on establishment of fuel cell devices on both residential and mobile sectors.

Fuel cell fundamentals

Figure 1.2 Japan’s program for FC vehicles sector.

Figure 1.3 Japan’s revised program.

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Figure 1.4 EU’s hydrogen and FC program.

H2 Power Plant and Mass Supply Chain starts around 2030 and focuses on H2 production. At the early stage, hydrogen is imported from overseas, and the program aims to make full-scale H2 power plants. CO2 -free Hydrogen starts around 2040 and aims to produce hydrogen from renewable sources of energy by water splitting.

1.1.1.2 Roadmap of EU The EU program on hydrogen and fuel cell is shown in Fig. 1.4. The program has two parallel rails, one on hydrogen production and the other on fuel cell development. From the figure we can conclude the following: Hydrogen Production starts by reformation of natural gas that has been developed and its infrastructure already exists. Hydrogen production technologies continue to be developed until H2 will become fully available from water dissociation using natural or green energy sources around 2050. Fuel Cell Development focuses on different purposes including usage of SOFCs and molten carbonate fuel cells (MCFCs) for stationary sectors such as power plants and PEMFCs for car industries. The technology aims to provide the energy required for aviation systems in about 2050.

Fuel cell fundamentals

Figure 1.5 The US hydrogen and FC program.

1.1.1.3 Roadmap of the United States The United States also have a similar roadmap on hydrogen technology, but the program gives a whole guide line up to 2040. At the early stages the government supports the technology, but in the second phase, the commercialization is given to private sectors. This roadmap is shown in Fig. 1.5, and as we can see, it has four distinct phases: 1. Technology Development 2. Initial Market Penetration 3. Expansion of Markets and Infrastructures 4. Fully developed Markets and Infrastructures

1.1.1.4 Roadmap conclusions Comparing the hydrogen program of the above three pioneers in the field reveals that hydrogen and fuel cell will play important roles in the near future. In addition to the above communities, other famous countries such as China, Canada, India, Brazil, and many others have their own roadmaps toward a hydrogen world concept in which H2 becomes the main source of energy. This study shows the importance of fuel cells and their role in human life.

1.1.2 Fuel cell operation As mentioned above, a fuel cell is an electrochemical device in which fuel and oxidant are combined to produce electricity. In addition to electricity, unavoidable products such as water and heat should be removed from the system. Note that the fuel can be

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oxidized in a chemical reaction in which the heat and water are produced. For example, the burning of hydrogen is expressed by H2 +

1 O2 −−→ H2 O + heat. 2

(1.1)

The burning reaction is considered to be chemical. The same reaction may also take place through a electrochemical reaction. So, what is the difference between the two? The answer is: • The electrochemical reactions take place at a solid surface. In other words, electrochemical reactions are surface phenomena. In contrast, chemical reactions can be done in the bulk. • The redox reactions of an electrochemical process take place at different locations. For example, the fuel is oxidized at the anode, and the oxygen is reduced at the cathode. However, in a chemical reaction, both redox reactions take place at the same location. • The electron transfer in an electrochemical reaction is performed via an external circuit, whereas for a chemical reaction, the electron is transferred directly from one species to another. By these characteristics burning of hydrogen is considered a chemical reaction because it first happens in a bulk. Second, the reaction takes place at a single location, in which a hydrogen molecule meets an oxygen molecule. Finally, the electron transfer does not happen through an external circuit. In contrast to burning process, the reaction of an FC is considered electrochemical because it meets all the above-mentioned characteristics. The process can be better understood by studying Fig. 1.6. The cell contains a negative electrode, which is separated by an electrolyte from a positive electrode. Each electrode is made of at least two layers, one of which is a porous substrate, and the other is a catalyst layer. Usually, the catalyst is coated on the substrate by different methods. The material used for making an electrode differs from technology to technology. Each technology uses a specific material for making the substrate and a different type of catalyst. In some FC types, there may be more layers such as sublayers for catalyst or other materials for enhancing the performance. The substrate is used for 1) backing the catalyst layer and 2) providing a uniform material concentration at the catalyst layer. Since in most of the FC types the fuel is in gas phase, the substrate is called a gas diffusion layer or GDL. From the name it is clear that GDL should be porous to let the fuel or oxygen diffuse through it and reach the catalyst layer (Cat). The materials used for making GDLs are electrically conductive, but it is worth noting that the reactions do not occur at GDL. The catalyst layer is where the electrochemical reactions occur and the electrons are produced or consumed. The electrons, however, can move through the GDL as well, since GDL is a conductive medium.

Fuel cell fundamentals

Figure 1.6 Schematic representation of a fuel cell and definitions.

In studying Fig. 1.6 the negative electrode is named anode, and the positive electrode is named cathode. In practice and in many papers, books, handbooks, and related media, the words anode and cathode are used, but it should be noted that these terms are not always scientifically true. By definition an anode is a place where electrons are released or produced, and a cathode is a place at which electrons are consumed. In a normal behavior of an FC, the anode and cathode are quite equivalent to the negative and positive electrodes, respectively. However, in regenerative FCs, in which the electrochemical reactions are reversed by applying an external voltage (just like when a battery is recharged), the negative and positive electrodes remain the same, but the anode and cathode change their positions. In the charging process the anode is the positive electrode, and the cathode is the negative electrode. Although the negative and positive electrodes are a better choice for distinguishing the electrodes (since they do not change during charge and discharge), and since FCs are mostly used as generators and are rarely used as a regenerative device, the terms anode and cathode are used in all the available literature. With this fact in mind, in this book, to be consistent with all the other literature, we also use anode instead of the negative electrode and cathode instead of the positive electrode. The electrolyte shown in Fig. 1.6 acts as a separator as well. It is a separator because it separates the electrodes so that they should not make a short circuit. In addition, it provides a medium through which the ion transfer occurs. All the available FCs can be categorized in two different types.

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1. The first types are those in which the electrical charge is carried by cations through the electrolyte. In these types, as shown in the figure, the cations are generated at the anode catalyst layer and move from the anode to the cathode. 2. The second types are those in which the electrical charge is carried by anions through the electrolyte. In these types the anions are generated at the catalyst layer of the cathode and, as shown in the figure, move from the cathode to the anode. In a first-type FC, the fuel is fed to the anode as shown in Fig. 1.6. A part of the fuel is consumed in the cell, and the excess is returned back and recirculated. The fuel is dissociated by the anode catalyst via the following electrochemical reaction: F −−→ Fn+ + n e− ,

(1.2)

where F stands for fuel. Since in this reaction the electron is directly removed from the fuel, advanced and expensive catalysts should be used. The produced electrons n e – cannot pass through the separator because it is a nonconductive material. Therefore the electrons move from an external circuit and pass through the load and reach the cathode. Meanwhile, the cations Fn+ move through the electrolyte, and reach the cathode. At the cathode catalyst layer, the oxygen which is fed through the cathode GDL will react with the cations and electrons via the following reaction: Fn+ + n e− +

1 O2 −−→ FO. 2

(1.3)

When using Eq. (1.3), depending on the fuel material, proper reaction balance should be carried on. The overall reaction is obtained by summing Eqs. (1.2) and (1.3): F+

1 O2 −−→ FO. 2

(1.4)

For a second-type FC, reactions completely differ from those of the first type. At the anode catalyst layer the fuel reacts by the anions that come from the cathode through the following reaction: F + An− −−→ FO + M + n e− .

(1.5)

In this reaction, An – is the moving anion, and M is an intermediate material. Note that since in reaction (1.5), the fuel electrons should not be directly removed, less expensive catalysts can also be used. Again, just like in the first-type FCs, the electrons move from the external circuit, pass through the load, and reach the cathode. At the cathode the oxygen is reduced on the catalyst layer via the following reaction: 1 O2 + M + n e− −−→ An− . 2

(1.6)

Fuel cell fundamentals

The overall reaction is obtained by summing Eqs. (1.5) and (1.6): F+

1 O2 −−→ FO. 2

(1.7)

The overall reactions of both types show that the final result of the processes is the burning of the fuel, but in an electrochemical manner. We reemphasize that in working with Eqs. (1.2) to (1.7), the reaction should be balanced according to the fuel, cation, anion, intermediate material, and oxygen. In the following subsections, we give the details for each FC technology.

1.1.3 Fuel cell types There are many different fuel cells available, which are categorized in different perspectives including the choice of separator, operating temperature, and fuel. Among the above-mentioned factors, the choice of their separator is the most important one, and most of the time the fuel cells are named after it, but in some types, their names are given after their fuel or operation. These cells can provide a very large range of power from microwatts to kilo- or megawatts depending on the type and application. Large-scale devices are manufactured by assembling a number of single cells in series since a single cell is not able to produce a large amount of power. Typically, the voltage of a cell in operation is around 0.7 Volts, which is not sufficient for producing large powers. Regardless of the type, in all the fuel cells, a fuel is oxidized at the anode, and the oxygen is reduced at the cathode. The oxidation and reduction take place on catalyst layers as mentioned before. For the redox reactions to become successful, the temperature should be kept within a specific limit. In this regard, all the cell types are categorized into three different levels: Low-Temperature FCs operate below 100 ◦ C including PEMFCs, direct methanol fuel cells (DMFCs), and microbial fuel cells (MFCs). Medium-Temperature FCs operate at higher temperatures around 150 to 250 ◦ C including AFC, PAFC, and HT-PEMFC. High-Temperature FCs work in very high temperature over 500 ◦ C and even may reach above 1000 ◦ C. For this category, we can name SOFC and MCFC, which are used in CHP power plants. There are lots of different technologies available both in the market and the labs. Many of these types are quite mature and commercially available. However, many different types are still under development. Here we introduce some important types, for which we discuss the main reactions, operating temperatures, and technology readiness.

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1.1.3.1 Proton-exchange membrane fuel cell (PEMFC) This technology is the most used FC, able to draw attention because of its unique characteristics. First of all, it is a low-temperature fuel cell whose operating temperature is less than 100 ◦ C (normally, around 90 ◦ C). Secondly, it uses a solid-state polymeric membrane, and hence its electrolyte does not suffer from leakage. Thirdly, its energy density is quite reasonable, which has made it a perfect choice for many portable highpower devices such as electrical vehicles. Finally, PEMFCs use hydrogen as fuel and combine it with oxygen taken from atmosphere to produce electricity as the main output and heat and water as by–products. Therefore they do not produce any hazardous materials such as acids, bases, or toxic gases. Consequently, their good characteristics have made them the best choice among the other low-temperature cells. The name of the PEMFCs is taken from their polymeric membrane made of perfluorosulfonic acid, which when absorbs water, becomes a good conductor of hydrogen protons, H+ , whereas it is an insulator for electron. Many efforts have been made to produce different material for these cells, but at the present time the only commercially available membrane is called Nafion, which is a brand name for the membrane discovered in the late 1960s by Walther Grot of DuPont. In some texts, PEMFC stands for Polymeric Electrolyte Membrane fuel cell as well. In other works, PEMFCs may be called solid-state membrane fuel cells or SSMFC. All these names refer to the material of the membrane. As a low-temperature device, PEMFCs have some advantages and suffer some disadvantages. The main advantage is that a PEMFC quickly becomes operational since its operational temperature is close to the ambient. The main disadvantage of being a lowtemperature device is that to accelerate the reactions, expensive catalysts are required. Usually, Pt is a very expensive choice. Moreover, Pt becomes poisoned when combined with carbon monoxide or CO. Therefore, for industrial scales, proper facilities should be mounted to remove the CO to recover the poisoned Pt. Fig. 1.7 shows the main components and reactions of a PEMFC. The fuel is hydrogen, which in practice is not completely pure. Fuel enters the anode and diffuses on the Pt catalyst through the anode GDL at which the platinum removes the hydrogen electron through the following reaction: H2 −−→ 2 H+ + 2 e− .

(1.8)

The electrons move toward the cathode via the external circuit, where it converts into work by the external load. Meanwhile, the protons or hydrogen ions move through the electrolyte to the cathode catalyst layer. At the cathode catalyst layer, the protons, electrons, and oxygen combine to produce water via the following reaction: 2 H+ + 2 e− +

1 O2 −−→ H2 O. 2

(1.9)

Fuel cell fundamentals

Figure 1.7 PEMFC configuration and reactions.

Thus the overall reaction of PEMFC is H2 +

1 O2 −−→ H2 O. 2

(1.10)

Needless to say, PEMFCs are of the first type since hydrogen directly converts into ions by means of catalyst.

1.1.3.2 Direct methanol fuel cells (DMFCs) Direct methanol fuel cells are another low-temperature FC technology whose construction is very similar to PEMFCs. They operate at temperature range less than 100◦ and use graphite as their electrode. The membrane is also made of Nafion, just like an PEMFC. There are mainly two differences between a PEMFC and a DMFC. First, in DMFC the primary fuel is methanol instead of hydrogen. Second, in DMFCs, Pt is not sufficient for dissociating methanol to electron and ion. Therefore, in addition to Pt, some other catalysts such as ruthenium, Ru, should be added. In contrast to many other FC types, the name of DMFCs is not after its membrane type, but it is the name of its fuel. The most advantage of a DMFC is that there is no need for hydrogen storage or reformer. Liquid methanol mixed with steam is directly fed into the FC anode. It is quite clear that storage of methanol is not a matter at all. A simple plastic tank can store as much as methanol as needed. Moreover, the methanol production facilities are quite mature, and the infrastructure already exists. The methanol factories can produce pure methanol out of coal, natural gas, oil, and, more importantly, from biomass.

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Fuel Cell Modeling and Simulation

Figure 1.8 Configuration of a DMFC.

DMFCs are promising candidates for medium-power devices, especially for mobile applications such as cell phones and laptops. The most benefit of DMFCs is that they can be recharged by filling a small tank by methanol. This is quite cheap and handy. In addition to small-scale devices, DMFCs are enhancing for becoming competitive to PEMFCs in car industries. The choice of fuel is its most important factor. Storage of methanol, as mentioned, is not a matter at all. DMFC is also a first-type FC, in which cations are moving from the anode to cathode. The configuration of a DMFC is shown in Fig. 1.8. In contrast to PEMFC, the fuel is not a gas but is a liquid consisting of liquid methanol mixed with water. Methanol reaches the anode catalyst and converts to proton according to the following reaction: CH3 OH + H2 O −−→ 6 H+ + CO2 + 6 e− .

(1.11)

In this FC, six moles of electrons are released for each mole of fuel. Moreover, carbon monoxide is released as a byproduct. The produced CO2 is solved in the fuel until it reaches the solubility limit. Then the excess CO2 creates gaseous bubbles, which in turn increases the pressure drop, blocks the alcohol mass transfer to the catalyst, and, in general, reduces the performance of the cell. For DMFCs, special facilities should be designed to remove the CO2 gas from the mixture before recirculating. At the cathode catalyst layer, the protons, the electrons, and the oxygen combine to produce water via the following reaction: 6 H+ + 6 e− +

3 O2 −−→ 3 H2 O. 2

(1.12)

Fuel cell fundamentals

Thus the overall reaction of DMFC is CH3 OH +

3 O2 −−→ 2 H2 O + CO2 . 2

(1.13)

Note that carbon monoxide is not balanced and should be vented from the cell.

1.1.3.3 Alkaline fuel cells (AFCs) Alkaline fuel cells or AFCs are one of the first industrialized FCs and were invented by Francis Thomas Bacon in 1959. AFC was the first technology that was used in actual programs such as Apollo space program, in which they were used as the main source of electrical energy. These cells are categorized as medium-temperature types since they operate at temperature levels between 100◦ to 250◦ . The high operating temperature makes them work even if nonprecious metals are used as catalyst. Therefore, in construction of an AFC, other inexpensive catalysts can be used, such as Ag, CoO, etc. Although to improve the efficiency, Pt can also be used if the price is not a matter. The name of AFC comes from the material used for construing the membrane. In this type the membrane is made of concentrated solution of an alkali metal such as NaOH or KOH. Alkaline solutions have suitable ionic conductivity in any temperature, whereas the temperature rise increases the conductivity even more. Therefore it becomes a good choice for the cell electrolyte or membrane. Recent advances on AFC introduced the low-temperature cells capable of working at from 25◦ to 70◦ . Since there is no need for precious metals in construction of AFC and their membrane is not very expensive, they are the cheapest FC technology at the present time. However, they suffer from electrolyte poisoning when carbon dioxide CO2 is solved in it. The formation of carbonate dramatically reduces the electrolyte conductivity, which is referred to as electrolyte poisoning. To overcome electrolyte poisoning, the KOH solution should be either refreshed or purified using carbon dioxide scrubbers. These limitations make them not quite applicable for large power plants, but efforts are being made to overcome the problems and reduce the cost. The electrodes are made of graphite and metals such as nickel. The fuel is almost always hydrogen, and the oxidizer is the oxygen taken from the atmosphere. The result of the cell operation is electricity, heat, and water. The water should be removed from the system because it solves in the electrolyte and reduces its concentration. The water removal takes place through evaporation from electrodes or should be evaporated using a specially designed system. AFC is a second-type FC as is shown in Fig. 1.9. Hydroxide ions are generated on the cathode catalyst layer and move toward the anode catalyst through the potassiumhydroxide, KOH, electrolyte. At the anode catalyst layer the hydroxide ions combine with hydrogen to produce water and electrons. The anodic reaction is 2 H2 + 4 OH− −−→ 4 H2 O + 4 e− .

(1.14)

15

16

Fuel Cell Modeling and Simulation

Figure 1.9 An AFC configuration and reactions.

In this FC, two moles of electrons are released for each mole of fuel. At the cathode catalyst layer the electrons, water, and oxygen combine to produce hydroxide ions via the following reaction: 2 H2 O + 4 e− + O2 −−→ 4 OH− .

(1.15)

As we can see, the hydroxide ions are conserved through the electrode reactions. However, OH – concentration will not be uniform from anode to cathode because it is consumed at anode and produced at cathode. As far as the diffusion process is able to balance the OH – gradient, the cell works fine, but at high current density, this gradient may affect the performance of the FC. Fig. 1.9 shows that at cathode inlet, pure oxygen is shown instead of air. For these types of FCs, as mentioned before, oxygen is purified before entering the cell since the carbon dioxide would degrade the electrolyte performance. If pure oxygen is not provided and air is fed into the cell, then the figure information should be modified. The overall reaction of AFC is H2 +

1 O2 −−→ H2 O. 2

(1.16)

It is clear that AFC is a hydrogen FC in which hydrogen reacts with oxygen through electrochemical reactions, but its mechanism completely differs from a PEMFC.

Fuel cell fundamentals

Figure 1.10 PAFC configuration and reactions.

1.1.3.4 Phosphoric acid fuel cells (PAFCs) Phosphoric acid fuel cells are another medium-temperature technology, which uses liquid phosphoric acid as an electrolyte. The technology was first introduced at 1961 by G.V. Elmore and H.A. Tanner. The operating temperature of these cells is between 150◦ to 200◦ . At this level, phosphoric acid exhibits good ionic conductivity and can be used as the electrolyte. Usually, acid is contained in a teflon-bonded silicon carbide matrix. The electrodes are made of carbon and are coated by platinum catalyst. These FCs are one of the best choices for CHP purposes because of their operating temperature and cost. They are usually used in stationary applications and equipped with CHP devices to harvest the thermal energy and its electricity. By this configuration the overall efficiency of PAFCs can reach up to 70%. Although the electrodes of PAFCs contain Pt as a catalyst, they are much less sensitive to CO poisoning of the platinum because they work at much higher temperature, at which CO poisoning becomes less important. Consequently, the fuels that contain CO can also be used as fuel in this FC type. This characteristic makes them quite attractive because they can be equipped with special reformers for producing H2 . Hence H2 storage tanks are not necessary in PAFC power plants. Just note that if the reformer is used to produce H2 out of gasoline, the sulfur must be removed because it affects the electrolyte and reduces its conductivity. A PAFC shown in Fig. 1.10 is a first-type FC whose reaction is the same as that of a PMFC. In other words, the anodic, cathodic, and overall reactions are described by Eqs. (1.8)–(1.11). The difference between the two is the membrane or electrolyte material and, of course, their working temperature. A higher temperature of PAFCs

17

18

Fuel Cell Modeling and Simulation

makes them more tolerant to CO poisoning, which in turn makes them better choices for power plants. The fact that they can be used for CHP systems means that their net efficiency can be reach up to 85%, which is quite remarkable.

1.1.3.5 Solid-oxide fuel cell (SOFCs) Solid-oxide fuel cells are one of the most used high-temperature FCs in stationary and mobile systems. The working temperature of these cells is usually between 800 to 1000 ◦ C, which is the highest range in all the other FC technologies. The high temperature makes them to operate in CHP systems and reach their efficiency more than 85%. The name of SOFCs comes from the material of their electrolyte, which is a solidstate ceramic. In fact, the invention of yttria-stabilized zirconia (YSZ) ceramic began a new horizon in FC technologies. This ceramic is able to conduct oxygen ions O2 – at very high temperatures. At this range, oxygen converts into O2 – at the cathode catalyst layer as shown in Fig. 1.11 through the following reaction: 1 O2 + 2 e− −−→ O2− . 2

(1.17)

The oxygen ions move from the cathode toward the anode through the YSZ ceramic electrolyte as shown in the figure. This means that SOFC is a second-type FC. At the anode side the oxygen ions are combined with hydrogen via the following reaction: H2 + O2− −−→ H2 O + 2 e− .

(1.18)

The overall reaction is 1 O2 −−→ H2 O. (1.19) 2 The difference between reaction (1.19) and the overall reaction of a PEMFC or Eq. (1.10) is that the generated water in SOFC is in vapor phase, whereas the water in PEMFC is in liquid phase. The high temperature of a SOFC makes it possible to use a wide range of fuel because it does not use Pt as a catalyst. Although the main reaction requires hydrogen, in practice, SOFC can be equipped with internal or external reformers to reform any hydrogen-rich fuel such as natural gas or other hydrocarbons to produce hydrogen for its functioning. It should be noted that in practice, it is observed that sulfur deteriorates the operation of SOFC, and hence if the fuel contains sulfur, then it must be removed. The high operating temperature of SOFCs and their role in CHP cycles make them quite attractive for stationary applications. These cells are combined to produce megawatts of heat and electricity, which is suitable for charging in grids. On the other hand, small-scale SOFCs are produced to incorporate in mobile applications including in FC H2 +

Fuel cell fundamentals

Figure 1.11 SOFC configuration and reactions.

trucks, buses, and similar heavy-duty vehicles. One of the main challenges for these applications is that the start of the cell requires high temperature. Hence sophisticated facilities should be incorporated to heat the cell for its startup. SOFCs are also used as auxiliary power units for heavy vehicles to provide the required energy on their idle time.

1.1.3.6 Molten-carbonate fuel cell (MCFC) Some salts including carbonate of alkali metals, such as lithium, potassium, or sodium, exhibit great ionic conductivity at their molten state at high temperatures. Therefore they are used as the electrolyte in molten carbonate fuel cells (MCFCs) as their name reflects. These FCs should be warmed up to 650 ◦ C during the operation; otherwise, the electrolyte cools down and becomes solid, which in turn looses the ionic conductivity, and the FC stops operating. For this reason, MCFCs are high-temperature devices suitable for CHP cycles. The efficiency of MCFCs is about 45% at their operating point which is considered as a high efficiency. By combining them with an CHP cycle their efficiency may exceed 85%. These cells are very useful for local power stations as well as large power plants. The high operating temperature means that MCFCs tolerate impurities, and hence a wider range of fuels can be used. In these cells, many hydrocarbons can be used as a fuel including methane, propane, marine gasoline, coal gasification products, and of course hydrogen. On the other hand, the high temperature nature of MCFCs results in mechanical failure of packing and parts. These cells should not repeatedly be turned on and off.

19

20

Fuel Cell Modeling and Simulation

Figure 1.12 SOFC configuration and reactions.

If a hydrogen-rich fuel is used, then it should be reformed externally to produce hydrogen for the main operation. In addition to hydrogen, MCFCs require carbon dioxide for their main reactions; however, carbon dioxide is conserved since it is produced in the anode and consumed in the cathode. In other words, it makes a closed cycle, and the overall reaction does not produce or consume any carbon dioxide. MCFCs are among the second-type FCs, or better to say, the moving ion is an anion that moves from the cathode toward the anode. In this type, the moving ion is carbonate, and as is shown in Fig. 1.12, when it reaches the anode catalyst layer, it combines with hydrogen fuel through the following reaction: CO23− + H2 −−→ H2 O + CO2 + 2 e− .

(1.20)

The anodic reaction, in addition to electron, produces carbon dioxide, which is required for cathodic reaction. The carbon dioxide and electron are transferred to the cathode catalyst layer, where they react with oxygen by the following reaction: CO2 +

1 O2 + 2 e− −−→ CO23− . 2

(1.21)

These electrode reactions show that carbonate and carbon dioxide are conserved, but as explained for AFC, the concentration gradient becomes important at high current density operation.

Fuel cell fundamentals

The overall reaction of MCFCs, as a hydrogen fueled FC, is the same as that of the other types: 1 (1.22) H2 + O2 −−→ H2 O. 2 The produced water is in vapor phase because of the high operating temperature. This removes the flooding problems that are important in low-temperature FCs.

1.1.3.7 Other technologies The above-mentioned technologies are all commercially available and can be found in the market. There are many other technologies available, some of which are in the market and mass produced, and some of which are just in research labs. Here we briefly introduce some of them. High-temperature proton-exchange FC (HT-PEMFC) is a variant of PEMFC that can work at a higher temperature range about from 100 ◦ C to 200 ◦ C. The key point in constructing an HT-PEMFC is the material used in construction of its membrane. In a low-temperature PEMFC the separator is made of Nafion, which cannot tolerate high temperature and is destroyed at temperatures above 130 ◦ C (above its glassy point). In HT-PEMFCs, the separator is made of acid-based polymers such as pure polybenzimidazole (PBI) or acid-doped PBI, which can work in higher temperatures without any failure. Other parts are more or less the same as in low-temperature PEMFCs. The advantage of HT-PEMFC over the ordinary ones is that the higher temperature makes them more tolerant to CO poisoning, and hence a wider range of fuels can be used (instead of pure H2 ), and also their efficiency can be enhanced using CHP cycles. Metal–air FCs are the most mature technology and are mass produced in different shapes and sizes. These FCs are considered as either a fuel cell or a battery. The main difference between an FC and a battery is that the available energy of a battery is limited to its stored active material and no material enters the cell, whereas an FC does not store any material, and its active material flows from an external storage device or atmosphere. A metal–air FC is something in between. It can be looked just like an AFC (see Fig. 1.9) with the same cathode material and the same electrolyte, but the only difference is that its fuel is a metal such as zinc, aluminum, or lithium. Therefore the active material of anode is limited, and when the metal is totally consumed, the FC stops working. In this view, the anode acts like a battery, and the cathode like an FC. Some textbooks put the metal–air FCs in battery sections, and some references consider them as an FC. Direct Ethanol FC (DEFC) is a variant of DMFC, in which ethanol is used instead of methanol. The most attractive point about DEFC is that ethanol is not toxic and can be produced from many sources. Although DEFCs are very promising devices, they are still being produced on a lab-scale and will not appear on the market in the near future. Microbial FCs (MFCs) are bio cells in which, as the name stands, microbes and special bacteria are used for removing electron from available fuels. Since the operation

21

22

Fuel Cell Modeling and Simulation

depends on bacteria, the power density is very low, which means that to obtain a specific amount of power, a large system should be built. Many different species exist for this purpose, some of which are good for anodic reactions, and some for cathodic. Usually, anode and cathode should be separated by a separator to stop the bacteria moving from one electrode to the other.

1.2. Thermodynamics Regardless of the anodic and cathodic reactions, the overall cell reaction determines the behavior of the cell. Whether a reaction takes place through a chemical or electrochemical process, the heat of reaction is the same. For example, burning hydrogen by chemical reaction or Eq. (1.1) releases the same amount of heat as when it happens in an electrochemical process such as Eq. (1.16). The overall process of an FC can be written in general form as aA + bB −−→ cC + dD + H ,

(1.23)

where the uppercase letters are chemical species, and the lowercase letters are the stoichiometric coefficients. As indicated by the symbol H, each reaction produces a specific amount of energy called the enthalpy of reaction. This is the highest available energy in a reaction. The enthalpy of reaction is the energy difference between the stored chemical energies of the products and reactants. In other words, every chemical species stores a specific amount of energy in their chemical bonds. When the bond breaks, the energy is released, and when a chemical bond forms, a specific amount of energy is stored. Hence, for a reaction in the form of Eq. (1.23), reactants and products store some energy in their chemical bonds. However, their amounts are not equal, and the difference is denoted H. By this definition the enthalpy of reaction is calculated using the equation H = chC + dhD − ahA − bhB ,

(1.24)

in which h is the enthalpy of each substance. Eq. (1.24) is an expression for conservation of energy. It states that the enthalpy of reaction is the difference between the enthalpy of products and that of reactants. By definition the enthalpy of any substance at standard pressure and temperature, i.e., 1 atmosphere and 25 ◦ C, is called the enthalpy of formation and is denoted by hf . In other, nonstandard conditions, the enthalpy of the material differs from the enthalpy of formation. For a pure substance, the enthalpy of formation is zero because the substance is at its stable condition, but this is not true for compounds. The enthalpies of formation of different materials are tabulated in thermodynamic tables. Note that these values are valid only at standard conditions. For other states, the enthalpy should be calculated using thermodynamic relations to be discussed later.

Fuel cell fundamentals

Example 1.1. Calculate the enthalpy of reaction (1.1) at standard conditions. Answer. According to Eq. (1.24), at standard conditions the enthalpy of reaction (1.1) is 1 2

H = hf H2 O − hf H2 − hf O2 = −286 − 0 − 0 = −286 kJ kmol−1 .

The negative sign indicates that the energy level of the products is less than the reactants. Hence this amount of energy is released, or in other words, the reaction is exothermic. Note that almost none of the FCs work at standard conditions. All the technologies work at temperatures higher than 25 ◦ C. Some of the FCs work at atmospheric pressure, but some of them work at a higher pressure. Consequently, the heat of reaction should be calculated at the actual state. The effect of temperature and pressure will be discussed later in this chapter.

1.2.1 Gibbs free energy As discussed above, the enthalpy of reaction is the utmost available energy. However, each reaction has some irreversibility, which decreases the useful energy. In theory the irreversible processes are calculated using the entropy of reaction or S. Just like the enthalpy, the entropy of reaction (1.23) is calculated as S = csC + dsD − asA − bsB ,

(1.25)

where s is the entropy of each species, which is tabulated in thermodynamic tables. Note that the tabulated values are at standard conditions. For nonstandard states, the entropy should be calculated. The method is discussed in Section 1.2.4. For some key substances in hydrogen FCs, the enthalpy and entropy of formation (i.e., the values at standard conditions) are given in Table 1.1. After calculation of entropy of reaction, the lost energy due to entropy change is T S, and the net available useful energy, called the Gibbs free energy, is G = H − T S.

(1.26)

As discussed, the Gibbs free energy is the most available useful energy that can be converted into useful work. Example 1.2. Calculate the entropy and Gibbs free energy of reaction (1.1) at standard conditions. Answer. According to Eq. (1.25), at standard conditions, the entropy of reaction (1.1) is 1 2 = −0.163285 kJ mol−1 K−1 .

S = sf H2 O − sf H2 − sf O2 = 0.06996 − 0.13066 −

1 × 0.20517 2

23

24

Fuel Cell Modeling and Simulation

Table 1.1 Enthalpy and entropy of formation for some key substances. Substance hf (kJ mol−1 ) sf (kJ mol−1 K−1 ) H2 0 0.13066 0 0.20517 O2 −286.02 0.06996 H2 O(l) −241.98 0.18884 H2 O(g)

Note that at 25 ◦ C the produced water is liquid; hence for the calculations, we use the data for liquid H2 O. Then the Gibbs free energy is G = H − T S = −286 − 298.15(−0.163285) = −237.3165 kJ mol−1 .

The comparison of H and G shows that 48.68 kJ mol−1 energy is lost due to the irreversible processes. As stated before, the Gibbs free energy is the net useful energy and can be converted to other sources of energy. In an FC, this amount of energy is converted into electrical form. The electric energy is defined as Wel = qE,

(1.27)

where Wel is the electrical work in units of Joule per mole, and q and E are electrical charge per Coulomb per mole and electric potential per Volt, respectively. We saw that in each FC technology, for each mole of fuel, a specific amount of electrons is released. If for each mole of fuel, n mole of electrons is released, then the electrical charge becomes q = nNA qel = nF ,

(1.28)

where NA = 6.02 × 1023 is the Avogadro number, which indicates the number of electrons in one mole, and qel = 1.60217662 × 10−19 Coulombs is the electrical charge of a single electron. Finally, F = NA × qel = 6.02 × 1023 × −1.602 × 10−19 = 96485 C mol−1

(1.29)

is called Faraday’s constant, which is the total electrical charge of one mole of electron. Using the above equations, the electrical work then is defined as Wel = nFE.

(1.30)

Since the total electrical work is equal to the Gibbs free energy, we can write Wel = −G,

(1.31)

Fuel cell fundamentals

where the negative sign is used because for a heat generating process, the Gibbs free energy is negative. From Eq. (1.31) we can find the theoretical potential of an FC: E=

−G

nF

.

(1.32)

This is called the theoretical voltage or the open circuit voltage (OCV) of the cell. Because in calculating the voltage or potential, we have neglected the potential drop due to ohmic resistance and other parameters. In other words, Eq. (1.32) is valid only for open-circuit conditions when no external current is passing the FC. Eq. (1.32) can be used to calculate the OCV of any FC at any operating condition. Just keep in mind that using this equation, G should be obtained at the same operating conditions. Example 1.3. Calculate the OCV of a hydrogen fuel cell. Answer. For a hydrogen FC, the Gibbs free energy is calculated in Example 1.2. Therefore the open circuit voltage can be easily calculated from Eq. (1.32): E=

−G

nF

=

237316.5 ∼ 1.230 V. 2 × 96485

Note that n = 2 is chosen because for all the hydrogen FCs, for each mole of hydrogen, two moles of electron are released. This can be checked by analyzing the anodic reactions of the hydrogen FCs, such as PEMFC, SOFC, and so on. Also note that E = 1.23 V is only valid at 25 ◦ C because G was calculated at that temperature. At other temperatures the open-circuit voltage differs.

1.2.2 Second law of thermodynamics and fuel cells The second law of thermodynamics states that the total entropy of a system must be increased over time. This statement is true for an isolated system, and if the process is reversible, then the entropy does not change. Each electrochemical reaction contains a specific amount of energy H. Any change in system entropy destroys a part of the available energy, and hence the amount of useful energy or Gibbs free energy decreases, which in turn, according to Eq. (1.32), is reflected in lowering the cell voltage. As we will see in future sections and chapters, there are many different parameters that cause the lowering of the cell voltage including: Crossover of fuel, in which a part of the fuel passes through the membrane and reaches the catalyst of the cathode. Activation polarization, which is the energy required for driving the electrochemical reactions. Ohmic resistance of the cell different parts, which according to Joule’s law, produces heat, which in turn destroys the useful energy (converts the useful energy into heat).

25

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Fuel Cell Modeling and Simulation

Concentration polarization, which happens when the concentration of fuel or oxygen decreases at the anode or cathode catalyst layers. Heat dissipation into environment results in destroying a part of useful energy, and if becomes too large, then the cell temperature decreases, which in turn decreases the cell voltage. The above-mentioned mechanisms and any other mechanism that destroys the energy level in any form, increase the entropy of the system and cause the voltage drop. Therefore, in practical works, experiments, and actual plant monitoring, the best way to measure the level of reversibility of the system is measuring the voltage drop. The final point about the entropy change is the role of temperature. Temperature has two different roles in electrochemical reactions: 1. It decreases G since the energy loss due to entropy change is defined as T S. Thus the higher the temperature, the higher the energy loss due to the entropy change. 2. It reduces the internal resistances, which in turn reduce the ohmic resistance and ohmic drop. Therefore temperature rise decreases the OCV and reduces the internal entropy change, which in turn increases the operational voltage. Therefore it has a dual effect on energy loss due to entropy change. This means that at lower current density the overall voltage drops, whereas at higher current density the overall voltage increases as the temperature rises.

1.2.3 Fuel cell efficiency The efficiency of any system is defined as the ratio of the output work to its input energy. For and FC, the input energy is the total energy H available from electrochemical reaction. On the other hand, the total output work is Wel , which according to Eq. (1.31) is equal to G. Consequently, the efficiency is defined as η=

G . H

(1.33)

Eq. (1.33) is valid for any state of operating condition if only H and G are calculated at the same state. Example 1.4. Calculate the efficiency of a hydrogen fuel cell at standard conditions and at OCV. Answer. For a hydrogen FC, we can easily calculate its efficiency at OCV from Eq. (1.33): η=

G 237.34 = ∼ 83%. H 286.02

Fuel cell fundamentals

The result shows that a hydrogen FC has a very high theoretical efficiency at its OCV condition. In practical and operational conditions, an FC does not work at OCV. When a specific amount of current is passed through the FC, its voltage drops due to many factors, which will be discussed in this chapter. The voltage drop is an indicator of irreversible processes, which in turn will reduce the cell efficiency. Calculation of all the irreversible processes is not an easy task, but fortunately a very useful relation can be derived from Eq. (1.33) for calculation of the efficiency. Dividing the numerator and denominator of the right-hand side of Eq. (1.33) by −nF gives η=

− nFG − nFH

(1.34)

.

The numerator is by definition the electrical potential or E. The nominator has the same unit as the electrical potential and is a fictitious voltage known as the thermoneutral potential and is denoted by Eth . Therefore Eq. (1.34) can be rewritten as a fraction of two voltages, the actual cell voltage and the thermoneutral potential: η=

E . Eth

(1.35)

The main advantage of Eq. (1.35) is that the efficiency of an FC can be calculated just by measuring its operational voltage. Example 1.5. Calculate the thermoneutral potential of a hydrogen FC. Answer. Thermoneutral potential of an FC is defined as Eth = −

H

nF

.

For a hydrogen FC at standard conditions, it becomes Eth = −

−286 ∼ 1.482 V. 2 × 96485

Note that for other temperatures or pressures, we have to calculate H at the same conditions. Example 1.6. Assume that in actual operating conditions, by passing a specific amount of current the voltage of a hydrogen FC reaches 0.85 V. What would be the efficiency of the cell at these conditions?

27

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Fuel Cell Modeling and Simulation

Answer. The efficiency of an FC is calculated using Eq. (1.35): η=

E 0.85 = ∼ 57%. Eth 1.482

This example shows that the actual efficiency at operating conditions decreases.

1.2.4 Role of effective factors All FCs work under nonstandard conditions. The standard condition rarely is operational in low-temperature cells. For example, metal–air FCs can be used at atmospheric pressure at 25 ◦ C. How often does it happen in practice? All the medium- and hightemperature cells work at nonstandard temperature. Even the low-temperature cells are preferred to work at temperature levels over 60 ◦ C. In addition to temperature, the operating pressure may increase to obtain higher efficiency. Therefore, to have more accurate calculations, the thermodynamic parameters such as H, S, and G must be calculated at the operational states. Temperature and pressure have very important effect on thermodynamic parameters. We discuss these parameters in more detail.

1.2.4.1 Effect of temperature The enthalpy and entropy of each substance are functions of temperature and are defined by the following equations: 

hT = hf + s T = sf +

T

cp dT ,

(1.36)

298.15  T

1 cp dT , T 298.15

(1.37)

where cp is the specific heat of the substance, and hT and sT , respectively, are its enthalpy and entropy at temperature T. The specific heat cp is a function of temperature, and hence to obtain hT and sT , this relation should be known. For most elements and compounds, the relation is given in thermodynamic tables. For example, for the substances involved in hydrogen FCs, the data are shown in Fig. 1.13. To work with the tabulated values, it is better to make a curve fit on the data and work with the relation, instead. The data shown in Fig. 1.13 indicate that they can be perfectly fitted by a second-order polynomial of the form cp = a + bT + cT 2 .

(1.38)

For different species involved in a hydrogen FC, the coefficients of Eq. (1.38) are given in Table 1.2. Using Eq. (1.38) and the data of Table 1.2, the integration of Eqs. (1.36) and (1.37) is straightforward.

Fuel cell fundamentals

Figure 1.13 Variation of cp for different species in a hydrogen FC. Table 1.2 cp temperature dependency coefficients for a hydrogen FC. Substance a b c H2 28.91404 −0.00084 2.01 × 10−0.6 O2 25.84512 0.012987 −3.90 × 10−0.6 H2 O(g) 30.62644 0.009621 1.18 × 10−0.6

Example 1.7. For an AFC working at 250 ◦ C, calculate the OCV and the theoretical efficiency. Answer. The enthalpy of reaction (1.1) is calculated using Eq. (1.24): 1 2

H = hH2 O − hH2 − hO2 .

At 250 ◦ C, the enthalpy of pure substances such as H2 and O2 is no longer zero and must be calculated using Eq. (1.36). Also, the enthalpy of H2 O changes from its value at 25 ◦ C and must be evaluated at the same temperature. Also note that at this temperature, water is in vapor state. Therefore, using the data of Table 1.1, we have to consider this fact. Hence, for H2 , substituting Eq. (1.38) into Eq. (1.36), we have 

h T = hf +

523.15



b c (a + bT + cT )dT = hf + aT + T 2 + T 3 2 3

523.15

2

298.15

. 298.15

Using the data of Table 1.2 for each species, we have 

hT ,H2 = 0 + 28.914T +

−0.00084

2

T2 +

2.01 × 10−0.6 3 T 3

523.15 = 6509.72, 298.15

29

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Fuel Cell Modeling and Simulation



0.01298 2 −3.90 × 10−0.6 3 T + T hT ,O2 = 0 + 25.845T + 2 3 

hT ,H2 O = −241980 + 30.626T +

0.009621 2 T + 2

523.15 = 7760.02,

298.15 −0.6 1.18 × 10

523.15 = −233173.49.

T3

3

298.15

Thus the enthalpy of the reaction is 1 2

H = −233173.49 − 6509.72 − 7760.02 = −243563.21 J mol−1 .

The same procedure must be done for calculation of sT : 1 2

S = sH2 O − sH2 − sO2 .

Substituting Eq. (1.38) into Eq. (1.37), we have 

s T = sf +

523.15

( 298.15

 a c 523.15 + b + cT )dT = sf + a ln(T ) + bT + T 2 . T 2 298.15

Then for different species, we have 

523.15



298.15 523.15

2.01 × 10−0.6 2 T sT ,H2 = 130.66 + 28.914 ln(T ) + −0.00084T + 2 sT ,O2 = 205.17 + 25.845 ln(T ) + 0.01298T +

−3.90 × 10−0.6



sT ,H2 O = 188.84 + 30.626 ln(T ) + 0.009621T +

2

= 147.12, = 221.90,

T2

1.18 × 10−0.6 2 T 2

298.15 523.15

= 208.44. 298.15

Thus the entropy of the reaction is 1 2

S = 208.44 − 147.12 − 221.90 = −49.63 J mol−1 K−1 .

Consequently, the Gibbs free energy of reaction becomes G = H − T S = −243563.21 − 523.15(49.63) = −217601.83 J mol−1 .

Having G at 250 ◦ C, the OCV of the AFC is E=

−G

2F

=

217601.83 = 1.1276 V, 2 × 96485

and its theoretical efficiency is η=

G −217601.83 = ∼ 89%. H −243563.21

Fuel cell fundamentals

Table 1.3 Results of Example 1.8. Temperature (K) H S 373.15 -242580 -46.57 -242941 -47.74 423.15 473.15 -243269 -48.74 523.15 -243563 -49.62 -243819 -50.39 573.15 623.15 -244037 -51.07 -244214 -51.67 673.15 -244349 -52.20 723.15 773.15 -244440 -52.67 -244486 -53.10 823.15 873.15 -244483 -53.47 -244431 -53.80 923.15 -244327 -54.10 973.15 -244170 -54.36 1023.15 -243958 -54.59 1073.15 1123.15 -243690 -54.79 1173.15 -243362 -54.96 -242974 -55.10 1223.15 -242524 -55.23 1273.15

T S -13239 -14603 -17377 -20201 -23065 -25961 -28883 -31825 -34782 -37751 -40728 -43709 -46692 -49673 -52651 -55623 -58586 -61539 -64480

G -225202 -222740 -220204 -217601 -214936 -212212 -209432 -206598 -203712 -200776 -197791 -194757 -191676 -188547 -185372 -182150 -178882 -175567 -172206

E 1.167 1.154 1.141 1.127 1.113 1.099 1.085 1.070 1.055 1.040 1.024 1.009 0.993 0.977 0.960 0.943 0.926 0.909 0.892

η

0.928 0.916 0.905 0.893 0.881 0.869 0.857 0.845 0.833 0.821 0.809 0.796 0.784 0.772 0.759 0.747 0.735 0.722 0.710

This example shows that both the OCV and theoretical efficiency of FCs decrease as the temperature increases. Note that the theoretical efficiency is the efficiency of the cell when no current passes through it or, in other words, when the cell is at its OCV state. Example 1.8. Calculate OCV for a hydrogen FC at temperature range from 100 to 1000 ◦ C. Answer. According to the problem, the temperature is over 100 ◦ C. Hence water is in vapor phase. Therefore the calculations are based on the same values of Example 1.7. The results of calculations for H, S, G, E, and η are tabulated in Table 1.3. The variation of these parameters is shown in Fig. 1.14. The graphs show that H increases up to about 900 K and then decreases, but G has a monotonic behavior and always decreases by increasing the temperature (note that the absolute value is important, not the signed value). This happens since the S increases and the energy loss, which is T S, increases by a higher slope. The net result is the increase in cell voltage and consequently cell theoretical efficiency. For low-temperature FCs, where the temperature range is lower than 100 ◦ C, the variation of cp can be ignored and assumed to be constant. For example, for PEMFC,

31

32

Fuel Cell Modeling and Simulation

Figure 1.14 Variation of different parameters.

Fuel cell fundamentals

which works at 70 ◦ C, the variation of cp due to temperature variations is very small and can be neglected. For these devices, Eqs. (1.36) and (1.37) have very simple answers hT = hf + cp (T − 298.15),   T . sT = sf + cp ln 298.15

(1.39) (1.40)

These equations give a much simpler relation for calculation of thermodynamic parameters and are also very accurate. Even for medium-temperature FCs, these equations can be used if cp is obtained at a medium temperature.

1.2.4.2 Effect of pressure Up to now, all the thermodynamic relations were obtained and discussed at atmospheric pressure. However, an FC may work at a higher pressure, at which the enthalpy and Gibbs free energy differs from their values at atmospheric pressure. To correct the thermodynamic relations, we must derive a relation of Gibbs free energy with respect to pressure. For an isotherm process, we can define the following relation for this purpose dG = Vm dp,

(1.41)

where Vm is the molar volume in m3 mol−1 , and p is the pressure in Pa. For a perfect gas, we can write pVm = RT .

(1.42)

The above equations yield dG = RT

dp . p

(1.43)

Integration of Eq. (1.43) gives the variation of Gibbs free energy with respect to pressure: 



p G = G◦ + RT ln , p◦

(1.44)

where G◦ is the Gibbs free energy at one atmosphere and 25 ◦ C, and p◦ is the reference pressure of one atmosphere in Pa. For any reaction in the form of Eq. (1.23), G is given as G = cGC + dGD − aGA − bGB .

(1.45)

33

34

Fuel Cell Modeling and Simulation

Substituting Eq. (1.45) into (1.44) gives ⎡ c d pC pD ⎢ p◦ p◦ G = G◦ + RT ln ⎢ ⎣ pA a pB b

p◦

k

p◦

⎤ ⎥ ⎥. ⎦

(1.46)

This relation is called the Nernst equation. In fact, in general, Nernst equation describes the Gibbs free energy in nonstandard states. For a hydrogen/oxygen FC, the Nernst equation is 



PH2 O . G = G◦ + RT ln 0.5 PH2 PO 2

(1.47)

Note that the uppercase Ps are nondimensional pressure described in bar. Substituting the Nernst equation into Eq. (1.32), we can obtain the cell OCV under nonstandard pressure: 



0.5 PH2 PO RT 2 . ln E = E◦ − nF PH2 O

(1.48)

Example 1.9. In a PEMFC, at standard conditions, atmospheric air is fed into cathode channel. What would be the OCV? Answer. The OCV of a hydrogen/oxygen FC was calculated in Example 1.3, E = 1.23 V. If instead of oxygen, air is fed into the cell, the cell voltage drops because the oxygen partial pressure is about 0.21. The voltage drop is calculated using Eq. (1.48): 

0.5 pH2 pO RT 2 ln E = E◦ − nF pH2 O







8.314 × 298.15 1 × 0.210.5 = 1.22 V. = 1.23 + ln 2 × 96485 1

The example shows that using air instead of oxygen reduces the OCV. Note that if air is used instead of oxygen, then the cell operational voltage dramatically drops, but its effect on OCV is negligible.

1.3. Electrochemical reaction kinetics Up to now, we studied the theoretical behavior of an FC. The theoretical behavior is the condition in which no energy loss takes place and all the chemical energy is converted into electrical energy of work. Such a condition happens only in OCV state when no current passes through the cell. At this state the cell reaches its maximum voltage and efficiency. However, when electrical current passes the cell, irreversible processes, such

Fuel cell fundamentals

as ohmic drop, Joule heating, concentration polarization, and other physical and chemical phenomena, reduce the available useful energy, resulting in voltage drop. Therefore the cell voltage is a function of current. In analyzing the relation between voltage and current, we have to pay special attention to electrode kinetics. The electrode kinetics determines the ability of the electrode to produce electrons in a unit time. The faster the kinetics is, the greater the potential of generating or consuming electricity the electrode has. This is because the faster kinetics is translated in lower entropy generation or less irreversible process. This means that the faster electrode encounters a lower voltage drop in comparison to a slower electrode. The electrode kinetics can be enhanced using proper catalysts. For example, in PEMFCs the conventional catalyst is Pt. If other catalysts such as Ni are used, then the electrode becomes slower and will generate a greater voltage drop if a specific current passes through the cell. As another example, in DMFCs the catalyst contains Ru in addition to Pt, since extraction of electron from methanol is not very fast at pure Pt. Hence if Ru is removed, then the electrode kinetics becomes slow and produces a higher voltage drop. At its simplest form, considering the voltage drop, the cell voltage is expressed by E = EOCV − iRint ,

(1.49)

where E and EOCV , respectively, are the FC operational voltage and its voltage at OCV, i is the current density passing through the cell, and Rint is the internal resistance. It is customary to express all the relations and equations in terms of current density to be able to compare different FCs. In terms of electrode kinetics, it is obvious that a larger FC is able to produce larger current because its active area is larger. Assume the situation shown in Fig. 1.15a, where an applied current Iapp passes through the cell. Increasing the projected area of the cell or h × w, the cell will be able to produce more current with lower voltage drop. Therefore, if we have two different FCs of different sizes, then we cannot compare their behavior. To solve the problem, if we consider just a unit area of the cell as shown in Figs. 1.15a and 1.15b, then the results of all the FCs will become comparable. In this situation, the current density is defined as i=

Iapp . h×w

(1.50)

In the rest of the book, all the relations will be expressed in terms of current density, unless it is explicitly specified. The internal resistance in Eq. (1.49) is a very complex parameter. The internal resistance is due to electron movement in solid parts, ionic transport though the membrane, electrical resistance at electrode/electrolyte interface, and many other parameters. These parameters themselves are functions of temperature, pressure, humidity level, species

35

36

Fuel Cell Modeling and Simulation

Figure 1.15 Illustration of current density.

concentration, and so on. Therefore Rint is very hard to find. Consequently, calculation of Eq. (1.49) is not as easy as it seems. In the following subsections, we will give a relation between these parameters.

1.3.1 Exchange current density All the chemical and electrochemical reactions are reversible, meaning that they take place in both forward and backward directions. Assume the extraction of electron in a hydrogen FC: forward

+ − −− −− −− − H2 −  −− 2 H + 2 e .

backward

(1.51)

As indicated by Eq. (1.51), the reaction may take place in both directions. In some cases the forward reaction is important, and in some cases the backward. In both cases, when one direction is important, it does not mean that the opposite direction is stopped. In fact, it means that the opposite direction is negligible, but it exists. To have a better understanding, consider the negative electrode of a hydrogen FC. In the present statement, it is preferred to use the term negative electrode over anode, because we are dealing with a reversible process, in which during the charge, the electrode becomes cathode instead. Therefore, using the word “anode” is not appropriate here. The electrode is under three different operational conditions, (a) discharge, (b) charge, and (c) rest. These three cases are studied in more detail. Discharge is a normal behavior of an FC during which the electrons are extracted from hydrogen and move out of electrode as shown in Fig. 1.16a. During this process, although the electrode is known to be an anode, but it does not mean that Eq. (1.51) takes place only in forward direction; in fact, both directions exist as usual, but forward

Fuel cell fundamentals

Figure 1.16 Negative electrode of a hydrogen FC.

direction is superior to backward direction. Therefore the backward reaction can be ignored because its amount is negligible in comparison to the forward reaction. For the negative electrode, the forward reaction is called the anodic reaction of the negative electrode because the reaction wants to make the electrode work as an anode. In the same manner the backward reaction is called the cathodic reaction of the negative electrode since it happens to make the negative electrode work as a cathode. Since the anodic reaction is much stronger than the cathodic reaction, the electrode is considered as an anode. Charge process is in opposite direction to discharge process. When an FC is working as an electrolyzer for splitting the water into oxygen and hydrogen, the situation is as illustrated in Fig. 1.16b, in which electrons enter the negative electrode from the external circuit and the electrochemical reaction of the negative electrode is the backward process of reaction (1.51). In this situation the negative electrode is the cathode of the electrolyzer because the electrons are consumed during the process. Again, as stated for the discharge process, the electrode does not work under the pure backward process or as the mentioned cathodic process. In fact, the anodic process also accompanies the cathodic process, or it means that the forward reaction also happens at the same time but its amount is negligible. Rest is the situation where no current enters or leaves the electrode or, in other words, where the FC is under open circuit conditions. In this case the net produced electron according to reaction (1.51) is zero, but it should be noted that both the forward and backward reactions take place at the catalyst layer. The fact that the amount of current in the external circuit is zero does not mean that the electrochemical reactions at the catalyst layers are stopped. It means that they exist but the rate of anodic reaction is equal to the rate of cathodic reaction, resulting in net zero electron production.

37

38

Fuel Cell Modeling and Simulation

Understanding the above statement is very important in analyzing the electrochemical behavior of any electrochemical system, especially, the FCs. Mathematically speaking, the rate of forward reaction is represented by the current produced by forward reaction with negative sign because the direction of electrical current is opposite to the direction of electron movement. Equally, the rate of backward process is determined by the electrical current produced by backward reaction. The net current that passes through the external circuit is the difference between these two values, i = if − ib .

(1.52)

In discharge process, if  ib , and in charge, ib  if . It is interesting that in rest, if = ib = 0. The nonzero value for forward and backward current is a very important parameter in electrochemistry and is called the exchange current density and is shown by i◦ = if = ib = 0.

(1.53)

This parameter determines the ability of an electrode to produce or consume electrons or, equivalently, the electrical current. The faster the electrode, the higher the value of i◦ . The exchange current density i◦ is a surface function because the electrochemical reactions are surface phenomena. Moreover, it is a strong function of temperature and also strongly depends on the composition of the catalyst layer and its construction process. For example, if we choose Ni instead of Pt in a PEMFC, then the electrode kinetics becomes strongly sluggish, meaning that its i◦ is decreased dramatically. A fast electrode, is an electrode able to generate and consume electrons as fast as it can during the rest process. Both forward and backward reactions according to Eq. (1.51) are very fast, meaning that in a unit of time, lots of electrons are produced, and in the same time the same produced electrons are consumed. A slow electrode has an opposite meaning. Therefore, in practice and modeling, the main electrochemical parameter indicating the behavior of the electrode is its exchange current density. By measuring i◦ in practice, we can determine how fast the electrode is.

1.3.2 Butler–Volmer equation So far, we understood that the electrode kinetics is very important and the faster the kinetics, the lower the voltage drop. How can we relate the electrode kinetics to voltage drop? Or what is the relation of electrical current and electrode polarization? The term electrode polarization or simply polarization is the same as the voltage drop. The only difference is that the former is more used by chemists and electrochemists, and the latter by electrical or mechanical engineers. The answer to the above question is that electrode kinetics is obtained using the so-called Butler–Volmer equation, which is the kinetic equation for electrochemical reactions. This equation in fact expresses the relation between

Fuel cell fundamentals

the current and voltage of a single electrode. This fact is very important to remember that the Butler–Volmer equation is written for a single electrode, not for the whole battery. To reemphasize, by the Butler–Volmer equation we can calculate the electrode polarization or voltage drop of a single electrode. So, to calculate the polarization of the whole FC, we need to use the Butler–Volmer equation twice, one for each electrode, and then calculate the whole polarization by summing up the polarization of both electrodes. Therefore the whole discussion in this section is for one single electrode. As discussed before, at any single electrode, both anodic and cathodic reactions take place. Therefore at a single electrode, we can write kf

−−  Ox + ne−  −− − − Red. kb

(1.54)

In this equation, Ox and Red are the oxidized and reduced materials, and kf and kb are, respectively, the forward and backward kinetic rates. By this notation the forward reaction is the cathodic reaction of the electrode, and the backward is the anodic, since the forward reaction consumes and the backward produces the electrons. Regardless of the electrode type (being the negative or positive electrode), reaction (1.54) describes its reactions. Example 1.10. For a hydrogen FC, express the electrode reactions in terms of Eq. (1.54) and discuss. Answer. For the anode of a hydrogen FC, we have kf

−−  2 H+ + 2 e −  −− − − H2 , kb

in which Ox is the hydrogen ions 2 H+ , Red is the pure hydrogen H2 , and n = 2. Note that for the anode under normal operation of an FC, the backward reaction is much faster than the forward. For the cathode of a hydrogen FC, we have 2 H+ + 2 e − +

kf 1 −  O2 −  −− − − H2 O. kb 2

This is in fact the oxygen reduction reaction known as ORR, which in its simple form is kf 1 2− −  O2 + 2 e − −  −− − −O . kb 2 Thus Ox is the pure oxygen O2 , Red is the oxygen ion O2− , and n = 2. Note that for cathode under normal operation of an FC, the forward reaction is much faster than the backward.

39

40

Fuel Cell Modeling and Simulation

The current density (see Eq. (1.50) and its discussions) that flows in the external circuit is the effect of electrical charges of electrons. Therefore the number of the moles of electrons that are released in a reaction is directly proportional to the electrical current. This relation was first discovered by the famous British scientist Michael Faraday, and the relation is also called Faraday’s law of electrolysis or Faraday’s relation: i = nFj,

(1.55)

where i is the external current density (A cm−2 ), n is the number of transferred electrons, F is the Faraday’s constant (C mol−1 ), and j is the number of moles of reaction that takes place in a unit of time in a unit area (mol cm−2 s−1 ). Faraday’s relation is important because it relates the electrical current to the number of moles of active material. It is well known that the number of moles of active materials is proportional to their concentration. In mathematical formulation, it is jf = kf COx , jb = kv CRed .

(1.56) (1.57)

Applying Eqs. (1.55)–(1.57) to Eq. (1.52), we find the net external current i = nF (kf COx − kb CRed ).

(1.58)

To be able to calculate the amount of external current, the kinetic rates should be known. These values are known as a function of temperature and Gibbs free energy: 

k=



G kB T , exp − h RT

(1.59)

where kB = 1.38 × 10−23 J K−1 and h = 6.626068 × 10−34 J s are the Boltzmann and Planck constants, and T and R are the absolute temperature and universal gas constant. The Gibbs free energy in electrochemical reactions in which an electron also exists has two parts, the chemical part, which as before describes the energy of chemical bonds, and the energy of electron and electrochemical bonds. Therefore the Gibbs free energy for forward and backward reactions is respectively expressed as G = Gch + αRed FE, G = Gch − αOx FE.

(1.60) (1.61)

In these equations, αRed and αRed are called the charge transfer coefficients of cathodic and anodic reactions. They play the same role as the heat transfer coefficient plays in the heat transfer phenomenon. More specifically, the higher the coefficients, the better the electron transfer.

Fuel cell fundamentals

Note that for a single electrode, these two coefficients are related to each other, and if one of them increases, then the other decreases. In general, we have the following relation: n αRed + αOx = . (1.62) ν

To define ν , we have to explain a little bit the reaction mechanisms. For any chemical or electrochemical reaction to happen, lots of intermediate chain reaction steps should take place. Among the many intermediate steps, some of them are very fast, and some are slow. Obviously, the reaction rate is determined by the slowest reactions. In some mechanisms the slowest step may happen several times, say ν ; for example, if the slowest step happens three times, then ν = 3. Examining and measuring αOx and αRed are not an easy task. These parameters in simulations are chosen so that the experimental and numerical values coincide. The above discussion yields the following relations for kinetic rates of reactions: 



Gch + αRed FE kB T , exp − kf = h RT   Gch − αOx FE kB T . exp − kb = h RT

(1.63) (1.64)

These equations can be simplified: 

kf = k◦,f exp −

αRed FE



RT  αOx FE , kb = k◦,b exp + RT

(1.65)

,



(1.66)

where all the chemical energy and constants are gathered in the coefficients k◦ . Substituting Eqs. (1.65) and (1.66) into Eq. (1.58), we obtain the following equation: 



i = nF k◦,f COx exp −

αRed FE

RT



  αOx FE − k◦,b CRed exp + .

RT

(1.67)

Note that αRed is the charge transfer for reduction reaction, in which an oxidized material Ox is reduced to a reduced material or Red. Thus αRed must be multiplied by Ox as shown in Eq. (1.67). The same argument is true for αOx . Eq. (1.67) expresses the relation of a single-electrode voltage E and its current density i; however, it is a little bit hard to use. First, k◦,f and k◦,b should be determined. Second, the electrode voltage needs a reference value since the potential is a relative parameter and has no absolute zero or reference. Finally, determining the surface local concentrations is not easy.

41

42

Fuel Cell Modeling and Simulation

To overcome the above-mentioned problems, the notion of exchange current density is very useful. We know from Eq. (1.53) that when the external current is zero, the interior current is called the exchange current density. This state is also called the open-circuit voltage when there is no polarization due to external current. Hence the electrode is at its highest voltage level, which is called the reversible potential, denoted Er . Thus mathematically we have 

i◦ = nFk◦,f COx exp −

αRed FEr



RT

  αOx FEr = nFk◦,b CRed exp + .

RT

(1.68)

Combining Eqs. (1.67) and (1.68) yields 



i = i◦ exp −

αRed F (E − Er )

RT



  αOx F (E − Er ) − exp + .

RT

(1.69)

Eq. (1.69) is one of the most famous relations in electrochemical systems and is called the Butler–Volmer equation. The equation expresses the relation between the cell voltage and exchange current density of a single electrode. It shows that for passing a specific amount of current such as i, the cell voltage must drop to E instead of Er to let the current flow. In other words, the so-called overpotential defined as η = E − Er

(1.70)

is the driving force for the current flow. The higher the η, the higher the current. Obviously, at OCV or reversible state when no current passes, η = 0. This fact can be easily seen from Eq. (1.69). Note that as stated, the Butler–Volmer equation is written for a single electrode only. Hence, to find the FC overall cell voltage, this equation should be written for each individual electrode to obtain the potential drop of each electrode. Then the net polarization is the sum of the two-electrode polarization. For this purpose, we use the following equations for anode and cathode:      αRed,a F (Ea − Er ,a ) αOx ,a F (Ea − Er ,a ) ia = i◦,a exp − − exp + , RT RT      αRed,c F (Ec − Er ,c ) α ,c F (Ec − Er ,c ) − exp + Ox . ic = i◦,c exp −

RT

RT

(1.71) (1.72)

The conservation of electrical charge indicates that the current produced by one electrode is exactly the same as consumed by the other electrode; or the electrons that move out of an electrode will enter the other electrode. No electron is generated or destroyed in the connecting wire. Since the electrical current is a directional parameter, the currents of cathode and anode have different signs: ia = −ic .

(1.73)

Fuel cell fundamentals

Example 1.11. For a PEMFC working at standard conditions, plot the cell voltage versus current density. Assume that the following data are available: Electrode

αa

αc

Hydrogen Oxygen

1 1

1 1

i◦ (A cm−2 ) 10−3 10−6

E (V) 0 1.23

Answer. The polarization of each electrode should be obtained separately. Note that for the present example, the anodic and cathodic currents of each electrode are equal. Hence Eqs. (1.71) and (1.72) are simplified as 



1 × F (Ea − Er ,a ) , ia = 2i◦,a sinh RT   1 × F (Ec − Er ,c ) . ic = 2i◦,c sinh RT Solving these equations for electrode potential yields 



RT ia , sinh−1 F 2i◦,a   RT ic . Ec = Er ,c + sinh−1 F 2i◦,c

Ea = Er ,a +

(1.74) (1.75)

Substituting the given data for OCV, we have Ecell = Er ,c − Er ,a = +1.23 − (0.0) = +1.23 V. If a current of i = 0.05 passes the cell, then the polarization of the anode and cathode is calculated as follows: 



0.05 = 0.042 V, 2 × 10−3   −0.05 = 1.011 V. Ec = +1.23 + 0.025691 sinh−1 2 × 10−6 Ea = 0.0 + 0.025691 sinh−1

Consequently, the overall cell voltage becomes Ecell = Er ,c − Er ,a = +1.01 − 0.042 = 0.969 V. The same procedure can be done for other current density values. The result is shown in Fig. 1.17. The results show that in this FC the most polarization is due to the cathode, as we expected, since the i◦ of the cathode is much less than the i◦ of the anode.

43

44

Fuel Cell Modeling and Simulation

Figure 1.17 Characteristic curve of Example 1.11.

Example 1.12. A constant current load of Iapp = 10 A is going to be powered by the FC of Example 1.11. The application requires the cell voltage to be 0.7 V. Determine the actual surface area for this application. Answer. Fig. 1.17 shows that for E = 0.7, the current density is i ∼ 0.9 A cm−2 . Thus from Eq. (1.50) we have i=

Iapp Iapp 10 =⇒ A = = ∼ 11.1 cm2 . A i 0.9

Therefore the actual surface area is equal to A = 11.1 cm2 . The Butler–Volmer equation gives only one part of polarization known as the activation polarization, which is the polarization required for generation of a specific amount of current. In practice, at high current density values, the concentration polarization dominates and makes a sudden drop in the voltage–current curve. Therefore the cell voltage shown in Fig. 1.17 is not an actual case because the voltage drop is not modeled perfectly.

1.3.3 Role of effective factors The kinetics of an FC is described by the Butler–Volmer equation, for which the main parameters are the exchange current density, anodic and cathodic charge transfer coefficients, and temperature. Since any electrochemical reaction is a surface phenomenon, all the parameters depend on the surface properties such as its morphology, porosity, tortuosity, and roughness. In addition to these parameters, the parameters strongly depend

Fuel cell fundamentals

on temperature, pressure, and concentration. In deriving the equations, it was shown that the parameters are obtained using the Gibbs free energy. Hence the dependency of the parameters is quite similar to that of the Gibbs free energy. However, since both i◦ and charge transfer coefficients are nonlinear functions of G, the relations cannot be obtained easily. Consequently, there are many different formulas for expressing the temperature and concentration dependency of these parameters on the temperature and pressure. The limitations and assumptions should be carefully considered when using the parameters.

1.4. Charge transfer As an electronic device, an FC looses some of its own generated energy during the charge transfer. The electrical energy is carried by electrons in solid conductive materials such as the catalyst layers, porous GDLs, and external circuits. In electrolyte phase the electrical energy is carried by ions. As explained before, in some FCs the cations are responsible for flowing the electrical charge, and in some FCs, the anions. Whether the electrons carry the electrical charge or the ions, their movement requires energy, and some part of the available energy is spent for their movement. The particles that carry the electrical charge in solid phase or electrolyte phase are called the carriers. By this definition, in solid phase the electrons are the carriers, and in electrolyte phase the carries are the ions. It is quite understandable that the size of ions is much larger than that of electrons, and hence there is more resistance for their movement. This means that the electrical energy loss due to the movement of ions is much larger than due to that of electrons. The most famous mechanism of energy loss is Joule heating, which happens when a charged carrier such as electron or ion moves through a conductive medium. The energy loss in this case is calculated using the relation QJoule = RI 2 ,

(1.76)

in which QJoule is the Joule heating, R is the resistance of the medium, and I is the amount of current passing through the medium. The energy loss results in voltage drop or polarization. The polarization caused by electrical resistance is called the ohmic drop or ohmic polarization. According to different mechanisms, the total polarization is the sum of electronic and ionic polarizations. This value is defined by the equation Eohm = IRohm = I (Relec + Rionic ).

(1.77)

Determining the electronic and ionic resistances is not easy. There are some efforts to give a rough estimation of these values, but in general it is not as straightforward as it seems. Here we briefly discuss these parameters.

45

46

Fuel Cell Modeling and Simulation

Figure 1.18 A conductive bar with cross-section of A and length L.

1.4.1 Electronic resistance The solid phase of each FC is composed of different parts including the bipolar plates, GDL, catalyst layers, external circuit, and cell interconnectors. These parts are made of solid conductive materials. As we know, the resistance of a solid bar such as that shown in Fig. 1.18 is obtained using the equation R=

ρL

A

(1.78)

,

where ρ is the resistivity of the material ( cm), L is the length of the bar (cm), and A is its cross-section area (cm2 ). The resistivity of the bar depends on its material and is the reciprocal of its conductivity σ : ρ=

1 σ

(1.79)

.

The unit of σ is S cm−1 (Siemens per cm). Note that an FC is a three-dimensional device, and hence the Eq. (1.78) cannot be simply used. However, in many problems the conductive solid phases can be assumed to behave as one-dimensional. Thus a good estimation of the resistance can be obtained. Example 1.13. Calculate the ohmic loss of an FC with A = 100 cm2 and the characteristics shown below. Anod GDL

Thickness, cm Conductivity, S cm−1

0.15 1.5 × 102

Anod Cat. 8.0 × 10−3

102

Membrane

7.5 × 10−3 0.2

Cathode Cat. 3 × 10−3 1.5 × 102

Cathode GDL 0.2 1.5 × 103

Answer. The overall resistance of the cell is the sum of the resistances of all parts since they are connected in series. Therefore the first step is calculating the resistance of each

Fuel cell fundamentals

individual part. For all parts, the area is the same and equal to A = 100 cm2 . So we have RGDL,a = RCat,a = Rmem = RCat,c = RGDL,c =

0.15 = 10−5 , 1.5 × 102 × 100 8.0 × 10−3 = 8.0 × 10−7 , 102 × 100 7.5 × 10−3 = 3.75 × 10−4 , 0.2 × 100 3 × 10−3 = 2.0 × 10−7 , 1.5 × 102 × 100 0.20 = 1.33 × 10−5 . 1.5 × 102 × 100

The overall resistance then becomes Rove. = 10−5 + 8 × 10−7 + 3.75 × 10−4 + 2 × 10−7 + 1.33 × 10−5 = 3.99 × 10−4 S cm−1 . The ohmic drop according to Eq. (1.77) becomes Eohm = 3.99 × 10−4 I , It is clear that the ohmic drop is linear as long as the internal resistance is constant. However, it is known that the internal composition of electrolyte, its concentration and concentration gradient, water content, and many other parameters affect the internal resistance. Thus in a real device the ohmic drop is not linear.

1.4.2 Ionic resistance Ionic resistance occurs inside the electrolyte where the ions move from anode to cathode or vice versa. Reconsidering all the fuel cell types, there may be three different available electrolytes: Liquid electrolyte, such as alkaline electrolyte in AFC, phosphoric acid in PAFC, and molten salt in MCFC. Solid polymeric used in PEMFC, HT-PEMFC, DMFC, and other polymeric membrane FCs. Solid ceramic such as SOFC and protonic ceramic membrane fuel cells PCMFC. There are may correlations defining the conductivity of the liquid membranes such as KOH, NaOH, H3 PO4 , and others in the handbooks. In general, the conductivity strongly depends on the concentration and temperature; hence for each solution, proper data must be used. For example, for KOH, Table 1.4 is obtained from [2]. For other materials, such tables or correlations can be obtained from the literature.

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Table 1.4 Properties of potassium hydroxide [2]. % KOH

0 5 10 15 20 25 30 35 38 40 45 50

Specific gravity at 15.6 ◦ C (-) 1.0000 1.0452 1.0918 1.1396 1.1884 1.2387 1.2905 1.344 1.3769 1.3991 1.4558 1.5143

Conductivity at 18 ◦ C ( −1 cm−1 )

0.17 0.31 0.42 0.5 0.545 0.505 0.45 0.415 0.395 0.34 0.285

Specific heat at 18 ◦ C (cal g−1 ◦ C−1 ) 0.999 0.928 0.861 0.801 0.768 0.742 0.723 0.707 0.699 0.694 0.678 0.66

The mechanism of conductivity in polymer electrolytes completely differs from that in liquid electrolytes. Actually, the polymeric membranes need to be hydrated to become conductive. Their conductivity is a function of their water content. The membrane water content can be explained as the ratio of the number of water molecules to the number of sulfonic acid (SO3 H) groups. Generally, this parameter has a maximum value of 14 (under 100% relative humidity), although higher values have also been reported. For instance, the values 22 and 23 were reported in the previous work by Mann et al. [3]. The water content can be obtained using the relation a=

pw , psat

(1.80)

where pw is the partial pressure of water vapor in FC, and psat represents the saturation water vapor pressure at the same temperature. Sharifi Asl et al. [1] proposed an equation that shows dependency of Nafion 117 water content to relative humidity. This correlation is as follows: σ = 0.0043 + 17.81 × a − 39.85 × a2 + 36 × a3 .

(1.81)

For solid ceramic electrolytes, different correlations are also defined in the open literature. Their values depend on the temperature and also on the medium shape and morphology. The most popular ceramic is yttria-stabilized zirconia (YSZ) used in SOFCs. This ceramic contains vacancies for moving ions such as O2− . Increasing yttria results in more vacancies, and hence the conductivity of the ceramic increases.

Fuel cell fundamentals

Needless to say, at this level of temperature the membrane does not contain any water, and the mechanism of ion transfer absolutely differs from other membranes.

1.4.3 Role of effective factors All the physical properties, including the electrical resistance, are temperature dependent and vary by temperature variations. In general, for a physical property such as σ , the temperature dependency is described by the Arrhenius equation 

Ea σ = σ◦ exp R



1 1 − Tref T

 ,

(1.82)

where σ◦ is the value of σ measured at a reference temperature Tref , Ea is the activation energy, and R and T are, respectively, the universal gas constant and desired temperature. To be able to use Eq. (1.82), the parameters σ◦ and Ea should be measured experimentally. The activation energy differs from parameter to parameter and must be obtained for each particular parameter that is involved in the calculations. The activation energy can be found by measuring σ in another temperature, rather than in the reference one, and from Eq. (1.83) the value of Ea is obtained. In addition to the temperature, the physical parameters such as σ strongly depend on the porosity and morphology of the medium. Assume that σ is measured for a solid bar shown in Fig. 1.18. Now if the bar becomes porous, then it is obvious that its resistance differs from its original state. Not only the value of porosity itself affects the parameter, the shape and morphology of the medium are also important. Fig. 1.19 is an illustration of a porous medium with two different possible morphologies. As we can see, in both figures the porosity of the medium is the same. However, the shape of solid particles and the morphology of the two media are not the same. Therefore, considering the electrical conductivity, the electric charge carriers (electrons or ions) encounter different path ways when traveling in these media. The result is that they experience different resistance, resulting in different electrical conductivity. To correct the electrical resistance, Brugmann’s relation is used. This relation modifies the conductivity due to the porosity and its morphology. The relation is σ = σ◦ ε ξ ,

(1.83)

in which ε is the porosity of the medium defined as ε=

Vvoid Vtotal

(1.84)

or in other words, the fraction of void fraction of the medium (shaded area in Fig. 1.19) over the whole volume (the area of the rectangle in the same figure).

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Figure 1.19 Effect of morphology on physical parameters.

In addition, in Eq. (1.83) the exponent ξ accounts the effect of morphology and may have any value, and it should be found according to an experimental test. However, for homogeneous porous medium composed of circular particles, the exponent can be assumed ξ = 1.5. In many papers and reports, the Brugmann relation is defined as σ = σ◦ ε 1.5 ,

(1.85)

however, it is not true for all the devices.

1.5. Mass transport Mass transport is very important in operation of FCs and greatly affects their performance. The main objective of studying mass transport is determining the species concentration at catalyst layers where the reactions occur. As studied before, the species pressure dramatically affects the Gibbs free energy, resulting in varying the net useful energy. More specifically, if the pressure drops, then the net useful energy follows that drop. In obtaining the Gibbs free energy, it is obvious that the net pressure itself is not important, but the important factor is the partial pressure of active species. For example, in the cathode side of a PEMFC, if air is used instead of pure oxygen, then the partial pressure of oxygen will be 0.21 atm. To understand the basic concepts of mass transport inside an FC, the most important phenomenon is shown in Fig. 1.20. Note that here only the anode of an FC is considered; the same argument will be true for the cathode side. As we can see in Fig. 1.20a, the fuel enters the channel with concentration Cin and flows along the channel. While passing the channel, a specific amount of fuel enters the porous GDL, reaches the catalyst layer, and reacts on the catalyst layer. Therefore the concentration of the fuel decreases as it moves toward the channel outlet. At the channel outlet the concentration of the fuel reaches Cout . Therefore the fuel concentration at each point along the channel is not the same and decreases from input to output. For clarity, the concentration of fuel at each point inside the channel is denoted Cch .

Fuel cell fundamentals

Figure 1.20 Mass transport mechanism from channel to catalyst layer.

The concentration of fuel at the GDL interface is denoted Cs , which differs from the bulk value. The concentration difference between the bulk value and the surface of the GDL results in a convective mass transport. The difference is shown in Fig. 1.20b with a linearity assumption. In reality the gradient is not linear and must be obtained using numerical techniques; however, for clarity, here we assume a linear gradient. The concentration at the catalyst layer shown by Cc also differs from the value of Cs . Inside the GDL, the main mechanism is diffusion because the GDL is a porous medium inside which the flow movement is very slow. Again, a linear assumption can be made for variation of concentration from Cs to Cc (see Fig. 1.20b). According to the illustration of Fig. 1.20, there are three important mass transfer mechanisms inside an FC: 1. Convective mass transport from flow channel to GDL. 2. Diffusive mass transport responsible for delivering active materials from GDL to catalyst layer. 3. Convective mass transport along the flow channel. In addition to these three mechanisms, the species are consumed or produced on catalyst layers resulting in sinks or sources of mass. In general, the mass transport should be analyzed using the governing equations of mass transport, which contains transient term, convection, diffusion, and source/sink of mass. The equation is in the form of a partial differential equation and should be solved numerically to obtain the exact values. However, in this section, we discuss the above-mentioned three mechanisms in a very simple form to have a better understanding of the mass transport mechanisms and to be able to have a good estimation of species concentration.

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1.5.1 Convective mass transfer from flow channel to GDL Fig. 1.20a shows the differences between the concentration Cch at bulk and the surface Cs of GDL. The difference causes a convective mass transport, which can be evaluated by the equation N = Aelec hm (Cch − Cs ),

(1.86)

where N is the mass flux (mol s−1 ), Aelec is the electrode area exposed to the channel, and hm is the convective mass transfer coefficient. Eq. (1.86) determines the specific amount of mass transfers from the bulk to GDL; hence, as the fuel flows along the channel, its concentration drops. Therefore Eq. (1.86) is a path function and has no constant value all along the channel.

1.5.2 Diffusive mass transfer Inside the GDL, there is no reaction, and hence no mass is produced or consumed inside the GDL. Therefore the mass that enters the GDL (see Eq. (1.86)) enters the catalyst layer. The main mass transport mechanism inside the GDL is diffusion because GDL is a porous medium inside which the flow velocity is very slow, so that the convection can be ignored. The diffusion of a species inside a medium is defined by Fick’s law or the equation N = −D

dC . dx

(1.87)

Since the thickness of GDL is very small, the linear assumption is almost appropriate. By this assumption Eq. (1.87) can be written as the equation N = −Aelec D

Cs − Cc δ

,

(1.88)

where δ is the thickness of GDL. Combining Eqs. (1.86) and (1.88) yields N=

Cch − Cc . δ 1 + hm Aelec DAelec

(1.89)

In practical applications, measurement of species concentration on catalyst layer is almost impossible. From theoretical point of view, we know that when the species reach the catalyst layer, they are consumed to produce electricity, and hence the rate of species flux is proportional to the electrical current of the FC. Consequently, by means of Faraday’s law or Eq. (1.55) the species flux can be related to the electrical current

Fuel cell fundamentals

according to the equation I iAelec = . nF nF Combining Eqs. (1.89) and (1.90) yields N=

i = nF

(1.90)

Cch − Cc . δ 1 + hm D

(1.91)

Eq. (1.91) relates the catalyst concentration to the electrical current. This relation is used in practice for calculating the species concentration on the catalyst layer. The higher the current density, the lower the Cc because the reactions must become fast enough to provide the required electrons. If the current density increases, then the species concentration on the catalyst layer decreases, and according to Eq. (1.89), the species flux from bulk to GDL increases. Since the mass flux is determined by diffusion, it has a limit controlled by D. This means that the mass flux cannot exceed a specific amount, and hence it cannot provide the consumed species at the catalyst surface. At this stage the species concentration at the catalyst surface becomes zero, and the FC cannot produce more current. The current at which this situation happens is called the limiting current and can be calculated using Eq. (1.91) by simply setting Cc = 0: iL = nF

Cch . δ 1 + hm D

(1.92)

The limiting current is one of the most important characteristics of an FC. This parameter determines the ability of the FC to produce a higher current density without loosing its voltage. The effect will be studied in more detail in Section 1.6.

1.5.3 Role of effective factors Different parameters affect the mass transport. The first parameter is pressure. As the pressure of the fuel or oxygen increases, the species concentration at inlet increases since almost in all the FCs, the fuel and oxidant are in gaseous phase, which can be properly described by the perfect gas law. An increase in Cch results in an increase in iL according to Eq. (1.92). The second effective factor is temperature. It is quite known that the temperature changes the physical properties, including the diffusion coefficients. In general, a physical property such as  varies by temperature according to the Arrhenius equation 

Ea  = ◦ exp R



1 1 − Tref T

 .

(1.93)

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Similarly to Eq. (1.82), the value ◦ is measured at Tref . Then the value of  at other temperatures can be found from Eq. (1.93). For obtaining the activation energy, Ea , another test must be done at a temperature other than Tref ; then it can be easily obtained using Eq. (1.93). In mass transport phenomena, the diffusion and convection coefficients D and hm must be corrected according to the Arrhenius equation. The values of these parameters at reference temperature must be obtained using handbooks or accepted references. In addition to temperature, hm strongly depends on the channel inlet velocity. The convective mass coefficient is a function of velocity and also its pattern. The shape of the channel, the obstacles inside the channel, and other geometrical issues strongly affect hm . This is why in many researches and industrial designs, the shapes of the channels are modified, and many ideas are being tested. By changing the geometry of the channel, hm may be enhanced. Moreover, the flow velocity is also important. These factors are being studied by many researchers all around the world. Finally, since the mass transport happens in a porous medium, all the physical properties such as D must be corrected according to the Brugmann relation discussed in Eq. (1.83). As mentioned before, for most of the works done up to now, the exponent in Brugmann’s relation is chosen 1.5, but it is not a universal value and may vary case by case.

1.6. Characteristic curve of a fuel cell The whole characteristic of an FC is depicted in a curve known as the characteristic curve or IV-curve. A sample IV -curve is shown in Fig. 1.21, in which the voltage of the cell is shown on the vertical axis, and the current density on the horizontal axis. Three different regions can be observed on the figure according to the cell actual curve shown by a thick solid line. The regions are the following: Crossover dominated region, in which the cell voltage drops due to the crossover of fuel from the membrane. This region is a very small area near the open-circuit conditions. According to fuel crossover, the cell voltage never reaches its theoretical value. Linear region, in which the voltage drop is almost linear due to ohmic polarization. Concentration dominated region is the last phase of polarization curve in which the cell voltage suddenly drops due to the depletion of species on the catalyst layers. The species depletion causes the cell to produce current not higher than the limiting current shown in the figure. In general, IV -curve can be assumed as the characteristic curve of an FC. Much different information can be obtained using an IV -curve. For example, since the efficiency

Fuel cell fundamentals

Figure 1.21 A typical IV-curve.

is proportional to the actual voltage, we can rescale the curve to obtain the efficiency profile of the FC. As another example, the curve can indicate which polarization is dominating, and then the designer can modify the cell to enhance its performance. For example, if the slope of the linear part is too steep, then it shows that the internal resistance of the cell is too high, or if the limiting current is very small, then it means that the cell design has mass transfer issues. Other conclusions can be obtained by studying the curve.

1.7. Summary In this chapter, we introduced the basic knowledge required for understanding the FCs. The chapter started by an introduction to FCs, their applications, and trends and also their importance in the future of energy. The roadmaps of different regions of the world show that FCs will play an important role in the future and many countries focusing on generation of different technologies. After that, we discussed the basic operation of FCs and introduced different technologies. Note that there are many different technologies that are not listed in this chapter. Hence the reader is encouraged to look for other technologies. The chapter continued by giving a brief introduction to thermodynamics of the fuel cell and formulating the FCs from different aspects. Finally, we introduced the characteristic curve of the FCs. In the following chapters, we discuss different FC types in more detail.

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1.8. Problems 1. Investigate the role of FCs on portable devices. What would be the future of FCs on these applications? 2. What does the theoretical voltage mean? 3. Discuss the OCV and its relation to the theoretical voltage. 4. List the differences between the theoretical voltage and the reversible voltage of an FC. 5. Calculate the theoretical voltage of a DMFC at standard conditions. 6. In practice, DMFCs are working at about 60 ◦ C. Recalculate the reversible voltage of DMFCs at actual temperature. 7. For an SOFC working at 600 ◦ C to 1000 ◦ C, plot the variation of OCV. 8. Repeat the previous problem for MCFC working at 500 ◦ C to 650 ◦ C. 9. For different technologies at their operating temperature range, plot the OCV and efficiency of the cell. 10. For and FC with IV -curve shown in Fig. 1.21, give an estimate for its internal resistance.

References [1] S.M. Sharifi Asl, S. Rowshanzamir, M.H. Eikani, Modelling and simulation of the steady-state and dynamic behaviour of a PEM fuel cell, Energy 35 (4) (2010) 1633–1646. [2] David Linden, Thomas B. Reddy, Handbook of Batteries, 3rd ed., McGraw-Hill, 2002. [3] Ronald F. Mann, John C. Amphlett, Michael A.I. Hooper, Heidi M. Jensen, Brant A. Peppley, Pierre R. Roberge, Development and application of a generalised steady-state electrochemical model for a PEM fuel cell, Journal of Power Sources 86 (1–2) (2000) 173–180.

CHAPTER 2

PEMFCs Contents 2.1. Introduction 2.1.1 Components and structure 2.1.2 Transport phenomena in PEMFCs 2.1.3 Hydrogen for PEMFCs 2.2. Microscale modeling and simulation of PEMFCs 2.2.1 Microstructure reconstruction 2.2.2 Pore-scale numerical simulation methods 2.2.3 Lattice Boltzmann simulation technique 2.2.4 Pore-network simulation of water transport 2.2.5 VOF simulation of water transport 2.3. Macroscale modeling and simulation of PEMFCs 2.3.1 1D modeling of a cell 2.3.2 Framework of finite volume method 2.3.3 2D/3D modeling of a cell 2.3.4 Modeling of a stack 2.3.5 Modeling and control of PEMFC system 2.3.6 Modeling of PEMFC cold start 2.4. Summary 2.5. Questions and problems References

57 57 75 130 138 140 147 148 178 186 190 191 204 215 219 221 225 227 228 230

2.1. Introduction 2.1.1 Components and structure As explained in the previous chapter, in a proton-exchange membrane (PEM) fuel cell (PEMFC), the hydrogen and oxygen gases are fed to the anode and cathode electrodes, respectively; they are incorporated in the following redox electrochemical half-reactions: H2 → 2H+ + 2e− ,

(2.1)

1 O2 + 2H+ + 2e− → H2 O. 2

(2.2)

The electrons generated by Eq. (2.1) in the anode electrode are transferred to the cathode electrode to be consumed by Eq. (2.2) and meanwhile to produce electric power (e.g., for the electric propulsion system in an FCEV). At the same time the produced protons in the first half-reaction are transferred to the cathode side through the Fuel Cell Modeling and Simulation https://doi.org/10.1016/B978-0-32-385762-8.00006-3

Copyright © 2023 Elsevier Inc. All rights reserved.

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Figure 2.1 Essential phenomena for an operating PEMFC.

polymer electrolyte to be consumed in the second half-reaction. Finally, the produced water in the second half-reaction must be drained out. Therefore, to have an operational PEMFC, the following events must happen (Fig. 2.1): 1- Fuel and oxidizer supply: Hydrogen molecules must be transferred from the fuel tank to the reaction sites in the anode electrode. Besides, the oxidizer molecules must be transferred from the environment (when the oxidizer is air) or the oxidizer tank (when the oxidizer is pure oxygen) to the reaction sites in the cathode electrode. 2- Electrochemical half-reactions: H2 and O2 molecules at the anode and cathode reaction sites experience oxidation and reduction electrochemical reactions, as previously stated by Eqs. (2.1) and (2.2), respectively. 3- Charge transfer: to accomplish the electrochemical half-reactions, charge transfer is necessary. Electrons and protons are the two types of electric charges, which are transferred between the reaction sites of two electrodes through the electric circuit and polymer electrolyte, respectively. Although the origin and destination of transfer phenomena for these two types of electric charges are the same in PEMFCs, the path differs. 4- Product drainage: the produced water in the cathode electrode must be removed from the reaction sites to the atmosphere or the water tank. Since no gas in a PEMFC anode is produced, product drainage at the anode side is not required. Although the above four steps seem simple and straightforward, to have an efficient and reliable PEMFC, they must be performed efficiently and adequately. However, these four essential steps in practice may face some challenges. The main possible challenges against these four steps are presented in Table 2.1.

PEMFCs

Table 2.1 Potential challenges against the necessary phenomena in a practical operating PEMFC. Fuel and oxidizer supply

Electrochemical reaction

Charge transfer

Product drainage

The long path from the tank to the reaction sites Nonuniform distribution of fuel and oxidizer on the reaction sites

Limited available reaction sites

Large ionic resistivity of electrolyte

Slow reactions, especially at the cathode side

Moisture-dependent resistivity of electrolyte

The long path from the reaction sites to the out of stack Liquid water formation through the path

Figure 2.2 Three-phase boundaries (TPBs).

To tackle these challenges, a special design and structure for a single cell are required. In addition, many technical notes must be considered when a few cells are to be stacked in a PEMFC pack, which is conventionally known as a PEMFC stack. Furthermore, stacking of the cells is not sufficient for generating and harvesting electricity; i.e., other auxiliary parts are also required for a reliable and efficient performance of a PEMFC system. In the rest of this section, we explain the components and structure of a PEMFC at three stages of cell, stack, and system. Simultaneously, the design points for tackling the mentioned challenges in Table 2.1 are described. a) Components and structure of a PEM cell Each PEM fuel cell consists of at least three main parts: anode electrode, electrolyte, and cathode electrode. To achieve the largest number of reaction sites, the electrodes must be highly porous with a large ratio between the real and apparent surfaces. In fact, each reaction site in a PEMFC is a three-phase boundary (TPB), on which the previously presented redox half-reactions can take place (Fig. 2.2). These three phases are gas, solid electrode, and solid electrolyte; the gas phase is necessary for the exchanging of gaseous reactants/products, whereas the solid electrode and solid electrolyte phases are necessary for the exchanging of the electrons and protons, respectively. All these three phases are necessary for the electrochemical reaction implementation; i.e., if a surface is in contact with two of these three phases, then nothing will happen!

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Figure 2.3 Complex porous structure of a CL [1,2].

i) CL: To construct the porous structure of electrodes, carbon black particles can be used as the solid electrode phase; they are good electron conductors with suitable heat conductivity and high resistance against corrosion. However, the reaction rate on the carbon particles is not fast enough (i.e., the exchange current density i0 is very small). In fact, the rate of redox half-reactions on the surface of these particles is much slower than the rate of mentioned reactions on the surface of precious catalytic metals such as Pt particles. On the other hand, substituting the carbon black particles with Pt particles will result in an extremely expensive electrode. Since the surface-to-mass ratio for a Pt particle is inversely related to its diameter, Pt nanoparticles with a tiny average diameter of 2–10 nm are used to employ catalytic metals more cost-effectively. In the most of practical commercial electrodes, the carbon black particles with an average diameter of 45–90 nm are used as the support for the placement of these Pt nanoparticles (Fig. 2.3), which results in lower Pt loadings for a specific cell’s power density. This complex structure, which includes carbon particles and Pt nanoparticles, is called the catalyst layer (CL), which is known as the heart of an electrode. Another type of CLs, which is less conventional, is constructed with nonprecious metals; these CLs are also recognized as PGM-free CLs, where PGM stands for platinum group metal. Although several efforts are performed for developing these CLs, they still provide less performance and durability in comparison to Pt-based CLs. ii) GDL: To minimize the concentration losses in an electrode, the CL thickness in a PEM cell is required to be very thin. The CL thickness in practical electrodes is 5–30 µm; therefore, to reinforce its mechanical strength, employing an attached mechanically strong layer is inevitable. However, this layer must be highly porous and permeable (for the ease of gas transport through it by diffusion), good electron conductor (for the ease of electron transport through it), and corrosion-resistant, the same as the CL. This layer is usually known as the gas diffusion layer (GDL) or diffusion media (DM) in the literature, according to its necessary attribute for facilitating the transportation of reactants

PEMFCs

and products in/through it via diffusion mechanism. The best choices for GDL are the thin layers constructed by carbon fibers, such as carbon paper, carbon cloth, and carbon felt (Fig. 2.4). The fibrous skeleton of these layers is treated with a carbonized resin binder that lessens the contact resistance between two intersecting fibers and improves the electrical conductivity of the layer. However, this resin treatment may change the micropores of carbon paper GDL, and consequently, it can affect the gas transfer pattern through the microstructure of an electrode. Both carbon particles in CL and carbon fibers in GDL are intrinsically hydrophilic. Since in conventional PEM fuel cells the operating temperature is less than 100 °C, and the operating pressure is about 1 atm, liquid water formation is probable, especially at the cathode electrode, where the only product is H2 O. The hydrophilic intrinsic of carbon materials in GDL and CL makes the drainage of liquid water from the electrodes sophisticated and sluggish. Therefore, to enhance the liquid water removal from these porous layers, they are often treated with a highly hydrophobic material. Polytetrafluoroethylene (PTFE) is the most common choice for the hydrophobic treatment of electrodes. Since implementing ideal hydrophobic treatment, in which all carbon microsurfaces are coated by the hydrophobic material, is almost impossible, the hydrophobic treatment in practice results in a complex microstructure consisting of both hydrophilic and hydrophobic pores. For example, if an immersion method is used for the PTFE treatment of GDL, then the PTFE through-plane distribution and, consequently, the distribution of hydrophobic pores in the through-plane direction will not be uniform. In fact, in the middle zone of a PTFE treated GDL, the PTFE content will be significantly less than the PTFE content near the two sides. The effects of the nonuniform distribution of PTFE through the GDL are recently investigated in a few microscale investigations [4–6]. Since thin and long carbon fibers are employed in the skeleton of all GDL types, the transport properties of GDL are anisotropic. More specifically, the transport properties of conventional GDLs along the two main directions of in-plane and through-plane are considerably different (through-plane direction is normal to the GDL while an inplane direction is parallel to the GDL). However, for different in-plane directions, the transport properties are not considerably different. The main properties of conventional GDLs are presented in Table 2.2. To accomplish the manufacturing of a cell, the CL must be added to the GDL to form a complete electrode. To this end, a slurry including carbon particles and Pt nanoparticles is sprayed on the GDL. Other methods such as tape casting can also be used for the placement of CL on the GDL [7]. When the anode and cathode electrodes are manufactured, they sandwich the polymeric membrane as the solid electrolyte to construct the final cell, which is also known as a membrane-electrode assembly (MEA). Another method for the production of MEA is placing the anode and cathode CLs on

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Figure 2.4 Microstructures of three types of employed layers as the GDL [2,3]: (a) Toray H-060 (carbon paper), (b) Ballard 1071HCB (carbon cloth), and (c) Freudenberg C2 (carbon felt).

PEMFCs

Table 2.2 Main properties of different GDLs [2]. Properties

Pore volume fraction Layer thickness, mm PTFE content, %wt Through-plane absolute permeability, m2 In-plane absolute permeability, m2 Through-plane electric resistivity,  cm In-plane electric resistivity,  cm

carbon paper

GDL type carbon cloth

carbon felt

0.6–0.8 0.2–0.4 5–30% 10−14 –10−12 10−12 –10−11 0.25–1 0.005–0.02

0.6–0.8 0.25–0.45 5–30% 10−14 –10−12 10−12 –10−11 0.75–1.5 0.005–0.01

0.5–0.7 0.2–0.4 5–30% 10−15 –10−13 10−13 –10−12 0.25–1 0.005–0.02

Figure 2.5 Schematic of different phases in the nanostructure of CL.

the membrane (via spraying, tape casting, or decal hot-pressing) and then sandwiching the membrane by the cathode and anode GDLs. By the way, the CL in a practical MEA is impregnated with a considerable amount of membrane ionomer material such as Nafion (up to about 40%). The schematic view for the nanostructure of an applied CL in an MEA is presented in Fig. 2.5. iii) MPL: To enhance the water management in the electrodes of many PEM fuel cells, two highly hydrophobic microporous layers (MPLs) are used between CL and GDL at the anode and cathode sides. Note that in a part of the literature, the MPL is recognized as a GDL sublayer. However, since recognizing MPL as an individual constructing layer of electrodes is more conventional, in this book, MPL will be recognized as an individual layer. It is believed that the hydrophobicity of MPL facilitates the ejection of produced liquid water from CL to GDL. However, the detailed effects of MPL on the cell performance in terms of water transport, electric-charge transport, and heat transport are not still well understood. MPL in commercial PEM FCs is 5–20 µm thick with small 100–500 nm pores and large microcracks with 2–10 µm width. There are two ways to manufacture and assemble MPL in a cell. In the first way a slurry including PTFE, carbon particles, and a polymeric binder is applied on the GDL, similarly to the CL manufacturing and assembling on the GDL. In the second way, MPL as a porous polymeric sheet is bonded

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Figure 2.6 Two types of MPL manufacturing method: (a) MPL manufacturing on the GDL; (b) MPL manufacturing on the CL.

to the CL surface, which is primarily placed on the membrane. By the way, the CL placement in the PEM cells encompassing MPL must be performed on the membrane and not on GDL (Fig. 2.6). iv) Membrane: One of the main constructing elements of PEMFCs is the solid-state polymer electrolyte, which is also known as the proton-exchange membrane (PEM). The main mission of PEM is providing a suitable path for the transportation of protons from the anode TPBs to the cathode TPBs, as it is clear from its name. The most usual material for employing as the PEM in low-temperature PEMFCs is an ionomer, which has a hydrophobic backbone with hydrophilic sulfonated sites. The commercial brand of Nafion, which is a good conductor of protons and, at the same time, a good insulator against electrons, is a famous material for such PEMs. In HT-PEMFCs, the PEM is made of acid-based polymers, such as pure polybenzimidazole (PBI) or an acid-doped PBI, which can work in higher temperatures without any

PEMFCs

failure. Among different proposed materials for the membrane of HT-PEMFCs, only the phosphoric acid-doped PBI is able to meet the US DOE criteria currently. Since the proton resistance of PEM is inversely related to the membrane thickness, producing PEMs as thin as possible is favorable. Today’s perflourosulfonated ionomer electrolytes for low-temperature hydrogen PEMFCs can be as thin as 18–25 µm and can operate below 120 °C. However, due to the humidity requirement and the degradation rate consideration for these PEMs, the operating temperature is almost always less than 90 °C in practical low-temperature PEMFCs. The proton conductivity of PEMs depends strongly on the water content and temperature. To have a better understanding of this dependency, the mechanism of proton conduction through the Nafion membrane (as the most commercially available membrane) will be briefly described. The chemical structure of a typical Nafion ionomer and its inert backbone (usually, PTFE) is presented in Fig. 2.7. As it is shown in this figure, an ionized sulfonated site (SO−3 ) exists in the structure, which can attract proton (H+ ). The primary mechanism for the transportation of protons is via these ionized sites. In this mechanism the molecular vibration and movement of the inert chain can displace the attracted proton. The attracted proton can also be displaced from its attracting site to the neighboring site. These displacements can result in proton transport from one side of the membrane to the other. However, the proton transport through the membrane is mainly due to a secondary mechanism, which is known as a vehicular mechanism. To describe this mechanism, it must be noted that there are pore spaces (also known as free spaces) among the Nafion chains, which can uptake water molecules. The water molecules and their movement can play the role of vehicles for transporting protons. In fact, a well-hydrated Nafion can include numerous H3 O+ , the same as an acidic aqueous solvent. For practical Nafion, the amount of uptaken water can be so great it causes the membrane to expand, even up to 25 vol. %! To quantify the amount of uptaken water, the parameter water content λ is used, which is defined as the ratio of water molecules to the sulfonated sites. The Nafion proton conductivity σ (T , λ) is correlated by water content and temperature (in Kelvin) through the following equation [8]:    1 1 , σ (T , λ) = σ303K (λ) 1268 −

(2.3)

σ303K (λ) = 0.5193λ − 0.326.

(2.4)

303

T

where

Experimental observations demonstrate that for most types of Nafion with different equivalent weights and thicknesses, the water content can vary from 22 (fully saturated

65

66

Fuel Cell Modeling and Simulation

Figure 2.7 The chemical structure of a Nafion ionomer and its inert backbone polymer PTFE.

Nafion) to almost 0 (dehydrated Nafion). For PEMFCs, we can sue the following equation for correlating the Nafion water content with its nearby humidity:  λ=

0.043 + 17.18aw − 39.85a2w + 36.0a3w if 0 < aw ≤ 1, 14 + 4 (aw − 1) if 1 ≤ aw < 3,

(2.5)

where aw is the water activity defined as aw =

Pw . Psat

(2.6)

In this equation, Pw is the water vapor partial pressure, and Psat indicates the water saturation pressure at the operating temperature. Note that the percentage form of aw reveals the relative humidity. By combining Eqs. (2.3)–(2.6) the proton conductivity of Nafion as a function of water vapor partial pressure for a PEMFC operating at 80 °C is presented in Fig. 2.8 (at 80 °C, Psat = 47.415 kPa). As it is evident, increasing the water vapor partial pressure will result in the increase of membrane proton conductivity.

PEMFCs

Figure 2.8 Proton conductivity of a Nafion vs. water vapor partial pressure.

However, this increase is more rapid for the pressures just below 47.415 kPa, which is the saturation pressure of water at 80 °C. In Fig. 2.9 the proton conductivity of a Nafion is presented as a function of temperature for a PEMFC operating under constant water vapor partial pressure of 47.415 kPa. Note that although from Eq. (2.3) it seems that the proton conductivity will increase exponentially by increasing the temperature, when the temperature increases, the saturation pressure of water will also increase, which leads to the decrease of water activity when the water vapor partial pressure is constant in the cell (here 47.415 kPa). Therefore, according to Eq. (2.6), increasing the temperature will lead to the decrease of σ303 . The resultant of these two opposite affecting features of temperature is the decrease of proton conductivity, as presented in Fig. 2.9. However, this decrease is more obvious for the temperatures above 80 °C, for which the saturation pressure will be 47.415 kPa, and according to Eq. (2.6), the water activity equals unity. Therefore the cell operating temperature above 80 °C is not recommended under such conditions. It is worth mentioning that for the derivation of Fig. 2.9, the relation between Psat and temperature is calculated via the following correlation: log10 Psat = −2.1794 + 0.02953T − 9.1837 × 10−5 T 2 + 1.4454 × 10−7 T 3 .

(2.7)

For calculating the electric resistance Rm of membrane, we must calculate the following integral: 

Rm = 0

tm

dz Am σ (z)

,

(2.8)

where tm and Am are the thickness and surface area of membrane, respectively. Since the proton conductivity of membrane can vary through the membrane, caution must be paid for incorporating σ (z). In fact, due to the different humidity at the anode and cathode sides and due to the dependency of membrane conductivity on the humidity, the proton conductivity is usually not constant through the membrane.

67

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Fuel Cell Modeling and Simulation

Figure 2.9 Proton conductivity of Nafion vs. temperature for a constant partial pressure of water vapor.

b) Components and structure of a PEMFC stack PEMFCs have a wide range of applications, from a few watts in portable electronics to a few kilowatts in electric vehicles and a couple of megawatts in CHP systems. However, in almost all of these applications, the required supply voltage is not 1.23 V (which is the theoretic voltage of a PEM single cell). Voltages as high as 600 V are required for the propulsion systems of electric busses. Even the operating voltage for the small portable electronics is usually 3 V and higher. Therefore connecting the cells in series to achieve higher voltages is inevitable. The simplest method for connecting the cells in series is applying electric wires and busbars, as illustrated in Fig. 2.10. As shown in this figure, a gap between cells for feeding the hydrogen fuel and air/oxygen on the surfaces of the electrodes is necessary. Since in this method the produced current on the entire surface of electrodes is drained from a single point connecting electrode and wire, this method will result in lots of electrical losses. Hence, in practical PEMFCs, a more efficient and reliable method for connecting cells in series is adopted. In this method, between every two neighboring cells, a bipolar plate (BP), which has a special design and structure, is used. The achieved set of bipolar plates and cells is known as a PEMFC stack. The special design and structure of these bipolar plates fulfill the following functions: • To provide cell-to-cell electric connection in series with minimum electrical losses • To uniformly distribute fuel and oxidizers on the surfaces of electrodes • To effectively drain the produced/accumulated water from the electrodes • To provide a suitable path for passing the coolant fluid and, consequently, cooling the cells, especially for the cells with high-power density • To guarantee the gas sealing and avoid fuel/oxidizer leakage Due to the porous structure of electrodes in the PEMFCs, if the initial assembly of anode–electrode–membrane–cathode–electrode is placed between the mentioned bipolar palates, then the reactant gases can leak from the edges of porous electrodes. Therefore the edges of the electrodes must be sealed. A common technique for sealing

PEMFCs

Figure 2.10 The simplest applicable method for connecting cells in series.

Figure 2.11 MEA with sealing gaskets.

the edges of electrodes is using a surrounding planar gasket for each planar electrode. Employing such gaskets requires employing membranes with a bit larger size in comparison to the electrodes, as the substrate for the gaskets (Fig. 2.11). To distribute the fuel and oxidizer on the apparent surfaces of electrodes, gas channels (GCs) are designed and engraved on the two sides of BPs. If the designed GCs on a bipolar plate are parallel to each other, the external manifolds shown in Fig. 2.12 can be used for the feeding of reactants to the GCs and draining the products from the GCs. The horizontal GCs and manifolds illustrated in this figure are for the fuel, and the vertical ones are for the oxidizer. The fuel/oxidizer gas that passes through the GCs can diffuse through the GDL (note that a straight GC is surrounded by the bipolar plate solid material from three sides and by the porous GDL of the electrode from one side).

69

70

Fuel Cell Modeling and Simulation

Figure 2.12 External manifolds for partitioning fuel/oxidizer between bipolar plates.

Figure 2.13 Inner manifolds for the distributing of reactants on the electrode surfaces.

Employing external arrangements for the manifolds will result in two shortcomings for the cooling and the sealing of the cells [9]. Since the external manifolds cover the edges of bipolar plates, it is hard to supply the circulating cooling water through the bipolar plates. Additionally, the sealing gaskets will not be sufficiently pressed down in the GC regions of bipolar plates, which can increase the probability of leakage. Therefore another arrangement known as an inner manifold is more applicable (Fig. 2.13). In the inner manifold the GCs engraved on the surfaces of each bipolar plate are limited to the inner part of plates, which is confronted with the surfaces of electrodes, i.e., the network of GCs does not extend to the edges of bipolar plates. To transfer the hydrogen fuel from the network of GCs on a face of one bipolar plate to the network of GCs on the face of the other, a pair of holes is placed at the two opposite corners of the plate (one for feeding the current network from the previous network and the other for the feeding of the next network from the current network). Similarly, for the

PEMFCs

feeding of the GCs network for the oxidizer, two other holes at two other opposite corners are used. These feeding holes and the GCs networks must be carefully designed to prevent the mixing of fuel and oxidizer. Employing the internal manifolds will result in a stack, which appears as a solid block with electric and hydraulic ports installed on the two opposite faces of it for connecting the electric power cables and gas feeding pipes. These two faces and endplates beneath them can have a complex design. The edges of bipolar plates for this arrangement can be easily employed to inject cooling water into the cooling jacket designed inside each bipolar plate. An interesting and precious feature of PEMFCs stemming from this type of stacking of cells is the ease of power scalability. By adding a few cells or removing a few cells the stack maximum power can be intentionally increased or decreased. To find out the benefits of this feature, examining the scalability challenges of other power sources such as internal combustion engines can be helpful. If a motor company wants to increase the maximum power of its produced engines for a specific product of an automaker company, several strategies can be followed, such as increasing the volume of the combustion chamber, increasing the compression ratio, employing effective turbo-charging, etc. However, the costs for reliable implementation of these strategies are rather high, which is not comparable with just adding a few cells to a stack. To perform suitable sealing of the stack, it is required to sufficiently compress the MEAs and bipolar plates together. For this purpose, four tie rods are usually passed through the four corners of the MEAs and bipolar plates. However, the applied compression for the stacking of the cells will result in the changes of porous electrode microstructure and, consequently, the changes of porous electrode transport properties. The bipolar plates are made from either polymer-sealed high-conductive graphite or noncorrosive metals. The polymer is used to make porous graphite impermeable against water. However, metal bipolar plates can result in higher power density and more robust stack design; but they can be expensive for limited productions. Besides, the metallic plates are more sensitive in corrosive conditions, which will lead to faster degradation of membrane and loss of electrical contact. An important feature of bipolar plates that can affect their key attributes is the design of the engraved network of GCs on its surfaces, which is also known as the flow field design in the literature. There are several proposed flow field designs in the literature (Fig. 2.14). The parallel flow field needs a small differential pressure for the flowing of gas, which means that low power is required for the gas exchange from the tank to the electrodes; however, it has a poor performance in the uniform distribution of gases on the electrodes and also in the removing of liquid water from a flooded GC. In fact, when a small droplet forms in one of the parallel paths, the gas flow rate through that path will be significantly reduced (always, a fluid tends to pass through the path with the smallest hydraulic resistance). On the other hand, the serpentine flow field has a better capability in removing liquid water from GCs and provides a more uniform gas

71

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Fuel Cell Modeling and Simulation

Figure 2.14 The proposed flow-field designs in the literature: (a) parallel, (b) serpentine, (c) parallel-serpentine, (d) interdigitated, and (e) pin-type.

distribution on the electrodes. However, the required differential pressure for the gas flow is considerably larger than the parallel flow field. Therefore today a combination of parallel and serpentine flow fields, known as parallel-serpentine (Fig. 2.14(c)), is used in most of PEMFCs, which has the advantages of both parallel and serpentine flow fields. Other particular flow fields, designed for particular purposes, are also proposed and manufactured. For example, if the cell operating condition is such that the liquid water formation is probable (i.e., electrode flooding is a severe challenge), then the interdigitated flow-field can be a good solution. In this flow field, two groups of inlet and outlet GCs exist, which are not directly connected to each other on the bipolar plate. However, the pressurized gas in the inlet GC can diffuse to the GDL, be transferred under the land (rib), and finally be ejected to the low-pressure GC. The capability of this flow field in removing liquid water from the porous electrodes is amazing. However, the required differential pressure required for purging the gas from the high-pressure GC to the low-pressure GC is significant; this means that a considerably larger blower/compressor is required.

PEMFCs

If the liquid water formation in the electrodes and channels is not a notable challenge (e.g., for the low-power PEMFCs), the pin-type flow field can be a good choice. The required pressure differential for this flow field is even less than for the parallel flow field. However, the pin-type land regions of the bipolar plate, which perform the role of electronic collectors from the electrodes, have a smaller area in comparison to the lands in the other flow fields. This fact represents that high current density cannot be achieved from this type of flow field. c) Components and structure of a PEMFC system A question may arise concerning the content of previous subsection: is having only a PEMFC stack sufficient for delivering electric power to the consumer? The answer is negative. Many auxiliary subsystems are still required, which may construct a large portion of cost breakdown in a PEMFC system. These essential subsystems are usually known as the balance of plant (BOP) or balance of system (BOS). The main parts of a BOP, which occupies a large portion of a PEMFC system (usually more than 50%), are as follows (Fig. 2.15): a) Gas supply subsystem: the functions of this subsystem are providing safe storage of hydrogen fuel, effective delivery of fuel and oxidizer to the GCs in the stack, and performing recycling if necessary. This subsystem includes the hydrogen storage tank, regulators for regulating the hydrogen pressure at the tank outlet, valves, pipes, and blowers/compressors to feed the stack with fuel and air streams with tuned flow rates, and also reutilizing the nonreacted gases. For the rare situations where pure oxygen is used as the oxidizer, the oxygen tank and regulator are also needed. b) Humidification subsystem: The function of this subsystem is humidifying the flow of reactants, if necessary. Usually, the anode side of the membrane needs humidification, and hence the hydrogen inlet stream is humidified. The reason for this required humidification will be later elucidated, when we will explain the water transport phenomena in the cell. These subsystems include a small water tank and a humidifier, which can be either of sparge type, membrane type, or spray type. In some systems with less complexity (e.g., portable power sources), this subsystem is not used, and instead, the generated water in the cathode electrode is used for the cell humidification; this kind of humidification (which is also known as passive humidification) can lead to lower performance due to the less control on the humidification operation [7]. c) Thermal management subsystem: The generated heat in a large PEMFC system may be significant, and hence active cooling may be required for the thermal management of the stack. There are two active cooling methods, air cooling and water cooling. In the air cooling method, excess air and fuel are fed to the stack just for cooling. However, the cooling capacity of this method is not sufficient for PEMFC systems with large power densities. In the water-cooling method, the cold liquid

73

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Fuel Cell Modeling and Simulation

Figure 2.15 Schematic of BOP for a PEMFC system.

mixture of water and ethylene glycol is used to cool the bipolar plates and, consequently, to cool stack. The thermal management system has other functions, such as preheating of the stack during cold start or heating of the stack when the system operates in an extremely cold environment. The thermal management subsystem includes the water tank, pump, valves, pipes, heater, radiator, and cooling fan. d) Hydrogen reformation subsystem: When a nonhydrogen fuel is the only available fuel for the PEMFC operation, hydrogen reformation for feeding the PEMFC is required. We will explain the reformation process and equipment in Section 2.1.3. For stationary PEMFC systems, a fuel reformer is often used. For automotive applications, the in situ reformation can lead to extra cost, complexity, and air pollution of the vehicle. Therefore this is usually avoided, and instead a large hydrogen storage tank, which can be refueled from a hydrogen fueling station, is employed. e) Power conditioning and control unit: Since the required electric current is typically alternating current (AC), the produced direct current (DC) from a PEMFC stack needs to be converted and conditioned by an inverter. Additionally, the PEMFC system must be carefully controlled to produce a suitable response for the demanded electric power. For the cases with transient operational intrinsic such as fuel cell electric vehicles (FCEVs), this subsystem can be quite complex; it receives feedback signals from various sensors installed for the monitoring of flow rate, pressure, voltage, current, and temperature. It also receives probable com-

PEMFCs

mands from the operator (such as acceleration or deceleration commands from the driver in an FCEV). Finally, it processes the inputs and sends the decision signals to the valves, blowers, fans, heaters, etc., to provide a stable and safe control of the system. Usually, this subsystem is accompanied by an energy storage system (ESS) with high-power but low-energy characteristic to ensure the startup of the system by running the necessary blowers and heaters.

2.1.2 Transport phenomena in PEMFCs In Fig. 2.16, we presented the cross-section of a PEMFC portion, including membrane, cathode and anode electrodes, GCs, and BPs. Each electrode consists of three main parts of CL, MPL, and GDL. Each illustrated part in this figure is a zone for which a transport phenomenon or a few transport phenomena can happen. To accomplish the PEMFC function in generating electric power, the required transport phenomena according to this figure are as follows (Table 2.1): 1- Reactants transport from the GCs to the TPBs in the CLs. More specifically, in the anode side, the hydrogen must be transported as GC → GDL → MPL → CL, where the only nonporous zone in this path is the GC. Similarly, in the cathode side, the air (or oxygen when the oxidizer is pure oxygen) must be transported as GC → GDL → MPL → CL. The reactants in the CL must diffuse through the porous nanostructure and reach the TPBs. To this end, diffusing through the solid electrolyte penetrating CL and surrounding TPBs is also required (Fig. 2.5). 2- Proton transport from the anode TPBs to the cathode TPBs. To this end, the protons must first be transported from the TPBs in the anode electrode to the solid electrolyte phase, which is penetrated into the anode CL. After that, the protons must be conducted via the solid electrolyte phase from the anodic face of membrane to the cathodic face of membrane. Finally, at the cathode side the protons must be delivered to the TPBs via transport in the electrolyte phase of the CL domain. Therefore the anode CL, membrane, and cathode CL are the three zones incorporated in the proton transportation. 3- Electron transport from the anode TPBs to the cathode TPBs. The electrons must be transferred from the TPBs to the solid electrode phase in the anode CL (this solid electrode phase consists of connected carbon particles). Then the electrons must be transported from the solid phase of anode CL to the solid phase of cathode CL. Finally, the electrons at the solid phase of CL must be transferred to the cathode TPBs, where the electrochemical redox half-reactions can happen. The transportation path from the solid electrode phase of the anode to the solid electrode phase of the cathode can be a long path. In fact, a portion of this path will be out of the cell, and even a portion may be out of the stack. The inner portion consists of the solid matrix of three porous layers of CL, MPL, and GDL, as well as the bipolar plates.

75

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Fuel Cell Modeling and Simulation

Figure 2.16 Main parts of a PEMFC in the cross-section of a portion of the cell.

4- Products transport from the cathode TPBs to the cathode GCs. The produced water at the cathode TPBs must diffuse to the gas phase of CL. After that, it must be transported to the GC. More specifically, in the cathode side the produced water must be transported as CL → MPL → GDL → GC, where the only nonporous zone in this path is the GC. If the produced water is in the form of only vapor, then the transport of water is based on the single-phase fluid flow, whereas if the liquid water also exists in the domain, then the multiphase fluid flow will be the basis of the transport of the products. 5- Heat transport from the TPBs to the cooling channels. There are three main sources of heat generation in a PEMFC: reversible heat generation in the TPBs (due to the entropy generation), irreversible heat generation in the TPBs (due to the activation loss), and irreversible heat generation in the electric charge conducting path, especially membrane (due to the ohmic loss). The generated heat can be dissipated through two general paths, the solid path (whose destination is the cooling channels) and the fluidic path (whose destination is the GC). Among these sources and paths, the most significant part of the heat is transported from TPBs to the cooling channels through the solid path, in both anode and cathode sides. This solid path includes the solid matrix of three porous zones of CL, MPL, and GDL, as well as the BPs, at both sides. We present the five above-mentioned transport phenomena and their incorporated zones in Table 2.3. To fully discover the mentioned transport phenomena in Table 2.3, we will describe the transport phenomena in the following structure: • Reactant transport through the GCs • Reactant transport through porous layers of electrodes

PEMFCs

Table 2.3 The transport phenomena and incorporated zones in a PEMFC. Zone

Cathode side

Anode side

Reactants

*

• • • • •

Cooling channel(s) Bipolar plate GC GDL MPL CL (gas phase) CL (solid electrode phase) CL (electrolyte phase) Membrane CL (electrolyte phase) CL (solid electrode phase) CL (gas phase) MPL GDL GC Bipolar plate Cooling channel(s)

Transported Item Protons Electrons Products

    

 

   

  

  

  

  

   

Heat*  

 

    



   

The most significant part of heat.

Proton transport through the membrane Electron transport in the cell Single- and multiphase water transport through porous layers of electrodes Single- and multiphase water transport through the GCs Heat transport in the cell

a) Reactants transport through the GCs Generally, there are two mechanisms for the mass transfer of a species, diffusion and convection. To have a better sense, consider a glass of water on the table; if we inject a red ink droplet in the water by a small syringe, then the molecules of ink will diffuse through the water molecules, and after a while, the glass will contain red water. Here the ink transfer mechanism from the syringe bevel to the entire glass, which stems from the random motions of molecules, is known as the diffusion mechanism. Now consider the air stream passing the surface of a sea. By the passing of air, liquid water at the surface will evaporate, and the produced vapor will be transferred by the air stream. This is an example of convective mass transfer from the surface to the bulk flow (here an air stream), which frequently happens in our environment. Now consider the humid air passing on a surface of a dehumidifying substance, such as silica gel, which absorbs

77

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Fuel Cell Modeling and Simulation

Figure 2.17 Convection mass transfer and its related boundary layer.

the vapor from the humid air. This is also an example of convective mass transfer, but from the bulk flow to the surface of the dehumidifier. To have a quantitative evaluation about the amount of mass transfer in this mechanism, consider a mixture of two chemical species S1 and S2 that passes on a surface, which selectively absorbs species S1 (Fig. 2.17). If the absorption occurs, then the molar concentration CS1,s of species S1 on the absorbing surface will differ from the molar concentration CS1,∞ of species S1 in the free stream. The difference between these two molar concentrations creates the concentration boundary layer, which has an analogy with velocity and thermal boundary layers. The concentration boundary layer is defined as a portion of fluid zone for which the surface affects the concentration field of the fluid, and, consequently, notable concentration gradients exist. More specifically, the boundary layer is a growing layer from the leading edge of the surface, whose thickness δC (x) at every longitudinal position x is defined as the height from the surface at which  (CS1 − CS1,s )/CS1,∞ − CS1,s = 0.99. To calculate the molar flux of surface absorption of species S1 (NS1,s with the unit of kmol m−2 −1 ), the Fick law, which demonstrates the mass transfer by diffusion mechanism, can be used for the surface. In fact, just close to the surface, the bulk velocity of the fluid is almost zero, and the random motion of the molecules, which is recognized as the diffusion mass transfer, is the origin of mass transfer. The law states NS1,s = −DS1,S2

∂ CS1

, ∂ y y=0

(2.9)

where DS1,S2 is the binary diffusion coefficient. Note that for any points of the boundary layer above the surface (i.e., y > 0 in Fig. 2.17), the species is transferred by both bulk fluid motion (advection) and diffusion, which is recognized as the convection mechanism (mode) of mass transfer. To calculate the convective mass transfer of species S1 on the surface, the following equation, which is analogous to Newton’s cooling law and is based on the difference of S1

PEMFCs

molar concentrations across the boundary layer, can be used: 



NS1,s = hm CS1,s − CS1,∞ ,

(2.10)

where hm (m s−1 ) is the convective mass transfer coefficient, which resembles the convection heat transfer coefficient. By comparing Eqs. (2.9) and (2.10) we can find out that

hm = −

DS1S2 ∂ C∂ yS1 y=0 (CS1,s − CS1,∞ )

(2.11)

.

This equation indicates that the situation in the concentration boundary layer in terms of surface concentration gradient can significantly affect the convection mass transfer coefficient and, consequently, the rate of species transfer in the boundary layer, which shows the importance of boundary layer. Similarly, for internal flows, there may be convection mass transfer. For example, consider a gas streaming in a tube with an interior surface coated with a dehumidifying substance. Then vapor absorption will happen, and, consequently, the internal concentration boundary layer will begin developing. Here the mean concentration CS1,m of species S1 plays a similar role for internal convection mass transfer, as CS1,∞ plays for the external convection mass transfer. It is defined as

CS1,m (x) =

Ax (CS1 u)dAx

um Ax

,

(2.12)

where Ax is the cross-section area of the tube at the longitudinal position of x, and um is the mean velocity of flow at that cross-section. By adopting CS1,m as the reference the local convection mass transfer between the internal flow and the tube surface can be calculated via 



NS1,s = hm CS1,s − CS1,m .

(2.13)

Moreover, in laminar and turbulent flows, fully developed concentration condition exists when ∂ C∗ = 0, ∂x

(2.14)

where, for a circular tube, C∗ =

CS1,s − CS1 (r , x) . CS1,s − CS1,m (x)

(2.15)

In this equation, CS1,s is the surface molar concentration of species S1 on the tube surface. Eqs. (2.10) and (2.13) can be written based on the mass concentration (e.g., ρS1 ),

79

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Table 2.4 Measured DS1,S2 at 1 atm for several pairs of gases [10]. Gaseous pair Temperature (K) DS1,S2 (m2 s−1 ) H2 –H2 O 307 9.15 × 10−5 H2 –Air 282 7.10 × 10−5 Air–H2 O 298 2.6 × 10−5 Air–O2 273 1.76 × 10−5 O2 –H2 0 308 2.82 × 10−5 N2 –O2 293 2.20 × 10−5 H2 –He 317 1.706 × 10−4 Air–He 282 6.58 × 10−5 O2 –He 317 8.22 × 10−5 H2 –CO2 298 6.46 × 10−5 H2 –CO 296 7.43 × 10−5

instead of molar concentration. In such a case the dimension of N on the left-hand side of these equations would be kg m−2 s−1 , but the convection mass transfer coefficient will be unchanged. To analyze the convection mass transfer coefficient, it is usual to analyze the corresponding nondimensional Sherwood number Sh, which is defined for an internal flow in a circular tube as ShD =

hm D . DS1,S2

(2.16)

For the fully developed laminar flow in a circular tube with a uniform vapor density at its surface, ShD = 3.66, and for the fully developed turbulent flow, it is 4/5 2/5 ShD = 0.023ReD Sc .

(2.17)

This relation is analogous to the Dittus–Boelter relation in heat transfer, whereas the Sherwood number and Schmidt number Sc = DS1ν,S2 are analogous to the Nusselt and Prandtl numbers in heat transfer. The values of DS1,S2 for some pairs of gases at 1 atm are presented in Table 2.4. Example 2.1. The internal wall of a circular tube is coated with a dehumidifier substance, for which ShD = 4.0. If humid air with 100% relative humidity, 25 °C, 1 atm pressure, and the mass flow rate of 5 × 10−4 kg s−1 enters the tube, then how long must the tube be to lessen the average relative humidity to 50%? The inner wall temperature is 25 °C, and its diameter is 10 mm. Solution. Known: Vapor is removed from the air, which has 100% relative humidity, 25 °C, and 1 atm at the entrance of a tubular humidifier, for which Sherwood number equals 4.0.

PEMFCs

Find: Required length to lessen the relative humidity to the half. Schematic:

1. 2. 3. 4.

Assumptions: The concentration of vapor on the surface is zero. The average concentration of vapor decreases linearly along the tube length. Sh is constant on the internal wall. The temperature of air stream remains equal to the wall temperature, which is 25 °C. Properties: Air (25 °C): ν = 1.57 × 10−5 m2 S−1 , DH2O,air = 2.60 × 10−5 m s−1 . Analysis: hm =

ShDS1,S2 4.0 × 2.6 × 10−5 = = 1.04 × 10−2 . D 0.01

At the inlet: Relative humidity = 100% → PH2O = PSAT ,25 °C = 3.169 kPa. CH2O,m,in =

PH2O 3.169 · 1000 Pa = = 1.2784 mol m−3 , 3 Ru T 8.3145 m Pa K−1 mol−1 · 298.15 K

ρH2O PH2O /RH2O T PH2O · MWH2O =m ˙ mix,in × =m ˙ mix,in × ρmix Pmix /Rmix T Pmix · MWmix 3 . 169 · 18 = 5 × 10−4 × = 9.706 × 10−6 kg s−1 . 101.325 · 29

m˙ H2O,in = m˙ mix,in ×

At the outlet: Relative humidity = 50% → PH2O = 0.5PSAT ,25 °C = 1.5845 kPa. CH2O,m,out =

PH2O 1.5845 · 1000 Pa = = 0.6392 mol K−3 , Ru T 8.3145 m3 Pa K−1 mol−1 · 298.15 K

ρH2O PH2O /RH2O T PH2O · MWH2O =m ˙ mix,out × =m ˙ mix,out × ρmix Pmix /Rmix T Pmix · MWmix 1 . 5845 · 18 = 5 × 10−4 × = 4.853 × 10−6 kg s−1 . 101.325 · 29

m˙ H2O,out = m˙ mix,out ×

81

82

Fuel Cell Modeling and Simulation

For the presented control volume in the schematic: m˙ H2O,in − m˙ H2O,out 

L





CS1,m (x) − 0 dx = (π D) hm

= (π D) hm 0

0

L

CS1,m,in − CS1,m,out xdx L

L = (π D) hm (CS1,m,in − CS1,m,out ) → 2  2 m˙ H2O,in − m˙ H2O,out 2(9.706 × 10−6 − 4.853 × 10−6 ) L= = π Dhm (CS1,m,in − CS1,m,out ) π ∗ 0.01 ∗ 1.04 × 10−2 · (1.2784 − 0.6392) = 4.648 × 10−2 m = 4.648 cm. The transport of reactants through the GC is an internal convection mass transfer. Common GCs have a rectangular cross-section. The internal flow in these GCs is restricted by the GDL from one side and by the bipolar plate from the other three sides; i.e., the fluid flow is restricted by three impermeable walls and a permeable wall. The reactants can diffuse through this permeable wall toward the CL and its TPBs. However, the situation on the permeable wall is somehow complicated due to the geometry and complex roughness of this wall. In the anode GCs, usually, a mixture of hydrogen and water vapor flows through the GCs, whereas in the cathode GCs a mixture of oxygen, water vapor, and nitrogen exists. In the anode side the hydrogen diffuses through the GDL, whereas in the cathode side the oxygen diffuses through the GDL. For a uniform current density distribution under a portion of GC, the rate of reactants diffusion (hydrogen in the anode side and oxygen in the cathode side) can be assumed constant on the permeable wall for a steady condition. To comprehensively analyze the reactants transport through the GC, all governing equations must be solved: the continuity equation, the momentum and energy conservations equations for the bulk flow, and the conservations of species for the hydrogen in the anode side and for the oxygen in the cathode side. For the rectangular GCs, the mentioned equations in the 3D Cartesian coordinate system are ∂ρ ∂(ρ u) ∂(ρ v) ∂(ρ w ) + + + = 0, ∂t ∂x ∂y ∂z

∂(ρ u) ∂(ρ u) ∂(ρ u) ∂(ρ u) +u +v +w ∂t ∂x ∂y ∂z        ∂u ∂ ∂u ∂ ∂u ∂P ∂ μ + μ + μ , =− + ρ gx + ∂x ∂x ∂x ∂y ∂y ∂z ∂z

(2.18)

(2.19)

PEMFCs

∂(ρ v) ∂(ρ v) ∂(ρ v) ∂(ρ v) +u +v +w ∂t ∂x ∂y ∂z        ∂v ∂ ∂v ∂ ∂v ∂P ∂ μ + μ + μ , =− + ρ gy + ∂y ∂x ∂x ∂y ∂y ∂z ∂z ∂(ρ w ) ∂(ρ w ) ∂(ρ w ) ∂(ρ w ) +u +v +w ∂t ∂x ∂y ∂z        ∂w ∂ ∂w ∂ ∂w ∂P ∂ μ + μ + μ , =− + ρ gz + ∂z ∂x ∂x ∂y ∂y ∂z ∂z ∂(ρ cp T ) ∂(ρ cp T ) ∂(ρ cp T ) ∂(ρ cp T ) +u +v +w ∂t ∂x ∂y ∂z        ∂ ∂ ∂T ∂T ∂T ∂ + + = k k k ∂x ∂x ∂y ∂y ∂z ∂z   ∂P ∂P ∂P ∂P + μ∅ + βT +u +v +w ∂t ∂x ∂y ∂z   Ns  ∂ Jx,i ∂ Jy,i ∂ Jz,i − , hi + + ∂x ∂y ∂z i=1

(2.20)

(2.21)

(2.22)

The first term in the right-hand side (RHS) of Eq. (2.22) denotes the heat transfer via conduction mechanism, the second term indicates the compressibility effects (β = − ρ1 ∂∂ρT P is the coefficient of thermal expansion), the third term μ∅ is the viscous dissipation rate, and ∅ equals 

   2  2   ∂u ∂v 2 ∂u ∂w 2 ∂v ∂w 2 ∂u 2 ∂v ∂w + ( + ∅=2 + + ) +( + ) +( + ) ∂x ∂y ∂z ∂y ∂x ∂z ∂x ∂z ∂y   2 ∂u ∂v ∂w 2 + + . (2.23) − 3 ∂x ∂y ∂z

The fourth term on the RHS of energy conservation equation (2.22) indicates the energy transfer due to the diffusion of species, where Ns is the number of species, and (Jx,i , Jy,i , Jz,i ) is the mass diffusion flux vector of species i, where ∂ρi 1 ∂T − D S ,i , ∂x T ∂x ∂ρi 1 ∂T − D S ,i , Jy,i = −Di,m ∂y T ∂y ∂ρi 1 ∂T Jz,i = −Di,m − D S ,i . ∂z T ∂z

Jx,i = −Di,m

(2.24) (2.25) (2.26)

83

84

Fuel Cell Modeling and Simulation

In these equations, Di,m represents the mass diffusion coefficient (also called the diffusivity) of species i in the mixture, DS,i implies the Soret diffusion coefficient of species i, and ρi is the mass concentration of species i. The Soret diffusion is a special kind of diffusion due to the thermal effects known as thermophoresis effects. Although the gas flow velocity through the GCs is often considerably less than 0.3 M, we cannot easily assume that the gas flow is incompressible. In fact, due to the electrochemical reactions in the CLs, the species densities will be changed along the GC, and, consequently, the gas overall density (which is the sum of species densities) does experience local changes in the flow domain. For the gas flow through the GCs, the second and third terms on the RHS of energy conservation equation (i.e., the terms representing compressibility effects and viscous dissipation rate) are not significant in comparison to other terms, and hence they can be neglected. Since the thermal gradients in the usual GCs are not considerable, the Soret diffusion can also be neglected. However, for the heat transfer term due to the diffusion of species (the fourth term on the RHS of Eq. (2.22)), this term cannot be neglected when the Lewis number Lei = ρ cP kDi,m is far from unity for most of the species. i ∂ρi ∂ρi , , ), which appear in the mass diffusion flux of species i In such a case, the terms ( ∂ρ ∂x ∂y ∂z (Eqs. (2.24)–(2.26)), can be determined by solving the conservation equations for the species   ∂ρi ∂(ρi u) ∂(ρi v) ∂(ρi w ) ∂ Jx,i ∂ Jy,i ∂ Jz,i , + + + =− + + ∂t ∂x ∂y ∂z ∂x ∂y ∂z

i = 1, . . . , Ns.

(2.27)

Neglecting the Soret effects and assuming constant diffusivities, these equations can be rewritten as  2  ∂ρi ∂(ρi u) ∂(ρi v) ∂(ρi w ) ∂ ρi ∂ 2 ρi ∂ 2 ρi , + + + = D i ,m + 2 + ∂t ∂x ∂y ∂z ∂ x2 ∂y ∂ z2

i = 1, . . . , Ns − 1. (2.28)

Since density and transport properties such as viscosity, conductivity, and diffusivity are temperature dependent, the governing equations (2.18)–(2.22) and (2.28) must be solved simultaneously to find out the following unknown parameters: ρ, u, v, w , T ρ1 , ρ2 , . . . , ρNs .      5

Ns

It is worth noting that since ρ1 + ρ2 + · · · + ρNs = ρ , solving only Ns − 1 equations in the set of equations denoted as (2.28) will be sufficient, and instead of conservation of species for the Nsth species, the equation ρ1 + ρ2 + · · · + ρNs = ρ can be solved. Usually, the species with the largest concentration is considered as the Nsth species. To obtain Di,m at pressures and temperatures that are not presented in the related tables and data sets (such as Table 2.4 in this book), the dependency of binary diffusion

PEMFCs

coefficients to the affecting parameters must be discovered. When the molecules of a gas species are diffusing into/among the molecules of another gas species, they collide with each other and also with the molecules of other species in their motions (which itself is due to their kinetic energy). Therefore we can expect the following facts: • By increasing the number of the molecules in the domain (i.e., by increasing the pressure) more collisions will happen, which decreases the average diffusion rate. • By increasing the temperature the molecules will have higher kinetic energy, and hence the average diffusion rate will increase. • By increasing the size of the molecules (i.e., molecule diameter) their motion will be accompanied by more collisions, and hence the average diffusion rate will be reduced. • By increasing the mass of the molecules (i.e., the species molar mass) their motion needs to overcome larger inertia, and as a result, the average diffusion rate will be reduced. The molecular dynamic theory predicts that for the self-diffusion of species S1 into a mixture of S1 [7], DS1 ∝

T 3/2 , 1/2 2 PMS1 σS1

(2.29)

which approves the above-mentioned facts. In Eq. (2.29), MS1 and σS1 are the molar mass and the molecular diameter of species S1, respectively. The above relation can be rewritten for a binary mixture of species S1 and S2 as  ref DS1,S2 = DS1 .S2

P ref P



T T ref

3/2

(2.30)

.

This equation can be used as a simple estimation when the binary diffusion coefficient at a reference pressure and temperature is known (e.g., from Table 2.4). However, when one of the species is a polar gas (such as water vapor), this estimation may not be accurate enough. Another useful estimation of binary diffusion coefficient, applicable for evaluating the diffusion rates of polar and nonpolar reactants in a PEMFCs, is the following relation [7]: 

1/3 a T  DS1,S2 = Pc,S1 Pc,S2 P Tc,S1 Tc,S2

b



Tc,S1 Tc,S2

5/12



1 1 + MS1 MS2

1/2 ,

(2.31)

where Pc and Tc are the critical pressure and temperature, whereas a and b are constants depending on the type of species. In this relation, all temperatures are in K, all pressures are in atm, molar masses are in kg kmol−1 , and the diffusion coefficient is in m2 s−1 . For a pair of two nonpolar gases, a and b are 2.745 × 10−8 and 1.823, respectively. For a pair

85

86

Fuel Cell Modeling and Simulation

Table 2.5 C1 and C2 constants for calculating DS1,S2 in Eq. (2.32). Gaseous air

C1

C2

H2 –H2 O H2 –Air Air–H2 O Air–O2 O2 –H2 0 N2 –O2

2.1355 × 10−10 2.4589 × 10−9 4.3656 × 10−11 6.3757 × 10−10 4.2099 × 10−11 6.3597 × 10−10

2.334 1.823 2.334 1.823 2.334 1.823

of H2 O and a nonpolar gas, a and b are 3.640 × 10−8 and 2.334, respectively. For the pairs of gases commonly existing in the GCs of PEMFCs, the above equation can be simplified by inserting critical properties and molar masses as DS1,S2 = C1

T C2 , P

(2.32)

where T is in Kelvin, P is in atm, DS1,S2 is in m2 s−1 , and C1 and C2 are presented in Table 2.5. This is a simple but useful equation for the modeling of gas transport in PEMFCs. When one of the binary gases is a gas with small molecule (such as H2 or He), the accuracy of Eqs. (2.31) and (2.32) for obtaining the binary diffusion coefficient may not be sufficient. However, there are other relations for calculating the binary diffusion coefficients with higher accuracy in the literature. We will not be discussed them here for brevity. Till now, we learned how to drive binary diffusion coefficients to be adopted in the governing equations of reactants transport through GCs. There is still a question? If the gas flow through a GC consists of more than two species (e.g., we know that at the cathode side of most of PEMFCs, at least three species of nitrogen, oxygen, and water vapor exist in the GC), how can we drive Di,m s to use them in Eqs. (2.24)–(2.26)? There are four general methods to deal with mixtures with more than two species, which are recognized as multispecies or multicomponent mixtures: • Neglecting inert species: In this method, we neglect the contributions of the inert (nonreacting) species. For example, in the cathode, nitrogen, as the inert species, is neglected in this method, i.e., the gas flowing through the GC is treated as a mixture of oxygen diffusing into water vapor. • Mole-fraction-based averaging of properties: In this method, the critical pressure and temperature of the mixture, as well as the molar mass of the mixture, are calculated by a mole-fraction-based averaging for the mentioned properties of constructing species of the mixture. After that, Eq. (2.31) or similar correlations can be adopted for the calculations of Di,m s. Note that for an intensive property like IP

PEMFCs

(such as critical pressure or critical temperature), the mole-fraction-based average of IP, denoted IPmix , can be calculated by •

IPmix =

Ns 

Xi IPi ,

(2.33)

i=1

where Xi is the mole fraction of species i in the mixture. For calculating the molar mass of the mixture, we can use the following two relations: Mmix =

Ns 

Xi Mi ,

(2.34)

i=1

1 , i=1 (Yi /Mi )

Mmix = Ns

•

(2.35)

where Yi is the mass fraction of species i in the mixture. Mole-fraction-based averaging of diffusivity inverses: In this method, the diffusivity of species i in the mixture can be calculated by the relation (1 − Xi ) Di,m = Ns , j=1 (Xj /Dij )

(2.36)

j=i

•

where Dij is the binary diffusion coefficient of species i and j, which can be computed via one of the already explained methods for the binary mixtures. The Maxwell–Stefan equations method: The above-mentioned three methods are straightforward but without high accuracy. A more reliable method to deal with multispecies diffusion is employing the Maxwell–Stefan equations. The Maxwell– Stefan set of equations for a nondense gaseous mixture containing Ns gases is Ns 1  Ci J x,j − Cj J x,i dCi = , dx Cmix j=1 D i ,j

i = 1, . . . , Ns,

(2.37)

Ns 1  Ci J y,j − Cj J y,i dCi = , dy Cmix j=1 D i ,j

i = 1, . . . , Ns,

(2.38)

Ns 1  Ci J z,j − Cj J z,i dCi = , dz Cmix j=1 D i ,j

i = 1, . . . , Ns,

(2.39)

j=i

j=i

j=i

where Cmix is the molar concentration of mixture, Cmix = C1 + C2 + · · · + CNs equal to P /Ru T. In these equations, Di,j is the binary diffusion coefficient of species i

87

88

Fuel Cell Modeling and Simulation

and j, and (J x,j , J y,j , J z,j ) is the molar diffusion flux vector for the species i with the dimension of mol m−2 s−1 . Note that this vector is related to the previously mentioned mass diffusion flux vector presented in Eqs. (2.24)–(2.26) via 







Jx,j , Jy,j , Jz,j = MWi J x,j , J y,j , J z,j .

(2.40)

In addition, Ci =

ρi

MWi

(2.41)

,

Examining Eqs. (2.37)–(2.39) reveals that there are 3Ns equations and Ns unknown vectors of (J x,j , J y,j , J z,j ) with 3Ns unknown components (if concentration gradients of species considered are known). Therefore, to solve the governing equations of reactants transport through GC (i.e., Eqs. (2.18)–(2.22) and (2.27)), Eqs. (2.37)–(2.39) can be used instead of Eqs. (2.24)–(2.26). However, since the components of molar diffusion flux vectors appear on the right-hand sides of these equations, dealing with these equations will not be an easy task. It requires a considerable computational cost in comparison to the three previously described methods, especially when the number of species is large. However, for tertiary mixtures, the Maxwell– Stefan equations can be simplified by writing each component of molar diffusion flux vector for species i as a function of binary diffusion coefficients and concentration gradients. Example 2.2. The gas flow through a GC at the cathode side consists of three species of oxygen, water vapor, and nitrogen. At a specific point of this GC, the gas flow can be assumed to be 1D (i.e., the local parameters only depend on the x-direction). At the mentioned point the molar fractions of oxygen, water vapor, and nitrogen are 0.15, 0.07, and 0.78, respectively, whereas their concentration gradients are −0.01, 0.02, and −0.004 mol m−4 . If the gas pressure and temperature is 1 atm and 308 K, what would be the mass diffusion flux of oxygen and water vapor? Solve the problem by all abovementioned methods. Solution. Known: gas flow in the cathode GC with 308 K and 1 atm consists of three species with the following mole fraction and concentration gradients at a point: XO2 = 0.15, dCO2 = −0.01 mol m−4 , dx

XH2O = 0.07,

XN2 = 0.78,

dCH2O = 0.02 mol m−4 , dx

dCN2 = −0.004 mol m−4 dx

Find: mass diffusion flux of oxygen and water vapor at that point by the four mentioned methods for calculating multicomponent diffusions.

PEMFCs

Assumptions: 1. The gas flow is 1D, along the x-axis. 2. The thermal diffusion is negligible. Properties: Pc,O2 = 50.14 atm,

Pc,H2O = 218.167 atm,

Tc,O2 = 154.78 K,

Tc,H2O = 647.27 K,

Pc,N2 = 33.54 atm, Tc,N2 = 126.2 K.

Analysis: i) Neglecting inert species: We assume that there are only oxygen and water vapor in the mixture. Therefore, regarding Eq. (2.32), DS1,S2 = C1

T C2 , P

where from Table 2.5 C1 = 4.2099 × 10−11

C2 = 2.334 → 3082.334 DO2,H2O = 4.2099 × 10−11 × = 2.7074 × 10−5 m2 s−1 1 and

Now, the Fickian mass diffusion flux is dρO2 dCO2 = −DO2,H2O MWO2 dx dx = −2.7074 × 10−5 × 32.00 × (−0.01) = 8.6637 × 10−6 kg m−2 s−1 ,

Jx,O2 = −DO2,m

dρH2O dCH2O = −DO2,H2O MWH2O dx dx −5 = −2.7074 × 10 × 18.02 × (0.02) = −9.7575 × 10−6 kg m−2 s−1 .

Jx,H2O = −DH2O,m

It is notable that the binary diffusion coefficient calculated from Eq. (2.32) differs from the experimental coefficient (2.82 × 10−5 m2 s−1 ) presented in Table 2.4 only by 4%. ii) Mole-fraction-based averaging of properties: Pc,mix =

Ns  i=1

Xi Pc,i = 0.15 × 50.14 + 0.07 × 218.167 + 0.78 × 33.54 = 48.95 atm,

89

90

Fuel Cell Modeling and Simulation

Tc,mix =

Ns 

Xi Tc,i = 0.15 × 154.78 + 0.07 × 647.27 + 0.78 × 126.20 = 166.96 K,

i=1

MWmix =

Ns 

Xi MWi = 0.15 × 32.00 + 0.07 × 18.02 + 0.78 × 28.01 = 27.91 kg kmol−1 .

i=1

According to Eq. (2.31), 

1/3 a T  Pc,O2 Pc,m DO2,m = P Tc,O2 Tc,m

b



Tc,O2 Tc,m

5/12





1 1 + MWO2 MWm

2.745 × 10−8 308 = (50.14 × 48.95)1/3 √ 1 154.78 × 166.96  1/2 1 1 × (154.78 × 166.96)5/12 + 32.00 27.91 −5 2 −1 = 2.1631 × 10 m s , 

1/3 a T  DH2O,m = Pc,H2O Pc,m P Tc,H2O Tc,m

b



Tc,H2O Tc,m



5/12



1/2

1.823

1 1 + MWH2O MWm

3.640 × 10−8 308 = (218.17 × 48.95)1/3 √ 1 647.27 × 166.96  1/2 1 1 × (647.27 × 166.96)5/12 + 18.02 27.91 −5 2 −1 = 2.6035 × 10 m s .

2.334

Therefore dρO2 dCO2 = −DO2,m MWO2 dx dx = −2.1631 × 10−5 × 32.00 × (−0.01) = 6.9220 × 10−6 kg m−2 s−1 ,

Jx,O2 = −DO2,m

dρH2O dCH2O = −DH2O,m MWH2O dx dx −5 = −2.6035 × 10 × 18.02 × (0.02) = −9.3830 × 10−6 kg m−2 s−1 .

Jx,H2O = −DH2O,m

1/2

PEMFCs

iii) Mole-fraction-based averaging of diffusivity inverses: Regarding Eq. (2.32) and Table 2.5, DO2,H2O = 4.2099 × 10−11 ×

3082.334 = 2.7074 × 10−5 m2 s−1 , 1

DO2,N2 = 6.3597 × 10−10 ×

3081.823 = 2.1881 × 10−5 m2 s−1 , 1

DH2O,N2 ∼ = DH2O,Air = 4.3656 × 10−11 ×

3082.334 = 2.8075 × 10−5 m2 s−1 . 1

Hence, according to Eq. (2.36), DO2,m =

(1 − XO2 ) = (XH2O /DO2,H2O + XN2 /DO2,N2 )

1 − 0.15 0.07 2.7074×10−5

+

0.78 2.1881×10−5

= 2.2232 × 10−5 m2 s−1 ,

DH2O,m =

(1 − XH2O ) = (XO2 /DH2O,O2 + XN2 /DH2O,N2 )

1 − 0.07 0.15 2.7074×10−5

+

0.78 2.8075×10−5

= 2.7909 × 10−5 m2 s−1 ,

and, as a result, dρO2 dCO2 = −DO2,m MWO2 dx dx = −2.2232 × 10−5 × 32.00 × (−0.01) = 7.1142 × 10−6 kg m−2 s−1 ,

Jx,O2 = −DO2,m

dρH2O dCH2O = −DH2O,m MWH2O dx dx = −2.7909 × 10−5 × 18.02 × (0.02) = −1.0058 × 10−5 kg m−2 s−1 .

Jx,H2O = −DH2O,m

iv) The Maxwell–Stefan equation method: Cmix =

P 101325 Pa = = 39.5667 mol m−3 . 3 Ru T 8.3145 m Pa K−1 mol−1 ∗ 308 K

According to the Maxwell–Stefan equations (here only Eqs. (2.37)), Ns Ns dCi 1  Ci J x,j − Cj J x,i 1  Ci Jx,j /MWj − Cj Jx,i /MWi = = , dx Cmix j=1 D i ,j Cmix j=1 D i ,j j=i

j=i

i = 1, . . . , Ns.

91

92

Fuel Cell Modeling and Simulation

Since

Ci Cmix

= Xi , Ns dCi  Xi Jx,j /MWj − Xj Jx,i /MWi = , dx D i ,j j=1

i = 1, . . . , Ns.

j=i

Therefore



1 XO2 Jx,H2O /MWH2O − XH2O Jx,O2 /MWO2 dCO2 = dx Cmix DO2,H2O  XO2 Jx,N2 /MWN2 − XN2 Jx,O2 /MWO2 , + DO2,N2 

dCH2O 1 XH2O Jx,O2 /MWO2 − XO2 Jx,H2O /MWH2O = dx Cmix DH2O,O2  XH2O Jx,N2 /MWN2 − XN2 Jx,H2O /MWH2O , + DH2O,N2 

dCN2 1 XN2 Jx,O2 /MWO2 − XO2 Jx,N2 /MWN2 = dx Cmix DN2,O2  XN2 Jx,H2O /MWH2O − XH2O Jx,N2 /MWN2 . + DH2O,N2 Now, since Di,j = Dj,i , using the binary diffusion coefficients calculated in the previous part of this example, we have 

−0.01 =

1 0.15Jx,H2O /18.02 × 10−3 − 0.07Jx,O2 /32.00 × 10−3 39.5667 2.7074 × 10−5  0.15Jx,N2 /28.01 × 10−3 − 0.78Jx,O2 /32.00 × 10−3 , + 2.1881 × 10−5 

1 0.07Jx,O2 /32.00 × 10−3 − 0.15Jx,H2O /18.02 × 10−3 0.02 = 39.5667 2.7074 × 10−5  −3 0.07Jx,N2 /28.01 × 10 − 0.78Jx,H2O /18.02 × 10−3 , + 2.8075 × 10−5 1 −0.004 = 39.5667 +

 0.78Jx,O2

0.78Jx,H2O 18.02

x,N2 × 10−3 − 28.01 × 10−3 2.1881 × 10−5  0.07Jx,N2 × 10−3 − 28.01 × 10−3 . 2.8075 × 10−5

32.00

0.15J

PEMFCs

Solving the above three linear algebraic equations yields the following three unknowns: Jx,O2 = 5.3839 × 10−6 kg m−2 s−1 , Jx,H2O = 1.0589 × 10−6 kg m−2 s−1 , Jx,N2 = 2.3336 × 10−5 kg m−2 s−1 . Comment: Comparing the results from the four applied methods demonstrates that the calculated mass fluxes by the Maxwell–Stefan equations are smaller than the mass fluxes calculated by the other three methods; this is more apparent for the water vapor mass flux. This fact shows that calculating the water vapor flux by the mentioned three simpler methods is usually overestimated. The mass transfer from the bulk flow of reactants in the GCs into the GDLs via convection mechanism is accompanied by friction, which leads to the pressure drop along with the GCs. This friction can be formulated by a friction factor. There are two friction factors in the literature, the Moody friction factor fM and the Fanning friction factor fF . These two friction factors are defined by the following relations for the internal flow through a channel: l P = fM Dh  τw = fF





1 2 ρV , 2

(2.42)



1 2 ρV , 2

(2.43)

where P, l, Dh , V , and τw are the frictional pressure drop, channel length, channel hydraulic diameter, fluid mean velocity, and the acted shear stress on the fluid from the channel wall, respectively. The hydraulic diameter of a channel equals four times the ratio of the channel cross-section area to its perimeter. Writing the force balance for a portion of channel length with l meters long results in   P × π Dh2 /4 = τw × π Dh l .

(2.44)

Hence τw = P ×

Dh . 4l

(2.45)

Substituting this relation in Eq. (2.43) yields fM = 4fF .

(2.46)

93

94

Fuel Cell Modeling and Simulation

Since these two friction factors are proportional to each other, in the rest of this book, we will use only the Moody friction factor; besides, for simplicity, we will drop the subscript M. To evaluate the factor, the fluid flow regime must be determined as the first step. h is less than 2000, the regime is lamGenerally, when the Reynolds number Re = ρ VD μ inar, and when it is larger than 3000, the regime is turbulent. For 2000 < Re < 3000, reliable determining of the regime needs further information. For laminar fluid flow, f=

64 . Re

(2.47)

Putting this relation into Eq. (2.42), we can find that for the laminar flow, the frictional pressure drop along the channel is P =

32μVl , Dh2

(2.48)

which is known as the Hagen–Poiseuille equation. The fluid flow through PEMFCs is usually laminar. However, for the rare case of turbulent flow in the GCs, we can solve the following implicit relation for f to determine the frictional pressure drop along the channel [11]: 



ε

2.51 1 Dh , = −2.0 ln + f 3.7 Ref 0.5

(2.49)

where ε is the average roughness of the channel wall. Note that the calculated pressure drop from Eq. (2.42) is only the frictional pressure drop, and there are other facts such as channel blockage by liquid water, the consumption of reactants, and convective flow of reactants in the underneath GDL, which can cause excessive pressure drop. However, the GC wall friction is almost always the most significant source of pressure drop in PEMFCs. b) Reactants transport through porous layers of electrodes Each PEMFC electrode consists of two or three porous layers: GDL, (MPL), and CL. The reactant transport in these porous layers, especially MPL and CL, is mainly performed via diffusion mechanism in the through-plane direction. Therefore the Fick law can be employed for evaluating the mass transfer rate of species, but with effective diffusion coefficients Deff . There are several correlations for evaluating the effective diffusion properties in the literature. As one of the simple correlations, the effective diffusivities for the reactant gases in the mentioned porous layers can be written as ∅ τ

Deff = D ,

(2.50)

PEMFCs

Figure 2.18 Two porous media with a) small tortuosity and b) large tortuosity.

where ∅ denotes the porosity (the fraction of void volume in the porous layers), D is the ordinary bulk diffusion coefficient of reactant gases, and τ denotes the tortuosity. A higher tortuosity of a medium indicates a more tortuous medium with longer effective average path across the medium (red paths in Fig. 2.18). The ratio of the average path to the length of the medium in the transport direction is defined as the tortuosity τ . As Eq. (2.50) indicates and as it is expected by increasing the tortuosity, the diffusive mass transfer will be more difficult. Since estimating the tortuosity of a porous layer is a difficult and complex task (direct investigation of porous structure is required), the Bruggeman correlation is often used for calculating the effective parameters, such as the diffusion coefficient. In this correlation, τ is assumed to be proportional to ∅−0.5 : Deff = D∅1.5 .

(2.51)

For gas diffusion layers (0.6 < ∅ < 0.8), Salem and Chilingarian [12] presented a useful and more accurate relation between the tortuosity and porosity: τ = −2.1472 + 5.2438∅.

(2.52)

95

96

Fuel Cell Modeling and Simulation

For a typical PEMFC GDL with ∅ = 0.7, the above relation yields τ ∼ = 1.5, and as a result, Eq. (2.50) becomes Deff =

D . 1.5

(2.53)

It is worth mentioning that the effective diffusion coefficient obtained from Eqs. (2.50), (2.51), and (2.53) would not be much different from each other, especially when the inherent uncertainties in the many other parameters are considered. After calculating the effective diffusion coefficients of reactant species, the mass fluxes can be calculated by either the Fickian relations (2.24)–(2.26) or the Maxwell–Stefan equations (2.37)–(2.39). Subsequently, the governing equations for the transport of reactants through the porous layers of electrodes can be solved. These governing equations in a 3D Cartesian coordinate system are presented by Eqs. (2.54)–(2.57), (2.59), (2.62), and (2.65) as follows: ∂ρ ∂(ρ u) ∂(ρ v) ∂(ρ w ) + + + = 0, ∂t ∂x ∂y ∂z

(2.54)

where (u, v, w ) is the volume-averaged velocity vector, also called the superficial velocity vector. In fact, the flow velocity vector may differ significantly from one point of the porous medium to another adjacent point; e.g., when a point locates in a pore, and its adjacent point locates in the solid matrix of the porous structure, the flow velocity in the pore will be nonzero, whereas it will be zero in the adjacent solid point. To handle this challenging issue, the volume-averaged properties such as the volume-averaged velocity vector are usually used. The transport of momentum in the porous medium is governed by ∂(ρ u) ∂(ρ u) ∂(ρ u) ∂(ρ u) +u +v +w ∂t ∂x ∂y ∂z        ∂ ∂ ∂P ∂u ∂u ∂u ∂ μe + μe + μe + Sx , =− + ρ gx + ∂x ∂x ∂x ∂y ∂y ∂z ∂z ∂(ρ v) ∂(ρ v) ∂(ρ v) ∂(ρ v) +u +v +w ∂t ∂x ∂y ∂z        ∂ ∂ ∂P ∂v ∂v ∂v ∂ μe + μe + μe + Sy , =− + ρ gy + ∂y ∂x ∂x ∂y ∂y ∂z ∂z ∂(ρ w ) ∂(ρ w ) ∂(ρ w ) ∂(ρ w ) +u +v +w ∂t ∂x ∂y ∂z        ∂ ∂ ∂P ∂w ∂w ∂w ∂ μe + μe + μe + Sz , =− + ρ gz + ∂z ∂x ∂x ∂y ∂y ∂z ∂z

(2.55)

(2.56)

(2.57)

PEMFCs

where μe is the effective dynamic viscosity, which is related to the bulk dynamic viscosity similarly to the relation of effective diffusivity and bulk diffusivity presented in Eqs. (2.51) and (2.53). The sink terms on the right-hand sides, stemming from Darcy’s law, are 

Sx = − 

Sy = − 

Sz = −

μ

Kxx μ

Kyx μ

Kzx

u+ u+ u+

μ

v+

Kxy μ

Kyy

v+

μ

Kzy

v+

μ

Kxz μ

Kyz μ

Kzz



w ,

(2.58)



w ,

(2.59)



(2.60)

w . ← →

In these sink terms, Kxx , Kxy , etc., are the permeability tensor elements, K . For an isotropic porous medium with permeability k, the above sink terms will be simplified as Sx =

μ

k

u,

Sy =

μ

k

v,

Sz =

μ

k

w.

(2.61)

Note that the permeability of a porous medium represents its tendency for the passing of gas flow through/in it; i.e., a larger permeability indicates a more facilitated gas flow through/in the porous medium and, consequently, a smaller sink against the momentum transport. A common unit for permeability is the darcy (also shown by d) represented by meters squared in SI units. Each darcy equals 9.869233 × 10−12 m2 . In a Darcy’s law modification, known as Forchheimer’s relation, a set of empirical terms such as bρ u |u|, bρ v |v|, bρ w |w | are added to the right-hand side of the sources Sx , Sy , and Sz , respectively, where b is an empirical constant. When the flow velocity magnitude is large, these added terms are more significant. As mentioned, in Eqs. (2.55)–(2.57), μe is the effective dynamic viscosity. Although there are a few correlations presented in the literature for the calculation of μe ([13], [14], [15]), in PEMFC models, it is usual to assume that μe = 0, i.e., the flow in/through porous layers is not affected by the second-order derivatives of velocity components. Neglecting the viscous dissipation rate and the compressibility effects, the transport of energy in/through PEMFC porous electrodes is governed by the equation γ

∂(ρf cp,f T ) ∂(ρf cp,f T ) ∂(ρf cp,f T ) ∂(ρf cp,f T ) +u +v +w ∂t ∂x ∂y ∂z           Ns ∂ ∂ ∂T ∂T ∂T ∂ Jx,i ∂ Jy,i ∂ Jz,i ∂ + + − , = keff keff keff hi + + ∂x ∂x ∂y ∂y ∂z ∂z ∂x ∂y ∂z i=1

(2.62)

97

98

Fuel Cell Modeling and Simulation

where the subscript f in ρf cp,f denotes the fluid portion of the domain, and u, v, and w are the components of superficial velocity. A porous domain consists of two portions, the fluid portion (also known as the pore volume) and the solid portion (also known as the solid matrix or solid microstructure), which will be denoted by subscript s. In the above equation, γ is a fluid–solid coupling factor, and keff is the effective thermal conductivity. They are defined as γ=

∅ρf cp,f + (1 − ∅)ρs cs , ρf cp,f

keff = ∅kf + (1 − ∅)ks .

(2.63)

(2.64)

In Eq. (2.62), it is assumed that the effective conductivity is homogeneous and ← → → keff − ∇ T ) must be isotropic. For a nonhomogeneous and anisotropic thermal medium, ∇.( ← → used instead of the first term on the RHS of the above equation, where keff is the tensor of effective thermal conductivity. Additionally, in the mentioned equation, it is assumed that thermal equilibrium exists between the solid and fluid phases. If this assumption is not correct, then two distinct equations must be solved for the fluid and solid portions to explain the energy transport in/through the porous medium, proposing Tf and Ts . Neglecting the Soret effects and assuming constant diffusivities, the conservation equations for the species mass can be written as  2  ∂ρi ∂(ρi u) ∂(ρi v) ∂(ρi w ) ∂ ρi ∂ 2 ρi ∂ 2 ρi eff , + + + = D i ,m + 2 + ∂t ∂x ∂y ∂z ∂ x2 ∂y ∂ z2

i = 1, . . . , Ns − 1. (2.65)

There are two differences between these equations and Eqs. (2.28), which were previously mentioned for the species transport through the GC. The first one is the fact that here the velocity components are superficial here, and the second one is the fact that here the diffusion coefficients are effective types. These effective diffusion coefficients can be calculated by the presented correlations for diffusion in porous media, such as Eqs. (2.50), (2.51), and (2.53). For MPL and CL, whose pores are relatively small, and, consequently, wall interactions are significant, the Knudsen diffusion may also play a role. When the characteristic length of a gas container (such as a pore volume) is very tiny, or the gas is too dilute, the Knudsen effects will be signified. More specifically, when Kn>10 (Kn is the Knudsen number and equals the ratio of the mean free path of the gas to the characteristic length of the container), the Knudsen diffusion regime dominates. However, when 0.01 4 (Fig. 2.19), the average value of λc and λa (which equals 9.4705+2 4.8905 = 7.1805) can be placed in DH2O,Nafion , which is the numerator of the above integrand. By this substitution, after integrating,  λx = 8.8C1 + 8.8C2 exp



0.1136 ∗ 4.8453 × 10−5 x . 1.3507 × 10−10

Assuming that the anode locates at x = 0, λx=0 = 4.8905 and λx=51×10−6 = 9.4705. Therefore the two constants in the above equation will be C1 = 0.4813,

C2 = 0.07446,

and, as a result,  λx = 4.2354 + 0.6552 exp 4.0751 × 104 x .

Now ASR of the membrane can be calculated from Eq. (2.8) and Eqs. (2.3)–(2.4): 

tm

ASRm = Am Rm = 0



51×10−6

ASRm = 0

dx = σ (x)



 0

tm

dx

 1 − (0.5193λx − 0.326) exp 1268( 303 

dx







1 358.15 )



→ 

1 − 1 ) 0.5193(4.2354 + 0.6552 exp 4.0751 × 104 x ) − 0.326 exp 1268( 303 358.15

= 9.2734 × 10−6  m2 = 0.092734  cm2 .

Therefore the ohmic voltage loss due to the proton transport through membrane is   ηohmic = jASRm = 0.85 × 104 A m−2 × 9.2734 × 10−6  m2 = 0.07882 V.

The presented method in this section for determining water content across the membrane by solving Eq. (2.77) is straightforward; however, there is a more advanced method, which can lead to more detailed and more robust results. In this method the following governing equation for the dissolved phase of water in the electrolyte material (which practically exists in membrane as well as CLs) is solved:    ρdry,Nafion cdrag  ∂ ∅MWH2O λ + ∇. im MWH2O ∂t MWNafion F  = ∇. MWH2O DH2O,Nafion ∇λ + Sλ + Sgd + Sld ,

(2.78)

where im is the protonic current density (which can be calculated by the electric potential field in the electrolyte phase via im = −σ ∇∅el , where σ is the conductivity of

PEMFCs

electrolyte), Sλ is the water generation rate according to the cathode CL reaction (i.e., this source term is zero for the membrane zone), and the two source terms Sgd and Sld on the RHS of Eq. (2.78) represent the rate of water mass change from the gas phase to the dissolved phase and the rate of water mass change from the liquid phase to the dissolved phase, respectively. By the dissolved phase we mean the water phase when it is uptaken by the electrolyte membrane material, such as Nafion. These two source terms can be calculated from Sgd = γgd (1 − sα )

Sld = γld sα

MWH2O ρdry,Nafion (λeq − λ), MWNafion

(2.79)

MWH2O ρdry,Nafion  λeq − λ , MWNafion

(2.80)

where γgd and γdl are the gas-dissolved and liquid-dissolved mass exchange rate constants, respectively, s is the saturation (which will be further explained in part e of this section), α is a user-defined parameter [18], and λeq is the equilibrium water content, which can be computed by 

λeq = 0.3 + 6aw 1 − tanh (aw − 0.5) + 0.69(λaw =1 − 3.52)aw0.5 + s(λs=1 − λaw =1 ),









aw − 0.89 1 + tanh 0.23 (2.81)

where aw is the water activity. Note that the above equation must be solved together with the governing equations for other water phases in the porous layers, as those presented in part e of this section. d) Electron transport in the cell When electrons are produced in the anode TPBs of a PEMFC, they must pass a long path to reach the cathode TPBs. A portion of this long path, which belongs to the cell domain, consists of the solid matrix of porous layers of electrodes (i.e., CL, MPL, and GDL) and the ribs of the bipolar plates at the two cathode and anode sides. To calculate the resistance against the transfer of electron in this portion of the path, the electrical resistance of different sections must be added together as a set of resistances connected in series. However, the contact resistances between these sections must also be taken into account. These contact resistances can be considerable, especially for a poorly assembled fuel cell or an aged one with ribs coated with some oxidized materials. Additionally, these contact resistances can be varied by the change of compression level during the stacking of the cells. Although the above-explained path for electron transport seems long, the path resistances are significantly smaller than the ionic resistance of the membrane. For instance,

105

106

Fuel Cell Modeling and Simulation

the ionic area-specific resistance of a PEMFC membrane is about 0.01–0.1  cm2 , whereas this parameter for a carbon paper GDL is about 0.001–0.01  cm2 , and for a bipolar plate, it is about 10−3 –10−5  cm2 ! Therefore the ohmic voltage loss due to the electron transport would be negligible, and hence in numerical modeling and simulations, this loss is often neglected. In rare cases where the voltage loss due to the electron transport is to be calculated and modeled, the total area-specific resistance against the electron transport from the first bipolar plate to the last bipolar plate in a PEMFC stack must be calculated. For example, if a stack consists of 100 cells, then the total electronic area-specific resistance of the stack would be approximately ASRelectronic,total = 100 × ASRBP + 2 × 100 × ASRBP −GDLContact + 2 × 100 × ASRGDL + 2 × 100 × ASRGDL−CLContact + 2 × 100 × ASRCL ∼ = 100 × 10−4 + 2 × 100 × 0.005 + 2 × 100 × 0.001 + 2 × 100 × 0.0005 + 2 × 100 × 0.0001 = 1.33  cm2 .

(2.82)

If the operating current density is 0.75 A cm−2 , the electronic ohmic loss in the mentioned stack will be ηohmic,electron = jASRm = 0.75 × 1.33 ∼ = 1 V.

(2.83)

This 1 V in comparison to the stack voltage, which is about 100 V, is only about 1%. e) Single-phase and multiphase water transport through porous layers of electrodes A severe potential challenge for low-temperature PEMFCs is the electrode flooding, which can lead to poor performance and even cell shutting down. Therefore in lowtemperature PEMFCs, it is always tried to manage the water in the form of vapor. In a well-managed low-temperature PEMFC system and in all high-temperature PEMFC systems, there is no liquid water, and therefore the water transport as the main product of cathode electrode would be a single-phase transport. In this section, we present the single-phase water transport in the form of vapor and afterward discuss the multiphase water transport through the porous layers. e1) Single-phase water transport phenomenon When water vapor is generated in the cathode TPBs, it must diffuse through CL (MPL) and GDL to reach the GC. However, to reach the pore volume of CL, it may be needed to diffuse through the Nafion film covering the TPBs. The diffusion coefficient of water through Nafion 1100-EW is a function of water content λ, and for λ > 4, it

PEMFCs

can be estimated by [19] 





1 1 − 303 T  × 2.563 − 0.33λ + 0.0264λ2 − 0.000671λ3 × 10−10 .

DH2O,Nafion = exp 2416

(2.84)

Note that when water diffuses through Nafion in a PEMFC operating under normal conditions, we recognize it as dissolved phase, which must be treated differently relative to the other two water phases in the GC and porous layers (vapor and liquid). The water transport through these electrolyte films is governed by the water transport equation (2.77), previously derived for the membrane. For the single-phase transport of produced water through electrodes, the governing equations are exactly the same as the governing equations previously presented for the reactant gas transport through the PEMFC porous layers, and hence we will not discuss it further here. e2) Multiphase water transport phenomenon1 Before discussing the multiphase water transport phenomenon through porous layers (GDL, MPL, and CL), let us consider a simpler case, water in a thin vertical tube. If we insert a thin glass tube into a water glass, we can see the water rise up. The reason for this fact is the hydrophilic property of the inner surface of a tube. This hydrophilic property stems from the surface tension between the liquid water molecules and the surface. The balance of forces acting on the water column, depicted in Fig. 2.20, results in h=

2σ cos θ , ρ gr

(2.85)

where σ is the surface tension from the inner surface of the tube to the water/air interface, θ is the contact angle, and r is the inner radius of the tube. Eq. (2.85) demonstrates that the amount of water rise up (h, also called capillary rise) depends on the strength of surface tension and also of the inverse of tube inner radius. Since the water wets the surface, here θ < 90◦ . If instead of water, a nonwetting fluid such as mercury was used, θ would be greater than 90◦ , and hence the free surface of nonwetting fluid would come down. In this case the inner surface of the glass tube is recognized as a hydrophobic surface for the fluid. Now we consider a porous medium through which the multiphase flow consists of gas and liquid water transfers. If the porous medium is hydrophilic, then the microsurfaces of its solid matrix tend to adhere more tightly to the liquid phase in comparison 1 As stated by Mench [7], the multiphase transport through porous media is not primarily developed for

fuel cells, but for the porous oil reservoirs and packed soil beds in oil engineering and civil engineering. In this section, we try to express the required science for fuel cell modeling without stating unrelated terminology and complex backgrounds in the oil engineering and civil engineering.

107

108

Fuel Cell Modeling and Simulation

Figure 2.20 Force balance for the column of water.

to the gas phase. On the other hand, a hydrophobic porous medium consists of microsurfaces, which tend to adhere to the gas phase more tightly, and hence, when the porous medium is hydrophobic, the water can be ejected more easily by a draining gas stream. Therefore in PEMFCs it is tried to implement hydrophobic treatment (such as the previously described PTFE treatment) for the porous layers of electrodes. Wettability of a porous medium is defined as the medium tendency to be wetted by the liquid water; i.e., a porous medium with a smaller contact angle and larger surface tension has a higher wettability, or a medium with more hydrophilic microsurfaces has a higher wettability. Another parameter related to the multiphase flow through porous media is the saturation. The saturation is defined as the fraction of pore volume occupied by liquid water (in some literature, the saturation is defined as the fraction of pore volume occupied by the nonwetting fluid; however, we do not use this definition in this book). The higher the saturation of a porous medium, the less void the volume available for gas transport. Hence the effective porosity can be defined as ∅eff = ∅(1 − s).

(2.86)

Another essential parameter for understanding the multiphase flow through a porous medium is the irreducible liquid saturation sirr ,l , which is the immobile part of saturating water. This part of liquid water is trapped in the isolated pores of the pore structure, and it cannot be removed by the inertial force, even with the entrance of a high-speed gas stream.

PEMFCs

The permeability (more specifically, the absolute permeability), which was previously used to relate the gas flow rate and pressure drop across a porous medium in single-phase flow, can be also used in multiphase flow by introducing the relative permeabilities. The absolute permeability is an intrinsic function of porous medium stemming from pore microstructure, not from flowing fluid and its phases. Nevertheless, the relative permeabilities are strong functions of saturation. To estimate the absolute permeability, the following Carman–Kozeny correlation can be used, especially when empirical data are not available: kabs =

r 2 ∅3 , 18τ (1 − ∅)2

(2.87)

where r is the average radius of pores, and τ is the porous medium tortuosity (Fig. 2.18). The relative permeability kr ,i of phase i is the ratio of the actual permeability ki of that phase at a given saturation to the intrinsic absolute permeability of the porous medium, i.e., kr ,i =

ki , kabs

(2.88)

where ki can be calculated using flow measurements and the relation Qi = −

ki A P . μi L

(2.89)

If the porous medium is dry (s = 0, or single-phase flow exists), then ki = kabs , and hence kr ,i = 1. Although various correlations for relative permeability are presented in the literature, for a hydrophobic porous medium, they all show the typical relation presented in Fig. 2.21 for the relative permeability of liquid water and gas vs. saturation (i.e., the liquid water is the nonwetting phase, and the gas phase is the wetting phase). Inspecting the figure reveals that increasing the saturation from sirr ,l will lead to the decreasing of the gas phase relative permeability and increasing of the liquid phase relative permeability. However, this increase is at first modest. This is due to the fact that at such saturations the liquid water fills only a small number of pores, and a large number of void pores still exist for the transport of gas phase. Therefore saturation has not a considerable effect on the liquid flow. On the other hand, at large saturations, a small increase of saturation leads to a significant increase of relative permeability of liquid phase. This is due to the fact that when liquid water occupies almost all the pores, the gas phase permeation is diminished, and the liquid phase permeation dominates. In the modeling and simulation of PEMFCs, the following correlation can be used (assuming that the liquid phase is nonwetting) [7]: kr ,liquid = s2.5 .

(2.90)

109

110

Fuel Cell Modeling and Simulation

Figure 2.21 Typical behavior of relative permeabilities vs saturation.

Figure 2.22 Liquid–gas interface and its curvatures and radiuses.

The saturation distribution through a porous medium is a function of capillary pressure, which is defined as the difference of pressures at the nonwetting and wetting sides (Pc = Pnw − Pw ). To have a better sense, again consider Fig. 2.20. The pressure on the gas/liquid interface and at the gas side is 1 atm, whereas on the interface and at the liquid side, it is 1 − ρ gh (according to Pascal’s law). The pressure difference, which is known as capillary pressure, is ρ gh, which equals the sensed pressure due to the weight of water column. According to Eq. (2.85), this capillary pressure can be written as 2σ cos θ/r. When two adjacent spherical particles (such as carbon particles) are wetted by the water as depicted in Fig. 2.22, the capillary pressure can be calculated by the following Laplace equation: 



1 1 Pc = Pnw − Pw = σ  +  . r r

(2.91)

For PEMFC applications, the porous medium is usually considered as a set of thin capillary tubes. Therefore the capillary pressure is stated as 2σ cos θ/r ∗ , where r ∗ is the average radius of pores. The liquid water surface tension in the air/liquid water multiphase flow is a function of temperature such as

PEMFCs

σ = −1.78 × 10−4 T + 0.1247,

(2.92)

where T is in K, and σ is in N m−1 . The capillary pressure is a positive variable, which is proportional to the inverse of average pore size. If the porous medium is hydrophilic (such as untreated GDL), at the interface of two phases the pressure of air as the nonwetting phase will be larger than the pressure of liquid water as the wetting phase. On the other hand, for a hydrophobic porous medium (such as a well PTFE-treated GDL), the pressure of liquid water as the nonwetting phase will be larger than the pressure of air as the wetting phase. In a PEMFC the average pore radius of CL is smaller than GDL, and both of them are usually hydrophobic. Hence the capillary pressure in CL is larger than GDL, and at the interface of two phases the liquid pressure is larger than the air pressure. Besides, since the air pressure in CL is almost the same as the air pressure in the GDL, the larger capillary pressure of CL means that the water liquid pressure in the CL is larger than the water liquid pressure in the GDL, and hence the liquid water in the CL will be transported to GDL by the pressure difference. On the other hand, if the CL and GDL were hydrophilic (which is not the case in the reality), the liquid water transport would be from GDL to CL. Since the cathode CL is a source of liquid water generation and usually water saturation is higher in the CL, the water transport from CL to the GDL (and after that to the GCs by the inertial force of air stream in the GC) is preferable. Therefore hydrophobic porous layers are employed in PEMFCs. Although the capillary force, stemming from capillary pressure and surface tension effects, is a serious driving force in the porous medium of PEMFCs, there are other possibly affecting driving forces (such as gravity, viscous, and inertia forces), which make analyzing the transport of liquid water a bit complex. To analyze and compare the magnitude of these driving forces, the following dimensionless numbers can be used. The Reynolds number, capillary number, bond number, and Webber number for liquid phase (as the nonwetting phase) can be estimated for CL with an average pore diameter of 0.05–0.1 µm as follows [20]: Re =

ρl ul D Inertia Force = ∼ 10−4 , μl Viscous Force

(2.93)

Ca =

μl ul Viscous Force ∼ 10−6 , = σ Capillary Force

(2.94)

Bo =

We =

g(ρl − ρa )D2 σ

=

Gravity Force ∼ 10−10 , Capillary Force

ρl u2l D Inertia Force = Re.Ca ∼ 10−10 . = σ Capillary Force

(2.95)

(2.96)

111

112

Fuel Cell Modeling and Simulation

Figure 2.23 Phase diagram and the nonwetting phase displacement regimes [21,22].

The subscripts l and a denote to the liquid and air phases, respectively. These numbers represent that for the liquid phase in the CL, Capillary Force > Viscous Force > Inertia and Gravity Forces.

(2.97)

Although the values of these dimensionless numbers for GDL with the average pore diameter of 20–30 µm are relatively different, the order of the driving forces is the same as that expressed in the above inequality. The phase diagram presented in Fig. 2.23 can be used to have a better understanding of the liquid water transport regime in a PEMFC. In this figure, M is the viscosity ratio, M = μl /μa . As it is marked in this figure, the liquid water transport through PEMFC is performed via capillary fingering. In general, for a porous medium, when the wetting phase displaces the nonwetting phase, the process is known as imbibition, and if the nonwetting phase displaces the wetting phase, the process is known as drainage. In the case of PEMFCs with hydrophilic porous layers, imbibition is the water uptake process, and drainage is the water release process. Fig. 2.23 demonstrates that when water is drained from the PEMFC (which is a desired phenomenon for recovering of a flooded electrode) by the capillary force, this drainage process is performed in a fingering pattern, i.e., there is no flat and stable front for the liquid phase. To have a quantitative representation of capillary pressure as a function of porous medium properties such as porosity, saturation, and permeability, we can employ the

PEMFCs

Leverett function approach. In this way, the capillary pressure can be obtained by Pc = σ

  12 ∅

k

Jm ( s) ,

(2.98)

where Jm (s) is the Leverett function derived experimentally for the drainage process in PEMFCs with hydrophobic porous media by Kumbur et al. [7]: 



Jm (s) = YPTFE 4.69 − 15.2 × YPTFE − 4.06s2 + 14.3s3 + 0.0561 ln s

for s < 0.5. (2.99)

In this equation, YPTFE is the mass fraction of PTFE in the solid matrix of the porous layer, which plays the role of contact angle in this approach. For example, for a GDL with 15%wt PTFE, YPTFE = 0.15. Note that Eq. (2.70) is valid for 0 < YPTFE < 0.2.  1 In Eq. (2.98), ∅k 2 represents the inverse of average pore diameter. The temperature and compression can also affect the capillary pressure, which is not incorporated in Eq. (2.98). Example 2.4. Consider a hydrophobic GDL with 200-µm thickness in a PEMFC cathode operating at 80 °C with 0.8 A cm−2 current density. Assume that in the adjacent CL, water is generated in only the liquid phase and the only water flux that comes into the GDL is this generated water. Suppose the porous GDL properties are ∅ = 80%,

k = 10−12 m2 ,

PTFE content = 15%wt.

By the aid of Darcy’s law and with the Leverett function approach, find the local position in the GDL (distance from in through-plane direction), where the water saturation is 0.25. Solution. Known: The operating condition of the cell (T = 80 °C and j = 0.8 A cm−2 ), the GDL thickness (200 µm), porosity (80%), permeability (10−12 m2 ), and PTFE content (15%wt). Find: the locations for which s = 0.25. Schematic:

113

114

Fuel Cell Modeling and Simulation

Assumptions: The liquid water flow through GDL is steady, incompressible, and laminar. The flow is one-dimensional along the through-plane direction. The air pressure in the pore volume of GDL is constant. The water transport from CL to the membrane is zero. Properties: At 80 °C the liquid water density and viscosity are 970 kg m−3 and 3.545 × 10−4 kg m−1 s−1 . According to Eq. (2.92), at this temperature the water/air surface tension is

1. 2. 3. 4.

σ = −1.78 × 10−4 T + 0.1247 = −1.78 × 10−4 × (273.15 + 80) + 0.1247 = 0.0618 N m−1 .

Analysis: Since the generated water is in the liquid phase, and 1D flow exists in the domain, the water generation rate equals the rate of liquid water passing through the GDL due to the conservation of mass for the illustrated control volume in the schematic. Therefore, according to the cell current density, the mass flux of water through GDL will be j = ρl ul → 4F (18.02 × 10−3 kg/mol) (0.8 × 104 A/m2 ) MH2O j ul = = × = 3.85 × 10−7 m s−1 . ρl 4F 970 (4 × 96485 C/mol)

JH2O,GDL = MH2O

According to Darcy’s law (2.89), the liquid velocity corresponds to the following pressure difference: ul = −

kl μl

∇ Pl = −

kr ,l kabs μl

∇ Pl .

By Eq. (2.90), kr ,l = s2.5 ; hence the above relation can be rewritten as  ∇ Pl = −

ul μl =− kabs s2.5

−5

3.85 × 10

s



m



 × 3.545 × 10−4

10−12 m2 s2.5

kg ms

 = −1.3648 × 104 s−2.5 .

Since Pc = Pnw − Pw and the GDL is hydrophobic, Pc = Pl − Pa . Therefore, by the 3rd assumption, ∇ Pc = ∇ Pl − 0 = ∇ Pl . Therefore ∇ Pc = −1.3648 × 104 s−2.5 .

(*)

PEMFCs

Now we employ the Leverett function approach to find another relation between ∇ Pc and saturation. Combining Eqs. (2.98) and (2.99) yields 

Pc = σ

∅

kabs

 12 





YPTFE 4.69 − 15.2 × YPTFE − 4.06s2 + 14.3s3 + 0.0561 ln s

→

1

 0.8 2 Pc = 0.0618 0.15 4.69 − 15.2 × 0.15 − 4.06s2 + 14.3s3 + 0.0561 ln s 10−12   dPc ds 0.0561 ds 4 2 5 3 ∇ Pc = = −3.366 × 10 s + 1.186 × 10 s + ds dx s dx   4 3 5 4 −3.366 × 10 s + 1.186 × 10 s + 0.0561 ds = . s dx

→

Since the left-hand side of this equation and the left-hand side of the equation marked by (*) are the same, we can build the following ordinary differential equation: 



 s ds × −1.3648 × 104 s−2.5 = 4 3 5 4 dx −3.366 × 10 s + 1.186 × 10 s + 0.0561  2.466s4.5 − 8.690s5.5 − 4.11 × 10−6 s1.5 ds = dx → 

s0





2.466s4.5 − 8.690s5.5 − 4.11 × 10−6 s1.5 ds =

0



x0

dx

→

→

0

x0 = 0.4484s05.5 − 1.337s06.5 − 1.644 × 10−6 s02.5 . For s0 = 0.25 → x0 = 5.57 × 10−5 m = 55.7 µm. There are other relations than Eq. (2.99) for the Leverett function presented in the literature for the imbibition and drainage processes. In these relations the imbibition and drainage processes for a porous medium in terms of capillary pressure vs. saturation are typically as the curves depicted in Fig. 2.24. SNW and SW are the nonwetting and wetting phase saturations, which are volumetric fractions of nonwetting and wetting phases in the pore volume, respectively. In this figure, SNW ,0 and SW ,0 denote the residual saturations of the nonwetting and wetting phases, respectively. In fact, SNW ,0 limits the maximum wetting saturation achievable by imbibition. Point A on this figure represents the breakthrough pressure, which is the minimum required pressure for initiating the drainage process, the displacement of the wetting phase by the nonwetting phase. Note that here we assume that water and air are immiscible phases. However, in reality, the situation may be a bit different, especially, when the water vapor pressure is considerably different from the saturation pressure. To analyze the multiphase transport of products in the porous layers of PEMFCs, as the first step, the saturation distribution must be resolved. To that end, the following

115

116

Fuel Cell Modeling and Simulation

Figure 2.24 Imbibition and drainage processes for a porous medium.

two coupled equations must be solved to obtain saturation(s) and liquid water (pressure) distribution in the porous layer [18]:   ∂ ρl kabs kr ,l ∇ Pl + Sgl + Sdl , (∅ρl s) = ∇. ∂t μl 

Pc = (Pl − P ) = σ |cos θ |

∅

kabs

(2.100)

 12

J ( s) .

(2.101)

In these relations the relative permeability of liquid water, kr ,l is a function of s, such as that previously presented in Eq. (2.90), P is the gas phase pressure, and J (s) is a Leverett function, J (s) = 1.417s − 2.120s2 + 1.263s3 . The two source terms of Sgl and Sdl on the RHS of Eq. (2.100) represent the rate of water mass change from the gas phase to the liquid phase and the rate of water mass change from the dissolved phase to the liquid phase, respectively. By dissolved phase we mean the water phase when it is uptaken by the electrolyte membrane material such as Nafion. These two source terms can be calculated from ⎧   P −Psat ⎪ H2O ⎨ γe ∅sDgl MW Pwv ≤ Psat , RT P ln P −Pwv , Sgl =   ⎪ ⎩ γc ∅ (1 − s) Dgl MH2O P ln P −Psat , Pwv > Psat , RT P −Pwv

(2.102)

PEMFCs

Sdl = γdl Sα

MH2O ρdry,Nafion  λ − λeq . MNafion

(2.103)

In the first source term, γe and γc are the evaporation rate and condensation rate coefficients, respectively, Pwv is the water vapor partial pressure, and ⎧  T 2.334  105  ⎪ ⎨ 3.65 × 10−5 343 for cathode, P Dgl =   ⎪ 1.79 × 10−4  T 2.334 105 ⎩ for anode. 343 P

(2.104)

In the second source term, γdl is the dissolved-liquid mass exchange rate constant, MNafion is the equivalent molar mass of Nafion, α is a user defined parameter [18], and λeq is the equilibrium water content defined by Eq. (2.81). Note that the saturation can have a noncontinuous distribution (such as at the interface of two porous layers), whereas the liquid pressure is always continuous. Once the saturation distribution is obtained, the gas-phase flow through the porous layers can be determined via the previously mentioned governing equations for the reactant gas flow in/through porous layers in part b of the current section. However, the available void space for the transport of gas and its species is here more restricted by the liquid water; hence, instead of the porosity, the effective porosity ∅(1 − s) must be used in the equations and employed properties. The above-explained procedure for multiphase flow through porous layers can be summarized as a two-step procedure: first, finding the saturation distribution by Eq. (2.100), accompanied by Eq. (2.101), and then using the achieved saturation to obtain the effective porosity for simulation of a gas-flow through porous layers. This straightforward two-step procedure is employed in many PEMFC models. However, there are more detailed methods for simulation of a multiphase flow, which are more accurate and more computationally expensive. One of these methods is using the following equations of volume of fluid (VOF) of a multiphase model for a porous medium [23]: n    ∂  m˙ pq + ∅Sq , ∅αq ρq + ∇. ∅αq ρq Vq = ∅ ∂t p=1

(2.105)

p=q

   ∂  q + ∇. ∅αq ρq V qV q ∅αq ρq V ∂t

⎛

 → μV f − ⎝∅2 αq2 q q + ∅3 αq3 = −∅αq ∇ Pq + ∇. ∅← τ q + ∅αq ρq B

kabs kr ,q

+∅

n  p=1

m˙ pq V pq + ∅F q ,





⎞

⎠

C2 ρq V q V q

2

(2.106)

117

118

Fuel Cell Modeling and Simulation

  ∂   q hq αq ∅ρq hq + (1 − ∅) ρs hs + ∇. ∅αq ρq V ∂t   ∂ Pq q − ∇. αq ∅kq + (1 − ∅) ks ∇ Tq + ∅Sh,q = −∅αq + ∅τ q : ∇ V ∂t n   ˙ pq + m +∅ Q ˙ pq hpq ,

(2.107)

p=1

which are equations of continuity, conservation of momentum, and conservation of energy for phase q, respectively. The description of variables in these equations is presented in Table 2.6. Table 2.6 Description of variables appeared in Eqs. (2.105)–(2.107). Variable

Description

∅

Porosity Volume fraction of phase q (which equals s for water liquid phase) Density of phase q Velocity of phase q Mass transfer rate from phase p to phase q Mass generation rate of phase q Pressure of phase q (note that the pressure differential between nonwetting phase and wetting phase is the capillary pressure) Shear stress tensor of phase q General body force (such as gravity) Viscosity vector of phase q Absolute permeability Relative permeability of phase q Constant in correction term due to inertia effects. For laminar flow, which is the case in PEMFCs, C2 ≈ 0 Relative velocity vector External body force acting on phase q Enthalpy of phase q Density of solid matrix in porous medium Enthalpy of solid matrix in porous medium Thermal conductivity of phase q Thermal conductivity of solid matrix in porous medium Temperature of phase q Heat source for phase q Heat transfer rate from phase p to phase q Enthalpy difference between phase p and q, (hpq = hq − hp )

αq ρq V q

m˙ pq Sq Pq ← → τ q f B μq

kabs kr ,q C2 V pq F q hq ρs

hs kq ks Tq Sh ,q Qpq hpq

PEMFCs

Figure 2.25 Multiphase flow regimes in a channel as a function of superficial velocities.

f) Single-phase and multiphase water transport through GCs The single-phase water transport through GCs is the same as the reactant gas transport through GCs, and hence we will not repeat it here. However, the situation for multiphase transport of produced water through GCs was not discussed before, and it will be explained in this section. This is especially important when the pressure drop through GCs is to be evaluated. When the water droplets are formed in GCs, the pressure drop through GCs will increase significantly. Generally, the multiphase flow through a channel can be categorized into different regimes such as bubble flow, slug flow, froth flow, and annular mist, according to the superficial gas and liquid velocities (Fig. 2.25). These two superficial velocities can be calculated by Vsg =

Qg , Ach

(2.108)

Vsl =

Ql , Ach

(2.109)

119

120

Fuel Cell Modeling and Simulation

where Ach is the channel cross-section area, Qg is the gas-phase volumetric flow rate, and Ql is the liquid-phase volumetric flow rate. The superficial velocity of a phase represents the velocity of that phase if it completely occupies the channel. Note that there is no distinct limit for switching from a multiphase regime to another one. Since in PEMFCs the liquid-phase superficial velocity is too low in comparison to the gas-phase superficial velocity, the multiphase flow regime is usually the annular mist regime. However, when the gas phase velocity is small (e.g., at the end part of anode electrode channels in a PEMFC operating with high current density), the slug flow may be also observed. To determine the multiphase flow field in the GCs, a multiphase model such as mixture model or VOF model can be employed. For example, in VOF model the following governing equations must be solved: n    ∂  q = αq ρq + ∇. αq ρq V m˙ pq + Sq , ∂t p=1

(2.110)

p=q

   ∂  q + ∇. αq ρq V qV q αq ρq V ∂t n     D f + pq + F q , F pq +m ˙ pq V = −αq ∇ Pq + ∇. ∅τ q + αq ρq B

(2.111)

p=1

  ∂  q hq αq ρq hq + ∇. αq ρq V ∂t n    ∂ Pq ← ˙ pq + m q − ∇. αq kq ∇ Tq + Sh,q + + → τ q : ∇V Q ˙ pq hpq . = −αq ∂t p=1

(2.112)

In these equations the only term that is not previously described in Table 2.6 is

D , which is the drag force acting on phase q from phase p and on their interface. F pq

When the medium is porous, this drag force is substituted by the viscous and inertia μ V resistance terms. The viscous resistance term and inertia resistance terms are ∅2 αq2 kabsq krq,q







C2 ρq V q V q

and ∅3 αq3 in the RHS of Eq. (2.106), respectively. More details about VOF 2 model are presented in Section 2.2.5. A simpler method to obtain the multiphase flow field in the GC is using the saturation approach, in which the following equation is solved to obtain the saturation:    ∂ l s = ∇. Dliq ∇ s , (ρl s) + ∇. ρl V ∂t

(2.113)

PEMFCs

where Dliq is the diffusivity of liquid water in GC, and V l is the superficial velocity of water. Adopting this equation enables us to solve the saturation equations in the entire fluidic domain of the cell; if we do not adopt this equation, then the saturation at the interface of GC and GDL must be known to solve Eq. (2.100) for obtaining the saturation in the porous electrodes, which is a challenging task. g) Heat transport in the cell The generated heat in TPBs of PEMFC has two different fluidic and solid paths to be transported throughout a PEMFC. A part of this heat is transported by the products from the TPBs (fluidic path), and the remaining part is transported by the solid matrix of the porous layers and current collectors. When the heat reaches the bipolar plates, it can be ejected from PEMFC by the coolant fluid of the thermal management subsystem, such as cooling air, water-glycol, or a refrigerant fluid. Among the three main mechanisms of heat transfer, radiation does not play a significant role in PEMFCs, because of their low operation temperature. Therefore it can be simply neglected. The conduction mechanism is the main heat transfer mechanism in the solid and porous zones of a PEMFC, whereas convection heat transfer is observed in the GCs and cooling channels of a cell (Fig. 2.16). However, in the regions of GDLs near the GCs, the situation is a bit complicated, and convection heat transfer can be observed. However, in many numerical models the heat transfer mechanism in the entire GDL is considered as the conduction. The conduction heat transfer through a solid material can be determined via the Fourier law, which states that − → 

← → q = − k ∇ T ,

− →

(2.114)

← →

where q is the heat flux vector, and k is the thermal conductivity tensor. When the solid material is an isotropic thermal conductor with thermal conductivity of k, the above relation can be rewritten as − → 

q = −k∇ T .

(2.115)

For porous materials with thermal equilibrium between the solid matrix and the fluid occupying the pores, the heat flux can be calculated by employing effective thermal conductivity − → 

q = −k∇ T ,

(2.116)

keff = ∅kf + (1 − ∅)ks .

(2.117)

where

121

122

Fuel Cell Modeling and Simulation

Table 2.7 Thermal conductivities of applied materials in PEMFCs at 25 °C [7]. Material

Thermal conductivity (W m−1 K−1 )

Hydrogen Oxygen Nitrogen Water vapor Liquid water Carbon Platinum Teflon Stainless steel Pure graphite

0.17 0.024 0.024 0.016 0.58 1.7 70 0.25 16 ∼2.0

In this relation the subscripts f and s denote to the fluid and solid portions of porous medium. When a multiphase fluid consisting of liquid water and gas exists in the pore volume of the porous medium, kf can be calculated by using the saturation as kf = skl + (1 − s)kg ,

(2.118)

where kl and kg denote the conductivities of liquid water and gas. The thermal conductivities of applied materials in PEMFCs at 25 °C are presented in Table 2.7. The thermal conductivity (also called the thermal conduction coefficient) can be varied by the temperature; however, the impact of temperature on the thermal conductivities of solids, liquids, and gases is different. The role of temperature for the temperatures near the operating range of PEMFCs (and not cryogenic temperatures, etc.) will be briefly and simply discussed in the following two paragraphs. Since in the metal solids the heat transfer conduction stems from the free electrons in the atomic structure of the metal, the temperature does not have a significant effect on the thermal conductivity of pure metals. However, for alloys such as stainless steel, which is a famous material for the production of PEMFC bipolar plates, the situation is a bit different, and the increase of temperature will lead to an almost linear increment of the alloy thermal conductivity. In the nonmetal solids the thermal conductivity is mainly due to the vibrations of atoms in the lattice. Since the increase of temperature intensifies these vibrations, the thermal conductivity will increase with the increase of temperature. In the gases, heat conduction stems from the elastic collision of gas molecules. Since the increase of temperature yields stronger collisions, the thermal conductivity of gases increases with the increase of temperature. The temperature dependency of nonmetal liquids is quite complex and is not well understood. The experimental investigations

PEMFCs

establish that under saturated conditions when the temperature increases, the thermal conductivity of refrigerating fluids such as Freon 12 and Ammonia will decrease, the thermal conductivity of glycerin will decrease, whereas the thermal conductivity of water presents a nonmonotonic behavior; it increases up to about 400 K, and after that it decreases [24]. When a liquid such as water or ethylene glycol is used as the coolant fluid in a cooling circuit, its larger thermal conductivity can benefit better cooling performance, and hence it is greatly desired. To increase the thermal conductivity of such fluids, a well-established strategy is adding nanoparticles to the fluid, and the resultant fluid is known as a nanofluid. The addition of nanoparticles to the fluid must be accompanied by uniform dispersing of particles through the base fluid, and any agglomeration of nanoparticles must be avoided. Example 2.5. By dispersing spherical nanoparticles in a base fluid we can obtain a nanofluid. If this dispersion is conducted uniformly, then the nanofluid thermal conductivity will be 

knf =



2kbf + kp + 2γ (kp − kbf ) kbf , 2kbf + kp − γ (kp − kbf )

(2.119)

where kp , kbf , and γ are the particle thermal conductivity, the base fluid thermal conductivity, and the volume fraction of added particles in the nanofluid, respectively. If Al2 O3 nanoparticles are added to water at T = 30 °C with γ = 0.02, how much the thermal conductivity will increase? If the nanofluid is applied in a medium containing copper foam with 70% porosity, what will be the thermal conductivity of medium? Solution. Known: The temperature of water as the base fluid (30 °C), the volume fraction of Al2 O3 nanoparticles in the nanofluid (0.02), the porosity of copper foam (70%), Find: the thermal conductivity of nanofluid relative to the thermal conductivity of base fluid and the thermal conductivity of the medium including copper foam and nanofluid relative to the base fluid. Assumptions: 1. The nanofluid thermal conductivity is a function of the thermal conductivities of base fluid and nanoparticles, as presented in Eq. (2.119). 2. The nanoparticles do not experience any agglomeration, neither in the nanofluid nor in the medium containing copper foam. Properties: At 30 °C the thermal conductivity of liquid water, Al2 O3 , and pure copper are 0.614 W m−1 K−1 , 18 W m−1 K−1 , and 365 W m−1 K−1 , respectively.

123

124

Fuel Cell Modeling and Simulation

Analysis: According to Eq. (2.119),





2kbf + kp + 2γ (kp − kbf ) kbf knf = 2kbf + kp − γ (kp − kbf )   2 × 0.614 + 18 + 2 × 0.02 × (18 − 0.614) = 0.614 2 × 0.614 + 18 − 0.02 × (18 − 0.614) = 1.055 × 0.614 = 0.648 W m−1 K−1 . This shows that adding 2% nanoparticles to the water will lead to 5.5% increase of fluid thermal conductivity. According to Eq. (2.117), keff = ∅knf + (1 − ∅) kcopper = 0.7 × 0.648 + (1 − 0.7) × 365 = 109.954 W m−1 K−1 . The effective thermal conductivity of the medium is more than 150 times greater than the thermal conductivity of the nanofluids. This establishes that applying porous material in a fluidic domain can result in an incredible increase of its thermal conductivity. Hence, when a significant increment of thermal conductivity is required, applying porous metal foams such as copper foam, aluminum foam, etc., can be a more attractive solution in comparison to adding nanoparticles to the fluid; however, note that this in turn can be more expensive with larger pressure drops. Thermal conductivity of Nafion is a strong function of water activity (which is equal to the relative humidity in the percentage form) and a weak function of temperature, as depicted in Fig. 2.26. The temperature variations of thermal conductivity of liquid water are also illustrated in this figure. As the figure shows, the thermal conductivity of Nafion is about from 25% to 50% of pure water, depending on its relative humidity. To calculate the amount of heat transfer rate by the convection mechanism from a surface with surface As , we use the convection heat transfer coefficient h: q = hAs (Ts − T∞ ),

(2.120)

where Ts is the surface temperature, and T∞ is the temperature of the faraway surrounding environment. If a fluid flows through ducts or channels such as GCs and the cooling channels of PEMFCs, then the mean temperature Tm of the fluid in that section of the channel must be used instead of T∞ for calculating the heat transfer rate at a channel section. To obtain the value of h, the correlations presented in the literature for calculating the Nusselt number Nu can be used. Once Nu is achieved, the convection heat transfer coefficient can be calculated by h=

kNu , L

(2.121)

PEMFCs

Figure 2.26 Thermal conductivity of Nafion 1100 EW vs. temperature at different water activities [25].

where k is the thermal conductivity of the fluid experiencing convection (and not the solid surface), and L is the length scale. For internal fluid flows such as the fluid flow through the GCs of PEMFCs, the hydraulic diameter of the channel is usually adopted as L. Nu correlations can be categorized based on the flow type (external or internal), flow regime (laminar or turbulent), development situation (developing or fully developed), convection type (forced or natural), number of phases (single-phase or multiphase), thermal boundary condition (constant temperature, constant heat flux, etc.), geometric parameters, etc. In PEMFCs the fluid flow through GCs and cooling channels (Fig. 2.16) is most probably laminar and fully developed due to the small hydraulic diameter and long length of channels. For such an internal flow, if the cross-section of the channel is a square, Nu will be 2.98 and 3.61 for constant wall temperature boundary condition and constant heat flux boundary condition, respectively. Since in a practical PEMFC the situation is between these two boundary conditions, an intermediate value 2.98 0, then ∅w = ∅W If Fw < 0, then ∅w = ∅P If Fe > 2De , then ∅e = ∅P If Fe < −2De , then ∅e = ∅E If−2De 2Dw , then ∅w = ∅W If Fw < −2Dw , then ∅w = ∅P If−2Dw 10De , then ∅e = ∅P

5 

Fe

aE = max 0, De 1 − 0.1 D +

e

If Fe < −10De , then ∅e = ∅E

 max {0, −Fe }  

Fw 5 aW = max 0, Dw 1 − 0.1 D +

w

If 0 < F < 10De, then  e

Fw ∅e = ∅P + 1 − 0.1 D w

5



max {0, Fw } aP = aE + aW + (Fe − Fw )

(∅P − ∅E )

If −10D < Fe < 0, then   e

Fw ∅e = ∅E + 1 + 0.1 D w

5

(∅E − ∅P )

and If Fw > 10Dw , then ∅w = ∅W If Fw < −10Dw , then ∅w = ∅P If 0 0, then ∅e = 68 ∅P + 38 ∅E − 18 ∅W If Fe < 0, then ∅e = 68 ∅E + 38 ∅P − 18 ∅EE

and If Fw > 0, then ∅w = 68 ∅W + 38 ∅P − 18 ∅WW If Fw < 0, then ∅w = 68 ∅P + 38 ∅W − 18 ∅E

Coefficients

If Fe > 0, then aE = De − 38 Fe & aEE = 0 If Fe < 0, then aE = De − 68 Fe − 18 Fw & aEE = 18 Fe If Fw > 0, then aW = Dw + 68 Fw + 18 Fe & aWW = − 18 Fw If Fw < 0, then aW = Dw + 38 Fw & aWW = 0 aP = aE + aW + aEE + aWW + (Fe − Fw )

Similarly, for cell 5, we can write      F F ∅4 = (2D − F ) + D + ∅5 . (2D − F ) ∅L + D +

2

(2.273)

2

In the above two equations, ∅0 and ∅L are known, and hence we can hypothetically consider that for cells 1 and 5, the following sources exist: S1 = Sp,1 ∅1 + Su,1 = − (2D + F ) ∅1 + [(2D + F ) ∅0 ]

(2.274)

S5 = Sp,5 ∅5 + Su,5 = − (2D − F ) ∅1 + [(2D − F ) ∅L ] .

(2.275)

and

Adding Eqs. (2.272) and (2.273) to the three equations of Eq. (2.270), we will have a system of five linear algebraic equations and five unknowns: ⎡

  D − F2 0 0 0 − D − F2 + (2D + F )   ⎢ F F D+ 2 2D D − 2  0 0 ⎢  ⎢ 0 D + F2 2D D − F2 0 ⎢ ⎢ F F ⎣ 0 0 D + 2  2D D − 2  0 0 0 D + F2 − (2D − F ) + D + F2 ⎡ ⎤ −(2D + F )∅0 ⎢ ⎥ 0 ⎢ ⎥ ⎢ ⎥ =⎢ 0 ⎥. ⎢ ⎥ ⎣ ⎦ 0 − (2D − F ) ∅L

⎤⎡ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎦⎣

∅1 ∅2 ∅3 ∅4 ∅5

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(2.276)

209

210

Fuel Cell Modeling and Simulation

Note that according to the central differencing scheme, in these three equations, aE s = D − F /2, aW s = D + F /2, and aP s = 0. The above system of equations can be easily solved by TDMA. If we a have a transient transport problem, then Eq. (2.261) must be also integrated over time as follows: 

 ρ∅dV

CV



=

t0 +t 

 



t0 +t



− CV



dSdt + nˆ . α ∇∅



CS

t0



t0 +t 

+

ρ∅dV t0

t0 +t 

t0

t0

CV

nˆ . (ρ∅ u) dSdt

CS

S∅ dVdt.

(2.277)

Therefore, in transient transport problems, time and space discretization is also required (also called grid or mesh generation). More specifically, after the generation of the grid, the above equation can be rewritten for the computation cell P in a 1D domain (Fig. 2.34), by incorporating F = ρ u and D = δαx and assuming that Ae = Aw = A, as follows:  (ρ∅P δ x)t0 +t − (ρ∅P δ x)t0 +  =

t0 +t



t0 +t

[Fe ∅e − Fw ∅w ] dt

t0

[[De (∅E − ∅P ) − Dw (∅P − ∅W )]] + δ x

t0



t0 +t t0

S∅,p dt.

(2.278)

Now the question is how to calculate the above three time integrals? In a more

general form, to calculate the integral of tt00 +t f (t)dt, we can use the following approximation: 

t0 +t

t0

f (t)dt ∼ = xf (t0 ) + (1 − x) f (t0 + t) .

(2.279)

This approximation can be used for making Eq. (2.278) further numerically solvable. However, if x = 1, then the obtained numerical equation will be explicit (i.e., the integrals are calculated based on the data at the previous time steps), whereas if x = 0, then the obtained numerical equation will be implicit (i.e., the integrals are calculated based on the data at the next time steps). Although explicit solution requires less computational cost, it has a lack of stability, especially for not enough small time steps. Hence all commercial software usually uses implicit schemes in transient problems. There are some semiimplicit schemes (such as the Crank–Nicolson scheme for x = 12 ) that have an interstitial feature.

PEMFCs

b) FVM for the numerical modeling of fluid flow Consider a fluid flow through a PEMFC GC. According to Eqs. (2.18)–(2.22), the governing equations for the steady-state flow are ∂(ρ u) ∂(ρ v) ∂(ρ w ) + + = 0, ∂x ∂y ∂z

(2.280)

       ∂P ∂ ∂(uρ u) ∂(vρ u) ∂(w ρ u) ∂u ∂u ∂u μ +μ +μ − + + = + Sx , ∂x ∂y ∂z ∂x ∂x ∂y ∂z ∂x

(2.281)        ∂P ∂ ∂(uρ v) ∂(vρ v) ∂(w ρ v) ∂v ∂v ∂v μ +μ +μ − + + = + Sy , ∂x ∂y ∂z ∂x ∂x ∂y ∂z ∂y

(2.282)        ∂P ∂ ∂(uρ w ) ∂(vρ w ) ∂(w ρ w ) ∂w ∂w ∂w μ +μ +μ − + + = + Sz , ∂x ∂y ∂z ∂x ∂x ∂y ∂z ∂z

(2.283) where Sx , Sy , and Sz are the momentum source terms, except the pressure gradient. Since the pressure gradient is the main source of momentum transport in many fluid flows, it requires special attention. If we know the pressure gradient, then finding the velocity field will be quite easy. It is only required to deal with u, v, and w in Eqs. (2.281)–(2.283) as the transported variable ∅ in Eq. (2.258). However, in most practical fluid flow problems, the pressure is also an unknown variable. If the flow is compressible (i.e., the density experiences local variations), then the continuity equation (2.280) can be used for finding the density field. By solving the energy conservation equation the temperature field can also be obtained, and, subsequently, the pressure field can be calculated based on the density and temperature fields via an equation of state. After that, the pressure gradients can be calculated and incorporated into Eqs. (2.281)–(2.283) to obtain the velocity field. However, in PEMFCs the fluid flow is incompressible, and hence the pressure is independent of density. The question is how to find the four unknown variables of p, u, v, and w from the four equations (2.280)–(2.283). This is not a simple task since these four equations are intrinsically coupled with each other via the velocity components and pressure (even the continuity equation is indirectly affected by the pressure); besides, the momentum conservation equations are nonlinear. To deal with these issues, an iterative procedure is usually used. The first iterative procedure was presented by Patankar and Spalding in 1972, known as SIMPLE (abbreviation of Semi-Implicit Method for Pressure-Coupled Equations). In

211

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Fuel Cell Modeling and Simulation

the first step of this method, a velocity field is guessed for the calculation of the convective flux per unit mass F. The pressure field and pressure gradients are also guessed. Now the discretized momentum conservation equations can be solved by the same method presented in Part a of this section (Fig. 2.35). After that, a pressure correction equation, stemming from the continuity equation, is solved to provide the required correction for improving the values of pressure initially guessed. The coefficients of this pressure correction equation are calculated by the obtained velocities from solving the mentioned discretized momentum conservation equations. These pressure corrections are also used to correct the values of the obtained velocities. After that, other transport equations (such as the energy equation or the species conservation equations) are also solved, and the convergence is checked. There are other iterative procedures for solving a fluid flow problem with velocity– pressure coupling, such as SIMPLER, SIMPLEC, and PISO. In SIMPLER (Fig. 2.36) the iterative loop starts by a preconditioning step, and this is the only difference between SIMPLE and SIMPLER. This preconditioning step contains the solution of the pressure equation (and not the pressure correction equation). Solving this pressure equation helps us to have a better guess for solving the discretized momentum equations, previously mentioned in the description of SIMPLE procedure. Although each SIMPLER iteration needs more computational cost in comparison to a SIMPLE iteration due to this extra preconditioning step, the overall computation cost of SIMPLER is often less than the computational cost of SIMPLE due to its considerably fewer required iterations for achieving the convergence. SIMPLEC can also be considered as an extension of SIMPLE. The only difference between SIMPLEC and SIMPLE is the fact that in solving the discretized momentum conservations equations, less significant terms are omitted to lessen the computational cost. In PISO procedure the mentioned pressure correction presented in the description of SIMPLE is performed two times at each iteration to speed up reaching the convergence. However, when in a numerical problem the adopted procedure cannot converge to a specific solution, the first technique to avoid numerical divergence is employing relaxation factors for the variables with improper convergence behavior (the convergence behavior can be achieved by analyzing the plot of residuals vs. iteration, provided by commercial software packages during the numerical solution process). For example, if in a numerical problem the component v of velocity does not have a proper convergence behavior, then after finding its value at an iteration (e.g., iteration n), the input value of it for the next iteration (iteration n + 1) will be adopted from the following relation: vn+1 = α vn + (1 − α)vn−1 ,

(2.284)

PEMFCs

Figure 2.35 The schematic of SIMPLE procedure.

where vn−1 is the value of v at the end of iteration n − 1, and 0 < α < 1 is the relaxation factor. In some severe numerical situations, a value as small as 0.001 for the relaxation factor may be chosen for running away from divergence. Now the question is which of these four famous procedures is more suitable? Unfortunately, this simple question has not a unique answer. In fact, many facts such as the degree of intrinsic coupling between velocity field and pressure field (or other scalar fields), the interpolation method (central, upwind, hybrid, QUICK, etc.), the amount of underrelaxation factors, etc., have an influencing role, and the answer of the men-

213

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Fuel Cell Modeling and Simulation

Figure 2.36 The schematic of SIMPLER procedure.

PEMFCs

tioned question differs case by case. Nevertheless, some general points are presented in the literature. For example, it is stated that although SIMPLE procedure is more straightforward, SIMPLER needs less computational cost, especially, obtaining the pressure field is a challenging task for SIMPLE procedure. When there is no scalar to be solved simultaneously (i.e., only fluid flow is to be solved), PISO procedure has the least computational cost among the four mentioned procedures. It is worth mentioning that in the transient form of procedures such as SIMPLE, the presented steady-state procedure in this section must be implemented at each time step; the results of each time step are fed as the initial guess for the next time step. However, there is a new source term in the pressure correction equation, which is a function of the time step; this source term causes the transient behavior of the fluid flow to be observable in the numerical results. More details about these procedures can be found in [110]. As the final point in this section, there are two types of staggered and colocated grids applicable in these four procedures. In the colocated grid the data of all variables are stored in the center of the computational cell, whereas in the staggered grid, only the pressure and other scalars are stored in the cell center; the velocity components are stored on the faces of computational cells. Dealing with colocated grids seems more straightforward and requiring less computational memory. However, from a historical point of view, the ordinary colocated grids can sometimes result in a false solution, e.g., for a fluid flow with highly oscillating pressure (see the checkerboard problem in [110]). This was the motivation for employing staggered grids for a period of time by computational researchers. However, the problem can be overcome by incorporating the contravariant system of coordinates for each computational cell in a colocated grid. Therefore most of current commercial CFD packages such as ANSYS Fluent and COMSOL use the colocated grid with the mentioned system of coordinates.

2.3.3 2D/3D modeling of a cell Several 2D/3D models based on FVM have been presented for a single PEM fuel cell in the literature until now, with different levels of complexity, accuracy, and reliability. However, all these models consist of the following transport submodels: a) Gas transport submodel b) Water transport submodel c) Proton transport submodel d) Electron transport submodel e) Heat transport submodel There is a coupling between all these transport submodels, which is the electrochemical reaction submodel, as the hearth of a PEM fuel cell. In this section, we will present more details about these submodels; however, beforehand, it is better to see Fig. 2.16 and Table 2.3.

215

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Fuel Cell Modeling and Simulation

a) Gas transport submodel As described in Section 2.1.2, the reactant and product gases (which usually are mixtures of hydrogen and water vapor in the anode and humid air in the cathode) must be exchanged between the gas channel and reactive sites (i.e., the anode and cathode TPBs). This fluidic path consists of both porous and nonporous parts. To model this gas transport, the governing equations presented in Parts a, b, e1, and f1 of Section 2.1.2 must be solved by FVM for the gas mixture. To that end, the velocity pressure coupling procedures such as SIMPLE must be used. Note that the governing equations for the porous parts (such as GDL and MPL) have additional source terms, as those on the RHS of Eqs. (2.55)–(2.57), and the velocity components are superficial ones in these governing equations. In addition, the mass transfer equations with effective diffusivities (2.65) must be solved for the species to determine the concentration of each species. In simple gas transport submodels, binary diffusion is considered, whereas in more advanced submodels, multicomponent diffusivities or even the Stefan–Maxwell equations are used. In some recent advanced submodels (such as that adopted in the PEMFC model of ANSYS Fluent commercial software package), the oxygen diffusion resistance through the Nafion and the liquid water surrounding each cathode Pt particle (which is a part of the path from gas inlet to the cathode TPB) is also taken into account. In isothermal PEMFC models the transport coefficients such as viscosity and diffusivity are determined for a specific temperature and do not experience time variation. However, in nonisothermal PEMFC models, these coefficients and all other properties are updated in each time step and at each computational cell, based on the temperature at that moment and location. b) Water transport submodel Water is the only species that can experience several phases, and it exists in all components of a PEMFC. As the simplest case (e.g., in an HT-PEMFC), water exists in at least two phases of vapor (anode and cathode electrodes) and dissolved (membrane); therefore it is treated as a separate submodel than the gas transport submodel. Although in HT-PEMFCs the liquid water and single-phase CFD models (for which the water in the electrodes experiences a single phase of vapor) are not representative, in lowtemperature PEMFCs the existence of liquid water is quite probable, especially at high current densities. Hence multiphase models must be incorporated. There are a few multiphase models, such as the mixture model and VOF model, that can be implemented in the FVM framework. The mixture model (which is sometimes denoted by M2 ) needs less computational cost, and hence it is more feasible for large PEM fuel cells and stacks. However, the VOF model has the advantage that it can capture the liquid–vapor interface, and this helps the researchers to better study the liquid water behavior in a cell (e.g., in a portion of GC).

PEMFCs

c) Proton transport submodel As mentioned in Section 2.1.2, Part c, the proton transport through the Nafion phase, which exists in the membrane and in CLs, is mainly conducted by the electrical potential gradients. However, the proton concentration gradients can also play a minor role. In most of proton transport submodels, the only driving force is the electrical potential gradients. i.e., it is assumed that the proton concentration is uniform (there is no concentration gradient). By this assumption the following equation must be solved:    σN ∇∅ N + iN ∇. = 0,

(2.285)

where σN is the proton conductivity through Nafion or any other membrane material,  ∅N is the electric potential of Nafion phase, and iN is the volumetric density of electric −2 current through the Nafion (in A m ). Since proton is generated in the anode CL, and it is consumed in the cathode CL, this current volumetric density must be positive in the anode CL and negative in the cathode CL. In addition, its value is proportional to the electrode current density via the  factor, which is the specific active surface area (in m2 m−3 ): 



iN =

ianode −icathode

for the anode CL, for the cathode CL,

(2.286)

where ianode and icathode can be calculated by applying the Butler–Volmer equation for the anode and cathode electrodes, respectively, as previously explained in Chapter 1. Note that according to the Laplace operator that appeared in Eq. (2.285), this equation is an elliptic partial differential equation, and its numerical intrinsic differs from the general transient transport equation for scalar φ , as we have previously seen in Section 2.3.2. However, this type of equation has a smooth behavior, and its solving is a more straightforward task. Its behavior is just like a steady energy equation for a conduction heat transfer problem with heat source. In fact, such an equation is a steady diffusion problem, which can be achieved by omitting the unsteady term from Eq. (2.258) and putting the velocity equal to zero (to remove the advection terms). The coefficients for the obtained discretized equation (the algebraic equation, previously represented by Eq. (2.267)) can be easily achieved by putting velocity components (i.e., Fs) equal to zero. d) Electron transport submodel In some PEMFC models, electron transport in the PEMFC is not considered, i.e., the solid phase electric potential in each electrode is assumed to be constant; here the solid phase consists of carbon black/Pt in CLs, carbon particles in MPL, carbon fibers in GDLs, and collectors (BPs). However, solving the following equation, which is similar to Eq. (2.285), is recommended for achieving a higher accuracy:    σSM ∇∅ SM + iSM = 0, ∇.

(2.287)

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Fuel Cell Modeling and Simulation

where σSM is the electron conductivity through the solid matrix of a porous layer such  as CL, ∅SM is the electric potential of solid phase, and iSM is the volumetric density of −2 electric current through the solid matrix (in A m ). An electron is generated in the anode CL and consumed in the cathode CL; however, since it is usual to determine the electric current direction based on the direction of positive electric charge displacement,  iSM has a negative value in the anode CL, a positive value in the cathode CL, and zero value in the other porous layers of electrodes, where no electron generation/consumption happens. This volumetric current density must be positive in the anode CL and negative in the cathode CL. Besides, its value in the CLs is proportional to the electrode current density via the specific active surface area :  

iSM =

−ianode icathode

for the anode CL, for the cathode CL.

(2.288)

Eqs. (2.287) and Eq. (2.285) must be solved to determine the proton and electron transport through the domain. However, these two equations are coupled to each other by their source terms, as is evident in Eqs. (2.286) and (2.288). These two equations play a considerable role in the simulation of a PEMFC. In fact, the obtained potential difference between electrolyte phase (Nafion) and solid phase (solid matrix) in each computational cell of a CL determines the activation overpotential in that cell:  η=

φSM − φN − Eanode φSM − φN − Ecathode

for the anode CL, for the cathode CL.

(2.289)

This activation overpotential affects the rate of electrochemical reaction (see the Butler–Volmer equation for an electrode), and the electrochemical reaction rate affects the mass sources in gas transport and water transport submodels, as well as the heat source in the heat transport submodel. Another required item for solving Eqs. (2.285) and (2.286) is determining the boundary conditions for solving these two equations. Consider the 3D computational domain previously presented in Fig. 2.16. Since there is no proton exchange from all exterior walls of this computational domain, ∂φ∂ nN = 0 on all exterior surfaces of this domain (here n represent the normal direction toward the outside of the domain). On the vertical exterior surface of this domain, ∂φ∂SM n = 0 is a reasonable boundary condition; however, on the horizontal top and bottom exterior surfaces of the domain, the electric current is exchanged, and hence the simple Newman boundary condition ∂φ∂SM n =0 cannot be implemented. Usually, there are two different boundary conditions applicable on these two faces: • If the cell voltage Vcell is known, then the condition φSM = 0 is used for the anode side (upper horizontal surface in Fig. 2.16) and φSM = Vcell for the cathode side (bottom horizontal surface in Fig. 2.16).

PEMFCs

•

icell If the cell current density icell is known, then ∂φ∂SM n = σBP for the cathode exterior i cell face and ∂φ∂SM n = − σBP for the anode exterior face are used.

e) Heat transport submodel In nonisothermal PEMFC models the energy equation is also solved. This governing equation was previously presented for fluidic zones in Section 2.1.2, Part a (Eq. (2.22) for nonporous fluidic zones) and Part b (Eq. (2.62) for the porous fluidic zones), accompanied by the source terms presented in Table 2.9 as Eqs. (2.126)–(2.132). The governing energy equation for the solid zones (which are the membrane and two current collectors, as presented in Fig. 2.16) can be simply achieved by setting the velocity components equal to zero in Eq. (2.22). This will lead to the following equation: ∂(ρ cp T ) = ∂t



      ∂ ∂ ∂T ∂T ∂T ∂ + + + S. k k k ∂x ∂x ∂y ∂y ∂z ∂z

(2.290)

To numerically solve this equation, the temperature must be treated as scalar ∅ in the FVM framework. Adopting this submodel will enable us to capture the temperature differences in a PEMFC (usually BPs are the coldest parts, and CLs, especially cathode CL, are the hottest layers), and this makes the design of TMS for the PEMFC more feasible. Besides, the transport properties such as viscosity, diffusivity, and conductivity, which depend intrinsically on the temperature, can be updated at each time step and for every computational cell. This will lead to a higher level of validity and reliability of numerical results.

2.3.4 Modeling of a stack In Section 2.3.1, we presented a 1D model for a single PEM fuel cell, by which the polarization curve and the mean concentrations of active species on the CLs vs. the current density of a cell can be determined. However, the model does not include the gas depletion effects, which is a key element to bridge from a 1D model of a cell to a 1D model of a stack. Fortunately, taking into account gas depletion effects is not difficult. It is only required to relate the reactant concentrations (more specifically, the reactant fluxes) at the inlet and the reactant concentrations at the outlet via the current density of the cell. For example, for the oxygen, C ,inlet C ,outlet nO − nO = 2 2

i . 4F

(2.291)

A similar relation can be written for the produced water vapor at the cathode: C ,outlet C ,inlet − nH = (γ + 1) nH 2O 2O

i . 2F

(2.292)

With the aid of the above relations, we can easily generalize a 1D cell model to a 1D stack model; e.g., if the cells are fed in series (Fig. 2.37), then the concentrations

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Fuel Cell Modeling and Simulation

Figure 2.37 Three main configurations for the cells feeding (fuel or air) in a stack: a) series, b) parallel Z-type, c) parallel U-type.

at the outlet of the first cell will be the same as the concentrations at the inlet of the second one. If the cells are fed in parallel, then the concentrations at the inlets of all cells will be the same. However, as depicted in Fig. 2.37, there are two types of parallel feeding, Z-type (in which fuel and air are fed from two opposite sides of the stack) and U-type (in which fuel and air are both fed from one side of the stack). In the Z-type, since the hydraulic resistance from the inlet to the outlet through all cells is the same, all the cells experience a similar inlet flow rate of fuel and air. However, in the U-type the inlet flow rate is not similar; the cells that are closer to the inlet will have a larger inlet flow rate. To determine the flow rate of fuel and air through the cells in the parallel or series configuration of cells in a stack, the notion of hydraulic equivalent circuit can be used. To simulate a PEMFC stack in 2D/3D, only more computational cost is required in comparison to the 2D/3D simulation of a single cell; therefore it is better to employ submodels with less computational cost in the 3D simulation of a PEMFC stack. The main note that we must be careful about here is that the gas depletion effects can play a significant role; hence the gas/water transport submodels must have sufficient accuracy

PEMFCs

to capture the species concentration changes along the gas channels on a BP, which are too long paths, and when we have several cells in the simulation process, these can be much more longer.

2.3.5 Modeling and control of PEMFC system Recall from Fig. 2.15 that a PEMFC system comprises several subsystems such as humidifier, gas supply, TMS, etc. However, the main subsystem is the stack, whose 1D model was previously presented. Adding the model of each of these subsystems makes it possible to build a more general model for the system. In the following, we present 1D modeling for these subsystems. a) Gas supply subsystem As mentioned in Part c of Section 2.1.1, the supply of hydrogen and air (or pure oxygen) for a PEM fuel cell is usually conducted via a fan (in atmospheric fuel cells with few cells) or a compressor (in high-pressure fuel cells or fuel cells with numerous cells). Different types of fans such as axial and centrifugal, as well as different types of compressors, such as piston, screw, and scroll, are different types of turbomachines, which acquires mechanical power and delivers hydraulic power in the form of fluid stream with a specific flow rate and pressure. In thermodynamics, for such turbomachines, the efficiency is defined as ηtm =

Wideal , Wact

(2.293)

where the ideal work (also called isentropic work) and the actual work can be written via the first law of thermodynamic as   2    U ˙ Wideal = ρ V in hout,ideal − hin +



 

U2 − 2 out 2 in  2    2     U U ˙ , = ρ V in cp Tout,ideal − Tin + − 2 out 2 in

  2    U ˙ Wact = ρ V in hout,act − hin +



(2.294)

 

U2 − 2 out 2 in  2    2     U U ˙ , = ρ V in cp Tout,act − Tin + − 2 out 2 in

(2.295)

where V˙ , h, and U denote the volumetric flow rate of fuel/air, enthalpy, and velocity magnitude, respectively, whereas the subscripts in and out denote the conditions at the   inlet and outlet of the turbomachine; therefore ρ V˙ in , which equals ρ V˙ out , is the

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Fuel Cell Modeling and Simulation

mass flow rate of the turbomachine. Note that the main parameter that differs between the RHS of Eq. (2.294) and RHS of Eq. (2.295) is the output temperature (enthalpy). On the other hand, for an isentropic compression work, we can write 





Tout,ideal Pout = Tin Pin

(γ −1)/γ

(2.296)

,

where γ is the ratio of the constant-speed specific heat to the constant-volume specific heat of the fluid. Combining the presented Eqs. (2.293)–(2.296), if we know the turbomachine efficiency (which is sometimes presented in a chart vs. pressure increment by some suppliers), then we can write   2    2    U U = ηtm Wideal − Wact = ρ V˙ in cp Tout,act − Tin + 2 out 2 in         2    Pout (γ −1)/γ U2 U ˙ , = ηtm ρ V in cp Tin − Tin + −

Pin

2

2

out

(2.297)

in

and hence the actual outlet temperature of the fluid will be 



Tout.act = 1 − ηtm + ηtm

Pout Pin

 γ γ−1 

Tin −

(1 − ηtm )

cp



U2 2



 −

ρU 2

2

out

  . (2.298) in

Sometimes, the outlet temperature is known, and the efficiency is the unknown parameter. In such cases the following equation can be used as the efficiency calculator of the model:   ηtm =









 

U2 U2 2 out − 2     in  U 2  U2 cp Tout,act − Tin + 2 out − 2 in

cp Tout,ideal − Tin +

.

(2.299)

b) Humidification subsystem In thermodynamics the specific humidity ω of air (which is a mixture of dry air and water vapor and is also the called absolute humidity or humidity ratio) is defined as the ratio of the water vapor mass to the dry air mass; the relative humidity of the air φ is defined as the ratio of the vapor pressure to the water saturation pressure at that temperature, Psat . These two humilities are related to each other by ω Pa  φ =  MW H2 O MWDryAir

Psat

=

ω Pa , 0.622Psat

(2.300)

where Pa is the partial pressure of dry air (the sum of dry air partial pressure and water vapor partial pressure is the total pressure of air). If we put φ = 1 into this equation,

PEMFCs

then the maximum achievable specific humidity in the equilibrium condition, which is usually called the saturation specific humidity, or ωs , can be obtained by  MW ωs =

H2 O

MWDryAir



Psat (2.301)

.

Pa

However, in PEMFCs the hydrogen stream is usually humidified, and the air stream is dry. Hence, similar to how we defined the specific humility for air, we can define this parameter for humidified hydrogen. More specifically, the saturation specific humidity for a humidified hydrogen (above which the saturation cannot be accessible under equilibrium conditions) will be as follows:  ωh,s =

MWH2 O MWDryH2



Psat

PH 2

=

8.937Psat , PH 2

(2.302)

where PH2 is the partial pressure of the dry hydrogen. Now consider a humidifier designed to humidify a dry hydrogen stream with the mass flow rate m˙ H2 ,in and known conditions PH2 ,in and TH2 ,in . If the humidifying process is performed completely and efficiently (i.e., the maximum specific humidity expressed in Eq. (2.302) is achieved, and no condensed water is generated), then what is the required water mass flow m˙ water for this humidifier (assume that the water condition Twater ,in is known)? Moreover, what will be the condition of the outputted humidified hydrogen from this system (assuming adiabatic boundaries for the system)? To answer the first question, we can use Eq. (2.302): m˙ water = ωh,s m˙ H2 ,in =

8.937Psat m˙ H2 ,in . PH2 ,in

(2.303)

To answer the second question, we can write the mass balance and energy balance for the humidifier as a control volume. More specifically, m˙ H2 &vapor ,out = m˙ H2 ,in + m˙ water ,

(2.304)

m˙ H2 &vapor ,out hH2 &vapor ,out = m˙ H2 ,in hH2 ,in + m˙ water hwater .

(2.305)

Since the outlet hydrogen contains water vapor and dry hydrogen, the above equation can be rewritten as 







m˙ H2 ,in cp,H2 Tout − TH2 ,in = m˙ water hwater − hvapor .

(2.306)

In this equation, since the humidified hydrogen is saturated, hvapor = hg @Tout ; thus the only unknown parameter in the equation is Tout ; therefore the temperature of the humidified hydrogen can be calculated by solving Eq. (2.306).

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c) Thermal management subsystem In a liquid-cooled PEMFC stack the liquid water after leaving the stack will be hot (the hotness degree of this water can be controlled by tuning the water mass flow rate, regarding the generated heat in the stack), and it is necessary to cool it down. A heat exchanger (HEX) such as a radiator is usually required to cool the water from Th,in to Th,out (here the subscript h denotes the water as the hot fluid in the heat exchanger). The cooling process is done by the convection heat transfer with the surrounding air with a specific temperature and mass flow rate (i.e., Tc,in and m˙ c ). The subscript c indicates the cold fluid, which is air here. To design a heat exchanger for this cooling process, ˙ UA, and Tc,out , which are the HEX heat we must find three unknown parameters Q, transfer rate, HEX thermal resistance inverse (U is the overall heat transfer coefficient of HEX), and the air temperature at the HEX outlet, respectively (the water mass flow, which depends on the heat generation rate in the stack, is assumed to be known, based on the stack condition). The first and last unknown parameters can be easily calculated by 



˙ = (mc ˙ )h Th,in − Th,out , Q

Tc,out = Tc,in +

(2.307)

˙ Q . ˙ )h (mc

(2.308)

However, calculating the UA is not too straightforward. There are several methods presented in the literature for analyzing HEXs and calculating parameters such as UA. These methods include the logarithmic mean temperature difference (LMTD), ε − NTU, P − NTUc , and ψ − P. Amongst these methods, LMTD is only applicable for cocurrent and countercurrent parallel HEXs; in fact, ε − NTU is more widely used in the HEX analysis. In the ε − NTU method, we have three interconnected parameters ε , NTU, and Crel ; i.e., if two of these parameters are known for a specific type of HEX, then the third one can be obtained via relations or charts presented in the HEX texts, such as in [112,113]. These three parameters are defined as ε=

˙ ˙ Q Q =  , ˙ max (mc ˙ )min Th,in − Tc,in Q

Crel =

˙ )min (mc , ˙ )max (mc

NTU = )

*

(2.309)

(2.310)

UA , ˙ )min (mc

(2.311) )

*

˙ )min = min (mc ˙ )max = max (mc ˙ )c , (mc ˙ )h and (mc ˙ )c , (mc ˙ )h . where (mc

PEMFCs

For calculating UA for a HEX in the thermal management system of a PEMFC stack, ε and Crel can be determined via Eqs. (2.309) and (2.310). To determine NTU and, subsequently, UA, the HEX type must be chosen beforehand. Crossflow HEXs are widely used in PEMFC TMSs. For such HEX with both fluids unmixed and for a single-pass case, the following equality must be solved to determine NTU:  ε = 1 − exp





)  * 1 (NTU )0.22 exp −Crel (NTU )0.78 − 1 . Crel

(2.312)

After obtaining NTU, UA can be calculated by Eq. (2.311). Although the above-presented 1D model for the analysis of a PEMFC TMS is quite advantageous, CFD simulations of HEXs are also performed to obtain a more detailed analysis of a HEX. ANSYS Fluent provides two specific modules entitled the macromodels (ungrouped and grouped) and the dual cell model. In these models the mentioned HEX analysis methods such as ε − NTU are performed for small computational cells, and not for the entire HEX; additionally, the HEX cores (consisting of compact fins and tubes) are considered as porous media to reduce computational cost in the simulation of a compact HEX [18].

2.3.6 Modeling of PEMFC cold start Since automotive PEMFCs can experience temperatures below zero, freezing is possible in a cold climate. In fact, under some severe cold conditions, when a dried PEMFC starts generating electric power, after a few moments, ice is formed in the cathode CL, where the water is produced; this results in the blockage of CL pores by ice and, consequently, suddenly diminishes the power. In the simulation of a PEM fuel cell during the cold start, the main modifications must be done on the water transport submodel, and after that, on the heat transport submodel. In a PEMFC during the normal operating conditions, free water in the pore spaces and in the either vapor or liquid phases, as well as the dissolved water in the electrolyte material and in the nonfrozen phase, can exist. However, for a PEMFC during cold start, other water phases such as ice (in the porous electrodes and GCs) or frozen water dissolved in the membrane can exist, as depicted in Fig. 2.38. When a PEMFC is started in a cold condition, the Jiao and Li [114] presented a CFD model for a PEMFC during cold start. In their model an ice saturation sice appears, which represents the fraction of void space filled by solid ice; similarly, slq represents the liquid saturation. Hence the available fraction of void space for the gas transport will be 1 − sice − Slq . The governing equations of their model are presented in Table 2.16, whereas the employed sources in these equations are presented in Table 2.17 [114]. The subscripts g, mem, nf, nmw, f, fmw, ele, ion, fl, and sl indicate gas, membrane, nonfrozen, nonfrozen membrane water, frozen, frozen membrane water, electron, ion, fluid phase, and solid phase, including electrolyte material and solid ice, respectively.

225

226

Fuel Cell Modeling and Simulation

Table 2.16 Governing equations in the presented cold start model by Jiao and Li [114].

Table 2.17 Source terms for presented equations in Table 2.16 [114].

PEMFCs

Figure 2.38 Different possible water phases for the different parts of a PEM fuel cell under normal operating condition vs. cold start [114].

2.4. Summary In this chapter, we explained in detail the components and the structure of PEMFCs, from the cell level to the stack and system levels, and the transport phenomena in these components. Besides, we presented different hydrogen production and storage methods. Afterward, based on the presented physics of transport phenomena in the PEMFC components, we presented the microscale simulation methods. Special attention is paid to LBM as one of the most capable numerical tools for the microscale simulations of PEMFCs; the capabilities of this method in extracting transport properties, investigating liquid water behavior in an electrode, and deep analyzing of electrochemical reactions on the CL are expressed, with a few examples (the applied codes for solving these examples are also provided in Appendix A). Additionally, the stochastic reconstruction of a PEMFC GDL microstructure, which is an inevitable requirement for most of the microscale simulations of PEMFCs, is well described. The implementation of the PNM and the VOF method for the microscale simulation of a PEMFC is also explained. For the macroscale simulation of a single cell in PEMFC, a 1D model and CFD models based on FVM are explained, and different submodels of them are introduced. After that, the modeling of BOP elements for a PEMFC system such as gas supply system, humidifier, and thermal management system is presented. Finally, the CFD simulation of a single cell in a PEMFC during cold start is discussed.

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2.5. Questions and problems Questions 1- What are the key parts of an electrode? Explain the roles of each one. 2- Propose a solution for more cost-effective use of the PGMs. 3- Explain the proton conduction mechanism through the Nafion membrane. 4- Discuss the pros and cons of different flow fields. 5- What are the main subsystems of BOP? Present the functions and constructional parts of each subsystem in a table. 6- Explain the role of temperature on the thermal conductivity of the following materials at room temperature: (a) ion, (b) aluminum, (c) mercury, (d) aluminum oxide, (e) hydrogen, (f) water vapor, and (g) liquid water. 7- Explain the potential role of hydrogen as an energy container in the future world. Provide a comparison between hydrogen and electricity as the two main clean energy carriers. 8- What are the pros and cons of water electrolysis as a hydrogen production technique? If you are going to produce hydrogen in a dry desert, do you choose this technique? Why or why not? 9- Provide a table and compare the pros and cons of different hydrogen storage techniques. 10- Provide a table and introduce five different versions of pseudopotential functions for the LB simulation of multiphase flow. Can you express the pros and cons of each one? 11- Describe the possible events during the cold start of a PEMFC from the transport phenomena aspect. 12- Describe the presented source terms in Table 2.17. Problems 1- For a binary mixture, write the mass diffusion fluxes by the Fick law and also the Maxwell–Stefan equations. Compare and analyze the attained fluxes by these two methods. 2- Estimate the time required for oxygen to penetrate and diffuse across a 2-µm liquid film on a TPB in the CL in a partially flooded PEMFC at 80 ◦ C. δ2 , where δ and Hint: The time required τD can be calculated from τD = DO2,liquid water DO2,liquid water are the film thickness and oxygen diffusion coefficient, respectively. 3- Capillary pressure can be calculated by 

Pc = σ |cos θ |

∅

kabs

 12

J (s) ,

where J (s) is the Leverett function, which for a hydrophobic GDL is equal to J (s) = 1.417s − 2.120s2 + 1.263s3 .

PEMFCs

Using this function, determine the capillary pressure at 80 °C for the GDL properties presented in the table below at s = 0.4. GDL Property

Contact angle Porosity Permeability

4-

567-

8-

910-

11-

115° 0.75 2.25 × 10−12 m2

Additionally, briefly explain the following: (a) If the contact angle is changed from 115◦ to 130◦ , how does the capillary pressure will be changed? Explain the reason. (b) If the porosity is increased to 0.85, how does the capillary pressure will be changed? (c) If the temperature is decreased, what is its effect on the capillary pressure? In a PEMFC, GC cross-section is a 0.5 mm×2 mm rectangle. If air with 350 K and 1.5 atm flows through the GC, then determine the average Nu and heat transfer coefficient for the following three cases: (a) ReDh = 1000, (b) ReDh = 15000, and (c) ReDh = 5000. Write the bounce-back implementation of the no-slip boundary condition for the four faces of a 2D lid-driven square cavity in a D2Q9 lattice. Prove Eqs. (2.185). By extending of presented LB code in Appendix A.2 for a 2D isothermal liddriven cavity, derive the temperature distribution for a lid-driven cavity being heated from the bottom edge and being cooled from the upper moving edge (while vertical edges are adiabatic) for three Re numbers 10, 100, and 1000. Consider the normalized temperatures of the bottom and top edges equal to 1 and 0, respectively. Redo Example 2.8 but assume that instead of a single fiber, a bundle of fibers is located in the channel. The bundle consists of 4 × 4 fibers with the same diameter, D. The space between the centers of two neighboring fibers is 2D. Regarding Example 2.9, investigate the role of fiber diameter and contact angle on the water invasion behavior in this example. Generate the microstructure of a Toray 09 carbon paper sample (1.0 × 0.2 × 0.2 mm3 ) and apply it as the GDL above CL in Example 2.10; now conduct the LB simulation using Appendix A.4 and extract the current density distribution on the CL. Compare the results with the results of Example 2.10. Use Appendix A.1 to generate the microstructure of a Toray 90 sample. Then generate a pore network for it via the method presented by Kuttanikkad [86]. Can you predict the total in-plane and through-plane permeabilities of a dry Toray 90 by the generated network? How much is the difference between these

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values and experimental data presented in Table 2.2? Why? Can you suggest a solution to lessen the departure? 12- Modify the code in Appendix B to provide the mole fraction distributions of species throughout the anode and cathode electrodes for the presented PEMFC in Example 2.11 and for i = 0.5iL . Recall that iL is the limiting current of the cell. 13- If we want to use a multispecies diffusion coefficient as presented in Part a of Section 2.1.2, and not a simple binary one, provide a 1D model by modifying Appendix B.

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[101] R.B. Ferreira, D.S. Falcão, V.B. Oliveira, A.M.F.R. Pinto, Numerical simulations of two-phase flow in proton exchange membrane fuel cells using the volume of fluid method – A review, Journal of Power Sources 277 (2015) 329–342, https://doi.org/10.1016/J.JPOWSOUR.2014.11.124. [102] S. Ge, C.-Y. Wang, Liquid water formation and transport in the PEFC anode, Journal of the Electrochemical Society 154 (2007) B998, https://doi.org/10.1149/1.2761830/XML. [103] J.M. Sergi, S.G. Kandlikar, Quantification and characterization of water coverage in PEMFC gas channels using simultaneous anode and cathode visualization and image processing, International Journal of Hydrogen Energy 36 (2011) 12381–12392, https://doi.org/10.1016/J.IJHYDENE.2011. 06.092. [104] D. Lee, J. Bae, Visualization of flooding in a single cell and stacks by using a newly-designed transparent PEMFC, International Journal of Hydrogen Energy 37 (2012) 422–435, https:// doi.org/10.1016/J.IJHYDENE.2011.09.073. [105] L. Chen, H. Luan, Y. Feng, C. Song, Y.-L. He, W.-Q. Tao, Coupling between finite volume method and lattice Boltzmann method and its application to fluid flow and mass transport in proton exchange membrane fuel cell, International Journal of Heat and Mass Transfer 55 (2012) 3834–3848, https:// doi.org/10.1016/j.ijheatmasstransfer.2012.02.020. [106] J.U. Brackbill, D.B. Kothe, C. Zemach, A continuum method for modeling surface tension, Journal of Computational Physics 100 (1992) 335–354, https://doi.org/10.1016/0021-9991(92)90240-Y. [107] T.F. Fuller, J. Newman, Water and thermal management in solid-polymer-electrolyte fuel cells, Journal of the Electrochemical Society 140 (1993) 1218–1225, https://doi.org/10.1149/1.2220960. [108] V. Gurau, F. Barbir, H. Liu, An analytical solution of a half-cell model for PEM fuel cells, Journal of the Electrochemical Society 147 (2000) 2468, https://doi.org/10.1149/1.1393555. [109] T.V. Nguyen, R.E. White, A water and heat management model for proton-exchange-membrane fuel cells, Journal of the Electrochemical Society 140 (1993) 2178–2186, https://doi.org/10.1149/ 1.2220792. [110] H.K. Versteeg, W. Malalasekera, An Introduction to Computational Fluid Dynamics: The Finite Volume Method, Pearson Education, 2007. [111] J.H. Ferziger, M. Peri´c, R.L. Street, Computational Methods for Fluid Dynamics, vol. 3, Springer, 2002. [112] S. Kakac, H. Liu, A. Pramuanjaroenkij, Heat Exchangers: Selection, Rating, and Thermal Design, CRC Press, 2002. [113] R.K. Shah, D.P. Sekulic, Fundamentals of Heat Exchanger Design, John Wiley & Sons, 2003. [114] K. Jiao, X. Li, Three-dimensional multiphase modeling of cold start processes in polymer electrolyte membrane fuel cells, Electrochimica Acta 54 (2009) 6876–6891, https://doi.org/10.1016/j.electacta. 2009.06.072.

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CHAPTER 3

Solid oxide fuel cells Contents 3.1. Introduction 3.1.1 Components and structures 3.1.1.1 The main reactor 3.1.1.2 Fuel-processing unit 3.1.1.3 Heat-transfer active components 3.1.1.4 Power regulator subsystem 3.1.1.5 Controllers 3.1.2 Transport phenomena in SOFCs 3.1.2.1 Charge transport 3.1.2.2 Mass transport 3.1.2.3 Heat transport 3.2. Microscale modeling and simulation of SOFCs 3.2.1 Microstructure reconstruction methods 3.2.1.1 Stochastic reconstruction technique 3.2.1.2 FIB/SEM reconstruction technique 3.2.2 Lattice Boltzmann simulation of reactive gas flow 3.3. Macroscale modeling and simulation of SOFCs 3.3.1 1D modeling 3.3.1.1 The first step, activation loss 3.3.1.2 Second step, concentration loss 3.3.1.3 Third step, ohmic loss 3.3.2 2D/3D models of SOFC cell, stack, and system 3.3.2.1 Gas transport submodel 3.3.2.2 Heat transport submodel 3.3.2.3 Electron transport submodel 3.3.2.4 Electrochemical reaction submodel 3.3.3 Modeling of a SOFC system 3.4. Modeling of a solid oxide electrolyzer cell (SOEC) 3.5. Summary 3.6. Problems References

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3.1. Introduction As a typical fuel cell, SOFCs are considered electrochemical reactors in which the oxidation of fuel occurs at the anode side. Therefore, we can analyze SOFCs through thermodynamic relations. In addition to electrochemical reactions, other physical phenomena are involved, and we have to be careful when dealing with analyzing them. For example, mass transfer, heat transfer, and conservation of chemical species are considered Fuel Cell Modeling and Simulation https://doi.org/10.1016/B978-0-32-385762-8.00007-5

Copyright © 2023 Elsevier Inc. All rights reserved.

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important physical phenomena. Each of these terms is effective in a specific component and can be neglected in some others. Thus, before we proceed to simulate SOFCs, a brief introduction to their major components and the involved phenomena is crucial.

3.1.1 Components and structures The basic structure of a SOFC is the same as the other FCs. The main reactor consists of an anode and a cathode with a solid-state separator sandwiched in between. In most low-temperature FCs, the hydrogen is converted into H+ ions at a proper catalyst layer. Then the H+ ions move through the membrane to reach the cathode surface. Consequently, the moving ion is H+ . In contrast to this phenomenon, in high-temperature SOFCs, the moving ion is O – – , formed at the cathode catalyst layer. Therefore the membrane of such an FC must be able to conduct the oxygen ions. The schematic of a SOFC is shown in Fig. 3.1. As illustrated and explained, the oxygen ions are formed at the cathode surface according to the following electrochemical reaction: 1 O2 2 e− −−→ O−− . 2

(3.1)

The reduced oxygen ions O – – move through the solid-state membrane to the anode, at which it reacts with hydrogen to form water: H2 + O−− −−→ H2 O + 2 e− .

Figure 3.1 Schematic of a SOFC.

(3.2)

Solid oxide fuel cells

Although the electrochemical reactions are different from the low-temperature FCs, the final result is the same. In other words, the overall reaction of SOFCs is H2 +

1 O2 −−→ H2 O + Energy + Heat. 2

(3.3)

One of the benefits of the SOFCs is that they can use carbon monoxide as their fuel. In such a case the anodic reaction becomes CO + O−− −−→ CO2 + 2 e− ,

(3.4)

which leads to the following overall reaction: CO +

1 O2 −−→ CO2 + Energy + Heat. 2

(3.5)

The fact that SOFCs can consume both hydrogen and carbon monoxide is of great interest. This helps them use reformers (internal or external types) to convert almost all types of hydrocarbons to produce their fuel. As we know, the reforming process generates hydrogen and carbon monoxide out of the hydrocarbon, and for the lowtemperature FCs, the carbon monoxide is useless. However, it is used as a fuel in high-temperature SOFCs. Consequently, the fuel of SOFCs can be stored as any hydrocarbon, which most of the time is quite easy to store or transport. For example, the feed of SOFCs can be natural gas, methanol, or similar fuels. Obviously, these fuels are easily transported and stored compared to hydrogen. In Fig. 3.1, we show the main reactor of the SOFC. However, a regular operation requires different components, which are schematically shown in Fig. 3.2. The picture is just an illustration, and in practice, there are different designs with different components. Generally, these subsystems are crucial for a smooth operation. As shown in Fig. 3.2, different subsystems are combined together. Here we briefly discuss each subsystem.

3.1.1.1 The main reactor The main reactor shown in Fig. 3.1 is the heart of an FC, where the fuel and oxygen react and produce water, carbon dioxide, energy, and heat. However, the main reactor cannot work without other subsystems.

3.1.1.2 Fuel-processing unit A fuel-processing unit processes the fuel to make it suitable for transferring to the main reactor. Depending on the stored fuel, the section may consist of different parts. For

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Figure 3.2 Illustration of a SOFC system and its subsystems.

instance, if the fuel is made of hydrogen and carbon monoxide, then it only requires some warming before transferring to the main reactor. The initial warmup is essential because if the inlet fuel has a significantly lower temperature than the main reactor, then the thermal stress becomes too high, which in turn results in cell breakdown. If the fuel contains hydrocarbon-rich substances such as natural gas or alcohol, then it requires a kind of reformer. Reformers convert the hydrocarbons to hydrogen and carbon monoxide to be used in the reactor. There are different types of reformers available in the SOFC industry. The reformers can be either external or internal. In an external reformer the fuel is reformed in a completely separate unit, and the results are fed into the FC main reactor. In an internal reformer the fuel is reformed to hydrogen and carbon monoxide in a built-in compartment located inside the anode of the FC.

Solid oxide fuel cells

In both cases the process requires heat and steam, which can be provided from the main reactor results. This design increases the overall efficiency of the FC and can be referred to as a CHP concept.

3.1.1.3 Heat-transfer active components Heat-transfer components are required in many different sectors. These thermally active components can be either heat exchangers or combustors. First of all, we need to warm up the feed to the temperature levels of the main reactor. Otherwise, the entering cold fuel creates a temperature gradient inside the anode channel. The temperature gradient creates different expansion in the fuel cell subsystem, which in turn creates mechanical stress on the anode, membrane, and cathode materials. The fact that the cell operates at very high temperatures means that all the cell components experience lots of expansions. Since different parts of the reactor are made of different materials with different thermal expansion coefficients, the components inherently have different elongations and produce mechanical stresses. This is a natural behavior, and the designers consider the normal expansion of different parts. However, if the cold fuel enters the cell, then it creates lots of mechanical stresses on the designed components. The stress is out of the tolerance threshold and results in the breakdown of the cell. The same argument is true for the cathode side. The entering oxygen of air must be warmed before entering the cathode channel. Otherwise, it produces mechanical stresses, as discussed for the fuel side. The required heat for warming up the inlet air also comes from the heat of reaction to increase the efficiency of the system. The above discussions show that we need some heat exchangers on the anode and cathode sides to regulate the inlet fluids. Therefore thermally active components are crucial in designing and building a SOFC. Apart from these components, we need other thermally active components for other purposes. For instance, we need an afterburner to burn the unused fuel and oxygen leaving the main reactor. This is an important component that has two different advantages: 1. it prevents the emission of the toxic carbon monoxide and hazardous hydrogen, and 2. the produced heat can be used in different processing units such as fuel reforming and so on. These functions show that the afterburner is crucial in the SOFC design and must be adequately designed. The above-mentioned thermally active components are crucial and cannot be avoided in the overall system. However, there are some optional components that play an important role in increasing efficiency. However, if they are missed, then the system works perfectly as before but with less efficiency. Specifically, the use of CHP in the system design increases efficiency by exploiting the wasted heat of the system. There are lots of different CHP designs for different purposes. Some concepts use the SOFC

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wasted heat for space warming; some use the heat for another process, some combine the system with desalination purposes, and other similar combinations exist to use the wasted heat of SOFCs. These combinations are attractive since the temperature of SOFC power plants is very high and can handle lots of different processes. Due to the importance of the CHP applications, a whole chapter is dedicated to the topic. Hence, for more information, we refer to Chapter 7.

3.1.1.4 Power regulator subsystem The voltage regulation sector is another part of a SOFC system. Obviously, SOFCs produce direct current (DC) power, but it is not useful for consumers. Therefore its produced power must be inverted into alternating current (AC) with proper frequency. In some countries, the grid frequency is 50 Hz, and in some others, it is 60 Hz. Depending on the grid specifications, proper inversion must be designed.

3.1.1.5 Controllers In addition to all the above-mentioned components, control and monitoring subsystems are required for smooth operation and safety. These facilities are necessary to monitor the following parameters: 1. cell temperature, 2. generated voltage and power, 3. controlling the mass flow rate of the fuel and oxidant, 4. regulation of the pressure and temperature of the inlet fluids, and 5. other similar important parameters.

3.1.2 Transport phenomena in SOFCs A SOFC system is a multidisciplinary system in which different physical phenomena exist. As mentioned above, there are various components in the whole system. At each component, different types of physics are important. Therefore, for analyzing the whole system, a good understanding of the involved phenomena is essential. Generally, the main involved transport phenomena are: 1. charge transport, 2. mass transport, and 3. heat transfer. Each phenomenon is important in some components. For instance, the main reactor includes all the mentioned phenomena, but the heat exchangers primarily work with heat transfer. As another example, the mass transport of the chemical species is not considered in heat exchangers, but it is crucial in the main reactor, as stated, as well as in the afterburner. In the following subsections, we give a brief study of the involved physical phenomena.

Solid oxide fuel cells

3.1.2.1 Charge transport The charge transport phenomenon occurs only at the main reactor. Because the electrochemical reactions take place at the anode and cathode surfaces, and the oxygen ion moves through the membrane, such transport occurs at no other components. Any conductor has an inherent resistance to charge transport. Whether the charge carrier is electrons or ions, the medium resists their movement. The result of this resistance is a voltage drop in the cell. A SOFC is composed of different porous components, including the anode, membrane, and cathode. These components are solid-state materials with different porosity. Since these components are porous, their resistance is higher than that of solid materials. The voltage drop due to the charge transfer is obtained using the following equation: vohmic = IRohmic = I (Relec − Rion ),

(3.6)

where Relec is the resistance of the solid porous electrodes to electron movement, and Rion is the resistance of the medium to the movement of oxygen ions. It is well known that the ionic resistance is much higher than the electronic resistance; hence, in most calculations, we can neglect Relec compared to Rion . The resistance of materials depends on the conductivity of the medium, its crosssection area, and its length. This relation discussed in Chapter 1, which is R=

L , σA

(3.7)

where σ is the electrical conductivity in terms of Ohm−1 cm−1 , L is the length, and A is the cross-section of the medium. By having the conductivity of the medium we can easily get a good estimate of its electrical resistance. For most materials, this property can be calculated using the equation v

σ = nq . ξ

(3.8)

Here a new term is defined called the mobility and is denoted by ui . It is the velocity of the charge carrier or v in the electrical field ξ . According to the definition, the mobility is v ui = . (3.9) ξ

This factor is important since it clearly defines the conductivity of the medium. The following equation defines another expression for the calculation of the conductivity of most media: σ =F



|zi |ci ui ,

(3.10)

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in which ui is the mobility as defined by Eq. (3.8), ci is the number of moles of charge carriers per unit volume, and zi is the charge number of the mobile carriers. For ceramic membranes in SOFCs, the mobile ion is O−− . The transport of oxygen ions happens through the yttria-stabilized zirconia (YSZ) ceramic, which is a good conductor at high temperatures. Doping zirconia with yttria creates sufficient oxygen vacancies and creates a good conductor. Increasing yttria increases the oxygen vacancies; however, it becomes saturated if the amount of the dopant is higher than about 8%. In ceramic membranes the mobility of the ions is related to the diffusion coefficient of the membrane according to the Einstein equation [2] ui =

zi D , kT

(3.11)

in which k is the Boltzmann constant (k = 8.61 × 10−5 eV/K). Comparing this relation with Eq. (3.11), we obtain the following definition for the ionic conductivity of the YSZ membrane: |zi |2 ci D σ= . (3.12) kT This equation is used to calculate the ionic resistance of the cell according to Eq. (3.7). Then the result is used to obtain the voltage drop via Eq. (3.6).

3.1.2.2 Mass transport Mass transport is in the form of either diffusion or convection. Diffusive mass transport is important in the main reactor and the afterburner. In these components, chemical or electrochemical reactions occur. Hence different species react to each other resulting in concentration buildup or depletion. The concentration gradient forms diffusive mass transport. In contrast, convective mass transport takes place almost in all the components. The convective mass transport is studied by means of fluid dynamics relations in the form of Navier–Stokes equations. Particularly, in the FC channels, where the fuel and oxidant flow from the inlet to the outlet, the pressure drop inside the channel becomes notable. The pressure drop causes a concentration decrease along the channel, affecting the diffusive mass transport. To explain the above phenomenon, we refer to Fig. 3.3, which illustrates a channel, GDL, and catalyst layer (CL). The present discussion for general FCs is presented in Chapter 1. In the present section, we further develop the same arguments for the case of SOFCs. Assume that the bulk concentration of a species is Ci, as it flows along the channel. Since the species are consumed or produced at the CL, their concentration is different from the bulk value. Hence a diffusive mass flux occurs, which is governed by Ficks’s law as m˙ = −D∇ c ,

(3.13)

Solid oxide fuel cells

Figure 3.3 Mass transport mechanism from channel to catalyst layer.

where m˙ is the diffused mass, c is the concentration of species, and ∇ c is the concentration gradient. For a simple one-dimensional diffusion, this equation reduces to m˙ = −D

dc . dx

(3.14)

We use this equation to explain the mass transport phenomena. In Eq. (3.13) or (3.13), D is the bulk diffusion coefficient and is valid at the channel. However, as we can see in the figure, the GDL and CL are porous media. Diffusion in porous media is not as straightforward as in bulk, since in a porous medium the solid parts of the medium work as obstacles and reduce the diffusion. It is customary to modify the diffusion coefficient to account for the porous media effect. The conventional method for modifying the diffusion coefficient is the wellknown Brugmann equation, by which the effective diffusion coefficient in a porous medium is defined as Deff = Dε1.5 .

(3.15)

Therefore, for the porous electrodes, Eq. (3.14) is modified to the equation m˙ = −Deff

dc . dx

(3.16)

We can use Eq. (3.16) to obtain a physical understanding of mass diffusion. Since the porous electrode and CL are very thin, we can assume that the concentration gradient

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is linear. Hence we can estimate Eq. (3.16) with the equation m˙ = −Deff

c i − cs δ

,

(3.17)

where cs and ci are the concentration of species at the GDL/Channel and GDL/CL interfaces, respectively. Also, delta is the mean diffusion length, which is the GDL thickness. These parameters are schematically shown in Fig. 3.3. It is clear that cs has a slight difference from the bulk concentration c0 . This difference can be viewed as the result of convective mass transport from the bulk to the GDL surface. The phenomenon is similar to the convective heat transfer from a surface to its surroundings. Using this similarity, we can relate cs to c0 by the following relation: m˙ = hm (c0 − cs ),

(3.18)

in which 1/hm is the resistance to the convective mass transfer. Combining Eq. (3.18) with (3.17), the mass transfer from bulk to the catalyst surface is m˙ =

c 0 − ci 1 hm

+

δ

.

(3.19)

Deff

This equation is analogous to heat transfer by conductive and convective mechanisms, where 1/hm is the convective resistance, and δ/Deff represents the conductive resistance. From Eq. (3.19) we deduce that mass diffusion depends on the bulk concentration inside the flow channel. Note that the direction of diffusive mass transport is perpendicular to the channel direction, where the convective mass transfer is dominant. From an electrochemical point of view, the mass transfer m˙ is due to the electrochemical reaction by which the species are consumed or produced. Therefore the amount of m˙ is proportional to the electrical current according to the Faraday equation i = nF m˙ .

(3.20)

Incorporating Eq. (3.19) into Eq. (3.20), we can connect the bulk concentration to the current density: c 0 − ci . (3.21) i = nF 1 δ hm

+

Deff

From this equation we can find the limiting current of a SOFC. It happens when the species concentration is zero at the catalyst surface; hence the cell cannot produce any further current. Using this concept and applying it to Eq. (3.20), the limiting current of a SOFC is c0 − ci . (3.22) iL = nF 1 δ hm

+

Deff

Solid oxide fuel cells

The limiting current not only shows the maximum ability of the cell to produce current but also is used for the determination of voltage loss due to the concentration. As we know, the voltage drop due to concentration is defined by the equation 

vconc =



RT iL . ln nF iL − i

(3.23)

Eq. (3.19) states that the bulk concentration changes along the channel. It is because, as the flow moves on, the species are consumed (or produced) at the CLs. Also, Eq. (3.22) shows that the concentration change depends on the current density of the FC. The higher the current density, the more material is consumed, resulting in more concentration changes. As a result of concentration change, the diffusive mass transport differs from the inlet to the outlet of the FC. Consequently, we have different current densities along the channels. The simulation of the flow movement in the channel and the combination of convective mass transport in the channel and diffusive mass transport inside the GDL and CL is rather complex. Therefore lots of research are carried out on the flow field simulation in the whole assembly. In the present chapter, we have used the lattice-Boltzmann method (LBM) to simulate such complex phenomena. It is shown that the GDL characteristics can be easily modeled using LBM.

3.1.2.3 Heat transport Heat transfer is almost dominant in all the different components of a SOFC system. It happens in all the heat exchangers. Also, the afterburner and the main reactor are severely affected by heat transfer. To analyze heat transfer, we need to have some information about the heat sources and sinks. In a SOFC the heat sources are: 1. the main reactor at which the fuel and oxygen react to produce heat and power, 2. the afterburner that burns the unconsumed materials, and 3. the electronic devices e responsible for regulating the power. The other components are not involved in heat generation, but they have a significant role in transferring the heat to different system sectors.

3.2. Microscale modeling and simulation of SOFCs 3.2.1 Microstructure reconstruction methods A key step before microscale modeling and simulation of SOFCs is to achieve the geometry of electrode microstructures. This process, called microstructure reconstruction, can be done via three main techniques: 1. stochastic reconstruction technique,

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2. focused ion-beam (FIB)/SEM technique, and 3. X-ray computed tomography (XCT) technique. The first technique is based on the stochastic approach, whereas the other two techniques are based on the image combination approach. The average pore size in a SOFC electrode is less than 1 µm. Therefore employing X-ray CT technique is not recommended due to its insufficient resolution and also its lack of ability for distinguishing two solid phases of YSZ and Ni. Most of the proposed electrode microstructure reconstructions in the literature are performed via either a stochastic reconstruction technique or a FIB/SEM technique. Hence, in the remaining part of this section, we will explain only these two techniques.

3.2.1.1 Stochastic reconstruction technique In this technique the microstructure of the electrode (usually, anode) is reconstructed based on a stochastic model, which is inspired from the manufacturing procedure. The tuning parameters of the model are selected from 2D images of a practical electrode. Some 2D and 3D stochastic reconstructions are presented in the literature [1,5,6,10,12]. The microstructure of a SOFC anode consists of two distinct solid phases: 1. YSZ (as the ionic conductor phase) and 2. Ni (as the electronic conductor phase). In some of the presented stochastic reconstructions, these two phases are considered as a single dense phase, which is far from reality. In some other ones the granulometry law is employed; according to this law, the porous anode is made of regular grains (either YSZ grains or Ni grains) with a fixed known shape. Such reconstructions do not consider the neighboring information. However, there are more advanced stochastic techniques recently proposed using multiple-point statistics, which can take into account such information and provide a higher level of reliability. Here we will explain one of these techniques, presented by Suzue et al. [12]. The stochastic reconstruction technique presented by Suzue et al. [12] requires two presteps: 1. sample preparation and 2. 2D imaging of the samples. They prepared anode samples by mixing YSZ and nickel oxide (NiO) powders with weight ratio of YSZ to NiO equal to 1:1.5; the final volume ratio of YSZ to Ni was 1.0:1.358. They molded samples into rods by extrusion molding. They presintered three sets of samples at 1000 ◦ C for 180 min and, subsequently, sintered each sample at one of the three different temperatures of 1300 ◦ C, 1350 ◦ C, or 1400 ◦ C for 180 min. Then they used hydrogen/nitrogen mixture with 2:1 molar ratio and 750 ◦ C to reduce samples for 10 h. The resulted porosity of the samples were 0.450, 0.393, and 0.335, respectively. Then they treated the three samples with epoxy and polished them with a diamond paste to produce a highly flat sample surface with the surface roughness less

Solid oxide fuel cells

Figure 3.4 The three prepared samples by Suzue et al. [12] at three sintering temperatures.

Figure 3.5 The final 2D image of the sample sintered at 1400 ◦ C by Suzue et al. [12]. The white, gray, and black regions denote Ni, YSZ, and pore phases, respectively.

than 80 nm, which is negligible compared to the grain size. Then they provided optical images of the polished samples via a confocal laser microscope with 41-nm resolution (Fig. 3.4). The white, gray, and black regions in this figure are Ni, YSZ, and pore phases, respectively. Comparison of the resulted images demonstrates that increasing the sintering temperature leads to the grain size increment (especially for Ni grains). After providing these 2D images, they analyzed the brightness value distribution for the samples and sharpened the brightness boundaries between phases via a weighted differential filter; subsequently, they used the scheme proposed by Lee et al. [7] to distinguish phases. They used the porosity and YSZ/Ni volume ratio to verify the final 2D images in which the phases are evidently distinguishable (for the sample sintered at 1400 ◦ C, such an image is presented in Fig. 3.5). As the third step, they defined the phase function Z (r ) based on the 2D images from the previous step at each pixel x as follows:

Z (r ) =

⎧ 0 ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

if the pixel at r is in pore phase (black),

1

if the pixel at r is in YSZ phase (gray),

2

if the pixel at r is in Ni phase (white).

(3.24)

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Using this phase function, we can define the following two-point statistical function: Rij (s) = Rij (|s|) =

δ(Z (r ) , i)δ(Z (r + s) , j) δ(Z (r ), i)

,

(3.25)

where δ denotes the Kronecker delta function. Based on this definition, Suzue et al. [12] found the two-point statistical function for YSZ–YSZ and Ni–Ni versus (i.e., R11 (s) and R22 (s)). Subsequently, they adopted the iterative scheme proposed by Yeong and Torquato [13] for the 3D stochastic reconstruction of anode microstructure in a lattice with 150 × 150 × 150 nodes (i.e., voxels). The lattice space step was considered 0.178 µm in each direction, which is around 0.1 of the averaged grain diameter. As the first step of this iterative scheme, each lattice voxel is occupied by a random phase, considering the porosity and YSZ/Ni volume ratio. After that, the phases of voxels are alerted in such a way that the following cost function E be minimized: E=

 i ,j

s0

0

2



Rij2D (s) − Rij3D (s) ds,

(3.26)

where Rij2D (s) is calculated for the distinguishable 2D image of the sample (e.g., Fig. 3.5), and Rij3D (s) is calculated for the reconstructed 3D image at an iterative step. By assuming the anode microstructure to be isotropic they calculated the integral in Eq. (3.26) (by considering an arbitrary direction to reduce the computational cost in Eq. (3.26)), which was equal to 7 µm in their study. As the final step, they used a surface correction scheme [3] to remove physically impossible subgeometries such as a solid part suspended in a pore region with no solid connection. They also checked the validity of their reconstructed microstructure by comparing both two-point statistical function and line-path function L (s) of the reconstructed 3D images with those of the 2D images of the practical samples. The line path-function was defined as Li (s) =

li (s) , Nvoxel

(3.27)

where li (s) denotes the number of line segments of length s that are located in the phase i of lattice, whereas Nvoxel is the number of lattice voxels. Their final reconstructed anode samples for the mentioned three sintering temperatures are presented in Fig. 3.6.

3.2.1.2 FIB/SEM reconstruction technique In this technique, sample preparation must be done in a similar manner to that explained about the stochastic reconstruction technique (including molding, sintering, epoxy treatment, and polishing) before starting the reconstruction technique. Afterward,

Solid oxide fuel cells

Figure 3.6 The final stochastically reconstructed samples of porous anodes by Suzue et al. [12] for three sintering temperatures: a) 1300 ◦ C, b) 1350 ◦ C, and c) 1400 ◦ C.

Figure 3.7 Schematic of a typical FIB/SEM observation setup.

a FIB/SEM observation setup is used to provide several 2D images from the sample sequential cross-sections. These sequential cross-sections are provided by a focused ion beam (FIB), as depicted in Fig. 3.7. At each observation, a layer from the front face of the target volume is removed by FIB milling. In addition, a gas injection system is used to deposit carbon on the outer surface of the target volume to protect it from undesired milling. After that, the SEM image is obtained from the milled surface; this cut-and-see procedure is repeated until the sufficient 2D images are obtained for the 3D reconstruction. Since the drifting of these 2D images is unavoidable during the acquisition process, reference marks should be used. To create reference marks, the FIB can be used to make marks on the carbon coating. After acquisition of 2D images, all of them must be aligned

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and enhanced in terms of phase distinguishability. The alignment is done by the position of reference marks in the images, which must be fixed. The phase distinguishability is enhanced via analyzing the brightness value distribution in the images and sharpening the brightness boundaries between phases (especially, between YSZ and Ni phases) via a weighted differential filter, similar to that presented in the stochastic reconstruction technique. As the final step, these 2D images are integrated in a 3D image, which is the reconstructed microstructure of the anode electrode.

3.2.2 Lattice Boltzmann simulation of reactive gas flow The framework of LBM is comprehensively described in the previous chapter. Hence we will not describe it here. The reader of this section is recommended to study LBM from Chapter 2 beforehand. To employ LBM for the simulation of a SOFC electrode, as the first step, the TPB edges in the reconstructed microstructure must be recognized. This can be done by checking all edges in the 3D image. These TPB edges are directly in touch with all three phases of YSZ (O2− ion conductor), Ni (electron conductor), and pore (the path for the gas transport). On these TPBs the electrochemical reaction happens. The electrochemical reaction rate on the TPBs (i.e., the volumetric current density ier due to the electrochemical reaction) is a function of activation overpotential, which in turn depends on the potential difference between the electron conducting phase (Ni) and ion conducting phase (YSZ). The electric potential on these phases can be obtained by solving the following charge transfer equations for the Ni and YSZ phases, respectively:       ∂ σe ∂ηe ∂ σe ∂ηe ∂ σe ∂ηe + + = −ier , ∂x F ∂x ∂y F ∂y ∂z F ∂z       ∂ ∂ ∂ σi ∂ηi σi ∂ηi σi ∂ηi + + = ier , ∂ x 2F ∂ x ∂ y 2F ∂ y ∂ z 2F ∂ z

(3.28) (3.29)

where σe and σi are the electronic and ionic conductivities, respectively, whereas ηe and ηi are the electronic and ionic potentials, respectively. In addition, the following gas diffusion equation through the pore phase must be solved:       ∂ ∂ ∂ ∂ CH 2 ∂ CH 2 ∂ CH 2 ier + + = D D D . ∂x ∂x ∂y ∂y ∂z ∂z 2F

(3.30)

The above three equations have the form of simple diffusion equations with source term, which can be resolved by the presented LB procedure in Chapter 2, for the first raw of Table 2.11. The volumetric current density can be calculated by the following equation [11,12]:     2F ηact F ηact ier = i0 lTPB exp − exp ,

RT

RT

(3.31)

Solid oxide fuel cells

where i0 is the lineal exchange current density [4], −0.03

i0 = 31.4PH2

PH0.24O exp

  1.52 × 105 . −

(3.32)

RT

To calculate (3.31), the following relation can be used [3]:



1 PH 2 O ηact = − 2ηe − ηi + G0 + RT ln 2F PH 2

 .

(3.33)

3.3. Macroscale modeling and simulation of SOFCs 3.3.1 1D modeling The presented 1D model for a SOFC cell in this section is based on the flux balance notion, similar to the presented 1D model in Section 2.3.1. Due to the analogy of these two models, here we do not mention all details for brevity. The reader is recommended to read Section 2.3.1 before studying this section. It is worth mentioning that in SOFCs the only species that is transferred through the electrolyte is the oxide ion O2− , in contrast to the PEMFCs, in which both the proton ion H+ and the water molecule H2 O are transferred through the electrolyte. This makes the modeling of SOFCs quite simpler. In addition, because of the high operating temperature, the water in SOFCs is always in the vapor phase, and there is no need to incorporate complicated multiphase models. However, modeling of thermal aspects in SOFCs is not as easy as that of thermal aspects in PEMFCs. Now consider a SOFC single cell as depicted in Fig. 3.8. The 1D of the model is based on the through-plane fluxes. For a specific point (x, y), the following balance of fluxes can be written: i C A = nEO2− = nA H2 = 2nO2 = −nH2 O , 2F

(3.34)

A where i represents the current density, nEO2− , nAH2 , nC O2 , and nH2 O are the molar fluxes of 2− O through the electrolyte, hydrogen at the anode, oxygen at the cathode, and water vapor at the anode, respectively (in some literature, molar fluxes are shown by J¯ instead of n; we choose a simpler notation n). Note that the interface of electrolyte with the anode and cathode electrodes are named section II and section III in Fig. 3.8, respectively. Here we assumed that the SOFC is not an anode-supported SOFC. Therefore the following assumptions can be taken into account: 1. Electrochemical reactions happen on the thin interfaces between the electrolyte and electrodes.

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Figure 3.8 The schematic of a SOFC cell structure and its fluxes for constructing the 1D model.

2. The only considered mass transfer mechanism is diffusion, and it only happens in the through-plane direction of electrodes (or z-axis) as the main direction in this 1D model. 3. For calculating the mass diffusion flux of oxygen and water vapor at the cathode, nitrogen can be considered an inert species. 4. The electronic resistance of the cell is negligible in comparison to its ionic. 5. The activation loss of the cell is only due to the electrochemical reaction at the cathode; the anode activation loss can be negligible. Now we try to formulate all voltage losses of the cell via the above assumptions and the balance of fluxes (Eq. (2.229)). We start from the activation loss.

3.3.1.1 The first step, activation loss Using the simplified form of the Butler–Volmer equation, we can write 



Ru T i 101325 ηact,c = ln × C III , 4α c F i0,ref P X O2

(3.35)

where P C and XOIII2 are the cathode pressure (in atm) and oxygen mole fraction on the cathode/electrolyte interface (section III of Fig. 3.8), respectively. Since XOIII2 is not known, it must be calculated by the oxygen through-plane flux. Based on assumptions 3 and 4, the species through-plane fluxes can be simply stated by the following Fickian

Solid oxide fuel cells

relation: nC O2 = −

eff P C DO 2 , N2 dXO2 , Ru T dz

(3.36)

eff is the effective binary diffusion coefficient of oxygen and water vapor. where DO 2 , N2 Therefore by solving the above differential equation we get

XO2 (z) = XOIV2 −

Ru TnC O2 z. eff P C DO 2 , N2

(3.37)

Choosing z equal to the cathode electrode thickness tC , from the above equation we can calculate XOIII2 = XOIV2 −

Ru TnC O2 C t . eff P C DO 2 , N2

(3.38)

Now via the balance of fluxes (Eq. (2.229)) we have XOIII2 = XOIV2 −

Ru Ti tC . eff 4FP C DO 2 , N2

(3.39)

By substituting the above relation into Eq. (3.12) we get  

Ru T i ln ηact,c = αRe,c F i0,ref







PC Ru T i − ln XOIV2 − tC eff 101325 4F P C DO , N 2 2

 .

(3.40)

Usually, all terms on the RHS of this relation are known, and thus ηact,c can be calculated. Note that the second logarithmic term in the RHS represents the effects of concentration loss across the cathode electrode (i.e., reducing the oxygen mole fraction from XOIV2 to XOIII2 ) on the kinetic feature of the SOFC. This concentration loss also has an effect on the thermodynamic feature of the SOFC, which can be calculated via the Nernst equation.

3.3.1.2 Second step, concentration loss The Nernst equation for the cathode electrode is 



PO0.25 Ru T Ec = Erev,c − . ln 2F 1

(3.41)

Note that the Nernst equation was previously presented in Chapter 1 for an entire cell (Eq. (1.48)). Due to the mass transfer limitations, the pressure of the reactants will be less than the reference state, and the pressure of products will be greater than that of the

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reference state. Therefore, using the above equation, we can present the concentration loss from the thermodynamic aspect by 

ηcon,c = Ec − Ec,ref



0.5 PO Ru T 2 = Erev,c − ln 2F 1







PO0.25 Ru T − Erev,c − ln 2F 1

 . (3.42) ref

Since the cathode situation at section IV (i.e., in the cathode channel) is the same as the reference condition, the above equation can be expressed as ⎡

Ru T ⎣ XOIV2 ηcon,c = ln 2F XOIII2

0.5 ⎤ ⎦.

(3.43)

Note that here the reference condition is the condition of the cathode/electrolyte interface if there are no mass transfer limitations, i.e., if the oxygen concentration on the mentioned interface is the same as the oxygen concentration in the channel. Among the two mole fractions that appeared on the RHS of this equation, XOIV2 is known, as they are the input parameters of the model. Eq. (3.16) can be used for calculating XOIII2 : ⎡⎛ ηcon,c =

⎜ Ru T ⎢ ⎢⎜ ln ⎢⎜ 2F ⎣⎝

⎞0.5 ⎤ ⎟ XOIV2 ⎟ ⎟ R Ti u C⎠ XOIV2 − t eff 4F P C DO 2 , N2

⎥ ⎥ ⎥. ⎦

(3.44)

In a similar way as we derived Eq. (3.16), the water vapor and also the hydrogen at the two sides of the anode electrode (sections I and II in Fig. 3.5) can be related by Ru Ti/2F A t , eff P A DH 2 , H2 O Ru Ti/2F A XHII2 = XHI 2 − A eff t , P DH2 , H2 O

XHII2 O = XHI 2 O +

(3.45) (3.46)

where tA is the anode thickness.

3.3.1.3 Third step, ohmic loss The thermal conductivity σ (T ) of the electrolyte can be obtained by the relation

σ (T ) =

Gact ASOFC exp − RT

T

! .

(3.47)

Solid oxide fuel cells

After that, using Eq. (2.8), the following relation can be used for the calculation of ohmic loss:  Mem  t cell Mem . ηohmic = iA R =i (3.48) σ (T ) After all, by subtracting the three calculated losses presented by Eqs. (3.40), (3.44), and (3.48) from the cell reversible voltage we can obtain the cell voltage cell − ηact,c − ηcon,c − ηohmic . V cell = Erev

(3.49)

The main output of the presented 1D model in this section is V cell , and the main input is i, which enables us to extract the polarization curve of the cell. If due to the fuel cross-over, etc., the cell experiences current leakage of ileak , i + ileak must be used instead of i as the main input of the model. Other inputs of the model are the operating conditions (the cell temperature and the pressures of electrodes), geometric properties (the thicknesses of electrodes and memeff eff brane), thermo-physical properties (such as DO , then DH ), the cathode kinetic 2 , N2 2 , H2 O properties (such as i0,ref and α ), and the mole fraction of species in the gas channels; i.e., XHI 2 in the anode side and XOIV2 in the cathode side. The mole fractions of water vapor in the anode side and nitrogen in the cathode side can be easily calculated noting that the sum of mole fractions of species must be equal to unity. The polarization curve is not the only output of this simple yet beneficial 1D model. The species mole fractions on the electrolyte/electrodes interfaces can also be calculated. These outputs can give insights into the design and control of a SOFC single cell. In the chart shown in Fig. 3.9, the procedures of this model for obtaining the outputs from the input parameters are illustrated.

3.3.2 2D/3D models of SOFC cell, stack, and system Several efforts have been made for modeling a SOFC cell with conventional CFD methods, especially FVM. The framework of FVM, as well as the governing equations of PEMFCs, represented as five transport submodels coupled with each other by the electrochemical reaction submodel, were discussed in the previous chapter. The transport submodels include gas transport, water transport, ion transport, electron transport, and heat transport; in SOFCs, there is no liquid water, and there is no electro-osmotic drag; i.e., there is no need for the complicated water transport submodel in SOFC modeling. This makes the simulations of SOFCs quite straightforward. Among different CFD models of SOFCs, here we will briefly explain the resolved electrolyte model of ANSYS Fluent software [8], which is a low-cost powerful CFD model. According to this model, the nonporous electrolyte and the porous interlayers on its two sides (depicted in Fig. 3.10) are considered as a pair of wall and wall-shadow faces, named “electrolyte interfaces” (i.e., these three layers are assumed to have zero thickness). By

257

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Figure 3.9 The 1D model and its procedures.

Solid oxide fuel cells

Figure 3.10 A SOFC cell and its components.

this way the species and the energy sources due to the electrochemical reactions are added to the adjacent computational cells. Electrical interconnects are used to connect the cells in series and build the SOFC stack. The electrical interconnects for a planar SOFC stack are the bipolar plates, labeled in Fig. 3.10 as the current collectors. In the SOFC unresolved electrolyte model, the fluid flow, as well as the heat and mass transfer through the GCs and porous electrodes at two sides of anode and cathode, must be solved, which can be implemented via a similar CFD technique presented in the gas and heat transport submodels of PEMFCs in the previous chapter. In addition, the electric charge transport must be determined by solving the potential field equations (note that only electron transport should be determined, and there is no need to solve the ion transport equation due to the electrolyte zero thickness). As the final step, the electrochemical reactions taking place at the TPBs must be modeled. Therefore the following submodels are required for the 2D/3D simulation of a SOFC single cell: 1. gas transport submodel, 2. heat transport submodel, 3. electron transport submodel, and 4. electrochemical reaction submodel. However, these submodels in the resolved electrolyte submodel are solved considering no volume for the electrolyte and interlayers. We will present more detail about these submodels.

259

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3.3.2.1 Gas transport submodel The reactant and product gases (which are usually a mixture of hydrogen and water vapor in the anode and a mixture of oxygen and nitrogen in the cathode) must be exchanged between the GC and reactive sites (i.e., the anode and cathode TPBs). This fluidic path consists of both porous and nonporous parts. To model this gas transport, the governing equations presented in Parts a, b, e1, and f1 of Section 2.1.2 must be solved by FVM for the gas mixture. Note that the governing equations for the porous zones have extra source terms, as appeared on the RHS of Eqs. (2.55)–(2.57), and the velocity components are superficial ones in these governing equations. In addition, the mass transfer equations with effective diffusivities (Eq. (2.65)) must be solved for the species to determine the concentration of each species. In ANSYS Fluent solver, multicomponent diffusivities are used; besides, the effective diffusivities are related to ordinary diffusivities via ε/τ factor, where ε and τ are the porosity and tortuosity of the porous electrodes. On the interface of anode interlayer/anode electrode, a hydrogen sink term (equal to −i/2F) and a water vapor source term (equal to i/2F) must be taken into account (assuming pure hydrogen as the fuel), whereas on the interface of cathode interlayer/cathode electrode, an oxygen sink term (−i/4F ) must be taken into account.

3.3.2.2 Heat transport submodel In this submodel the energy equation is solved. This equation was previously presented for the solid zones (which are the two current collectors in Fig. 3.10) in Section 2.3.3, Part e (Eq. (2.290)), for the fluidic zones (which are the GCs at the two sides in Fig. 3.10) in Section 2.1.2, Part a (Eq. (2.22)), and for the porous zones (which are the two electrodes in Fig. 3.10) in Section 2.1.2, Part b (Eq. (2.62)). This equation must be solved accompanied by a source term in the solid zones and porous zones, which equals i2 Rohmic , where Rohmic is the area-specific ohmic electronic resistance of the zone (see Eq. (3.6) and the related discussions). To incorporate the heat generation due to the ionic resistance in the electrolyte and also other reversible and irreversible heat generations in the anode and cathode interlayers, the corresponding heat sources must be added together. Subsequently, the sum of these sources must be applied on the two interfaces at the two sides (i.e., the interface of anode electrode/anode interlayer and interface of cathode electrode/cathode interlayer); ANSYS Fluent resolved electrolyte model applies half of this sum on one interface and the other half on the other [8].

3.3.2.3 Electron transport submodel To determine the current vectors in the solid and porous zones, the following electric potential field must be solved:  · (σ ∇φ)  = 0, ∇

(3.50)

Solid oxide fuel cells

where σ is the electron conductivity of the zone (equal to the electric conductivity of the solid matrix of a porous layer in the electrode zones and the electric conductivity of solid material of electric interconnectors in the current collectors). The obtained  will represent the current vectors. To solve Eq. (3.50), boundary conditions at two σ ∇φ sides must be determined. For the exterior surfaces of the cells (i.e., horizontal faces in Fig. 3.10), two different boundary conditions are applicable: • If the cell voltage (Vcell ) is known, then the condition φ = 0 is used for the anode side (bottom horizontal surface in Fig. 3.10), and φ = Vcell for the cathode side (top horizontal surface in Fig. 3.10). ∂φ icell • If the cell current density (icell ) is known, then the conditions = for the ∂n σCC ∂φ icell cathode exterior face and =− for the anode exterior face are used (σCC ∂n σCC denotes the electric conductivity of current collector). Between the two interfaces of the resolved electrolyte model (interface of electrode/interlayer at the anode side and interface of electrode/interlayer at the cathode side), a potential jump must be considered: φintf, c = φintf, a + φjump ,

(3.51)

anode cathode elec φjump = E − ηact − ηact − ηohmic .

(3.52)

where anode , ηcathode , In the last equation, E is the reversible voltage of the cell, whereas ηact act elec and ηohmic are the anode activation overpotential, cathode activation overpotential, and ohmic loss of electrolyte, respectively. In fact, the difference between φjump and the cell actual voltage is equal to the ohmic loss of electrodes and current collectors, which can be determined by solving Eq. (3.50) when the cell current density on the current collectors is known.

3.3.2.4 Electrochemical reaction submodel ANSYS Fluent resolved electrolyte model solves the full version of the Butler–Volmer equation for two electrodes in the electrochemical reaction submodel [8]. According to this model, the current density is related to the anode and cathode activation overpotentials by 







XH2 γH2 XH2 O γH2 O XH2 , ref XH2 O, ref  "  anode anode  anode # 2αa F ηact 2α anode F ηact − exp − c , (3.53) × exp RT RT   "  cathode cathode   cathode # XO2 γO2 4αa F ηact 4αccathode F ηact c exp − exp − , (3.54) i = i0,ref XO2 , ref RT RT i = i0a ,ref

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Table 3.1 Sink or source terms of species mass for the anode electrode supplied by an H2 –CO mixture. Species

Sink/source

H2

−

β

2Fi

CO (1 − β) −

2Fi

H2 O β +

2Fi

CO2 (1 − β) +

2Fi

where αa and αc denote the transfer coefficients for the forward and backward reactions, Xs are the species mole fractions, and γ s are the species concentration exponents. In the electrochemical reaction submodel, the current density i is known, whereas the anode and cathode activation overpotentials are unknown. To determine these overpotentials, Eqs. (3.53) and (3.54) can be solved via a root-finding procedure such as the Newton method. The presented electrochemical reaction submodel is based on using pure hydrogen as the SOFC fuel. However, one of the key advantages of SOFCs is fuel adaptation. If a mixture of hydrogen and carbon monoxide is introduced in the anode side, the following hydrogen content ratio can be defined: β=

XH2 . XH2 + XCO

(3.55)

This hydrogen content ratio can be used to modify the source/sink terms of species mass in the mass transport equations. By this ratio the sinks of hydrogen and carbon monoxide, as well as the sources of water vapor and carbon dioxide on the anode side, are related to the current density, as presented in Table 3.1. Recall that the carbon monoxide oxidation reaction in the anode electrode is defined by Eq. (3.4). The kinetic of this oxidation reaction can be taken into account by the Butler–Volmer equation similar to Eq. (3.53). If multiple SOFC cells are stacked together, the mentioned submodels in this section must be implemented for each cell. However, since this can lead to an increase in computational cost, the following approaches can be followed to reduce the computational cost: 1. using binary diffusivities in the gas transport submodel (DH2 , H2 O in the anode side and DO2 , N2 in the cathode side), 2. neglecting the electronic ohmic losses due to their smaller values in comparison to the ionic ohmic losses (i.e., adopting a constant uniform electric potential for electrode and current collector of each side instead of employing the electron transport submodel), and

Solid oxide fuel cells

3. using simpler forms of Butler–Volmer equations in the electrochemical reaction submodel. To further lessen the computational cost and build a stack model that can be used for predicting and controlling a SOFC stack, 1D models of cells can be used instead of 2D/3D CFD-based models of cells. However, note that these 1D models provide less detailed output information in comparison to the CFD-based models. The gas depletion in a stack is considerably higher than in a single cell, and hence, when 2D/3D models are used, the gas transport submodel must have sufficient accuracy, and when 1D models are used, the 1D model must be able to incorporate the gas depletion effects. The arrangement of the cells in the stack and the type of fuel and air manifolds alter the degree of gas depletion in the GCs. The effect of these parameters on the gas depleting is quite similar to the case of PEMFC stacks, previously discussed in Section 2.3.4.

3.3.3 Modeling of a SOFC system Employing only a SOFC stack is not sufficient for power supply; in fact, auxiliary subsystems such as the gas supply subsystem, electric power converter, system controller, and thermal management subsystem are required for the suitable operation of a SOFC system, which is called the balance of plant (BOP) (see Fig. 3.11). Most of these systems are similar to the PEMFC BOP subsystems. However, due to the high-temperature operation of SOFCs, the thermal management system is essentially more complicated. Hence, in the remaining part of this section, we discuss only this SOFC BOP subsystem.

Figure 3.11 Schematic of the BOP for a SOFC system.

To make the YSZ electrolyte of SOFCs become ion conductive, its temperature should reach a minimum value; i.e., below a threshold temperature, there is no oper-

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ation, and, consequently, there is no internal heating. Hence a preheating system such as a preburner is mandatory in the thermal management subsystem of a SOFC system. However, starting the operation, the heat is generated in the cells due to the exothermic nature of SOFC overall reaction and also the voltage losses, which emerge as the generation of heat. This generated heat in the cells must be actively ejected from the cells to prevent thermal abuse, especially in large stacks. Fortunately, this heat has a high quality (due to the high temperature of the exhaust gas stream) and can be effectively recovered, e.g., in a CHP system, which will be explained in Chapter 7. Another function of the thermal management subsystem in a SOFC system is to lessen the temperature gradients in the stack. In fact, due to the brittle attribute of SOFC electrolyte, such temperature gradients can lead to destructive stresses, causing crack emergence and durability fading of the system. In some traditional SOFC systems, a considerable excess air is delivered to the stack to lessen these gradients, which increases the parasitic loads of blowers and compressors. Highly effective heat transfer elements such as high-temperature heat pipes can be used for enhancing the heat transfer from the stack to the environment and providing a more uniform temperature distribution in the stack. The heat recovery from a SOFC stack can have several cold and hot streams, especially when the SOFC is equipped with a reformer, which usually experiences an exothermic reaction. This makes the analysis of the thermal management subsystem a bit complicated. Pinch point analysis is a famous and powerful tool for the numerical evaluation of such a case. This analysis consists of the following steps [9]: 1. identifying the cold gas and hot gas streams, 2. thermal data acquisition about each stream, 3. setting acceptable minimum temperature difference (Tmin,set ) between a hot and a cold stream (often, from 3 to 40 ◦ C temperature difference is recognized as the acceptable range), 4. Constructing temperature–enthalpy diagrams and checking that the pinch point temperature is observed between hot and cold streams (Tmin ≥ Tmin,set ), 5. if Tmin < Tmin,set , then changing the HEX orientation, and 6. conducting scenario analysis of HEX orientation until Tmin ≥ Tmin,set . More details about this method can be found in Ref. [9].

3.4. Modeling of a solid oxide electrolyzer cell (SOEC) Today, solid oxide electrolyzers are widely used for the pure and clean production of high-pressure hydrogen. The components of a SOEC are the same as the components of a SOFC. However, a SOEC is a power-consuming device and not a power-supplying one. In fact, in SOECs the electrochemical half-reactions are conducted in the reverse

Solid oxide fuel cells

direction. For SOECs, the overpotentials are negative, i.e., the applied voltage on the cell is larger than the thermodynamic reversible voltage. The 2D/3D modeling of SOECs is similar to SOFCs, with only the following differences to consider: 1. In the gas transport submodel, the sign of species mass sinks/sources must be reversed. 2. In the heat transport submodel, only the sign of reversible heat generation source (which equals T s) must be changed. 3. In the electron transport and electrochemical reaction submodels, the signs of activation overpotentials and ohmic voltage losses must be reversed.

3.5. Summary In this chapter, we presented and explained two famous techniques for the microstructure reconstruction of SOFC anode, the stochastic and FIB/SEM techniques. The presented stochastic technique is based on the two-point statistical function, which can take into account the neighboring information. Afterward, the LB simulation of reactive fuel flow through a SOFC anode is described, which is a well-known numerical tool for the microscale simulation of SOFC electrodes. In the next section, we presented a 1D model of a SOFC single cell. This simple model can provide the polarization curve of a SOFC single cell with sufficient accuracy for controller design applications. After that, a CFD-based model of a SOFC single cell, named electrolyte resolved model, is presented, which includes four submodels. A discussion about employing these cell models for the simulation of a SOFC stack is also expressed, and then SOFC BOP, with a focus on the thermal management subsystem analysis, is presented. In the last section of this chapter, we explained the CFD-based simulation of a solid oxide electrolyzer cell (SOEC).

3.6. Problems 1. Why the X-ray CT is not a suitable technique for the reconstructions of SOFCs? 2. How is FIB/SEM reconstruction technique implemented? 3. Summarize in a few steps the stochastic reconstruction procedure for the SOFC anode presented by Suzue et al. [12]. 4. What does “unresolved electrolyte” mean in the model presented in Section 3.3.2? 5. What are the main differences between the 2D/3D modeling of a SOFC and a SOEC?

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6. Using the presented 1D model in Section 3.3.1, provide the polarization curve for the SOFC single cell with the features shown in the following table: Physical properties

Values

Unit

cell Reversible voltage of the cell, Erev A Anode thickness, t Cathode thickness, tC Electrolyte thickness, tM Temperature, T Inlet mole fraction of hydrogen, XH2 Inlet mole fraction of oxygen, XO2 Inlet mole fraction of water vapor (cathode side), XH2 O Pressure at cathode, P C Pressure at anode, P A eff Effective hydrogen (or water) diffusivity, DH 2 , H2 O eff Effective oxygen (or water) diffusivity, DO2 , H2 O Transfer coefficient, α Exchange current density, j0

1.0 75 750 25 1023 0.92 0.19 0.1 1.2 1.2 1 × 10−4 2 × 10−5 0.5 0.15

V µm µm µm K atm atm m2 s−1 m2 s−1 A cm−2

7. Can you generalize the presented 1D model in Section 3.3.1 to take into account the gas depletion effects?

References [1] Pietro Asinari, Michele Calì Quaglia, Michael R. von Spakovsky, Bhavani V. Kasula, Direct numerical calculation of the kinematic tortuosity of reactive mixture flow in the anode layer of solid oxide fuel cells by the lattice Boltzmann method, Journal of Power Sources 170 (2) (2007) 359–375. [2] Allen J. Bard, Larry R. Faulkner, Electrochemical methods: Fundamentals and applications, Surface Technology 20 (1) (1983) 91–92. [3] Dale P. Bentz, Nicos S. Martys, Hydraulic radius and transport in reconstructed model threedimensional porous media, Transport in Porous Media 17 (3) (1994) 221–238. [4] B. de Boer, SOFC anode: Hydrogen oxidation at porous nickel and nickel/zirconia electrodes, PhD thesis, Faculty of Science and Technology, University of Twente, The Netherlands, October 1998, https://research.utwente.nl/en/organisations/inorganic-materials-science. [5] Kyle N. Grew, Abhijit S. Joshi, Aldo A. Peracchio, Wilson K.S. Chiu, Pore-scale investigation of mass transport and electrochemistry in a solid oxide fuel cell anode, Journal of Power Sources 195 (8) (2010) 2331–2345. [6] Abhijit S. Joshi, Aldo A. Peracchio, Kyle N. Grew, Wilson K.S. Chiu, Lattice Boltzmann method for continuum, multi-component mass diffusion in complex 2D geometries, Journal of Physics D: Applied Physics 40 (9) (2007) 2961. [7] K.-R. Lee, S.H. Choi, J. Kim, H.-W. Lee, J.-H. Lee, Viable image analyzing method to characterize the microstructure and the properties of the Ni/YSZ cermet anode of SOFC, Journal of Power Sources 140 (2) (2005) 226–234. [8] UDF Manual, ANSYS fluent 12.0. Theory Guide, 2009. [9] Ryan O’hayre, Suk-Won Cha, Whitney Colella, Fritz B. Prinz, Fuel Cell Fundamentals, John Wiley & Sons, 2016. [10] Jacques A. Quiblier, A new three-dimensional modeling technique for studying porous media, Journal of Colloid and Interface Science 98 (1) (1984) 84–102.

Solid oxide fuel cells

[11] Naoki Shikazono, Daisuke Kanno, Katsuhisa Matsuzaki, Hisanori Teshima, Shinji Sumino, Nobuhide Kasagi, Numerical assessment of SOFC anode polarization based on three-dimensional model microstructure reconstructed from FIB-SEM images, Journal of the Electrochemical Society 157 (5) (2010) B665. [12] Yoshinori Suzue, Naoki Shikazono, Nobuhide Kasagi, Micro modeling of solid oxide fuel cell anode based on stochastic reconstruction, Journal of Power Sources 184 (1) (2008) 52–59. [13] C.L.Y. Yeong, Salvatore Torquato, Reconstructing random media, Physical Review E 57 (1) (1998) 495.

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CHAPTER 4

Hydrogen storage systems Contents 4.1. Introduction 4.2. High-pressure tanks 4.3. Hydrogen-absorbing tank 4.3.1 1D modeling 4.3.2 CFD simulation 4.4. Summary 4.5. Questions and problems References

269 270 272 276 278 280 280 281

4.1. Introduction To use hydrogen as an energy carrier in various applications, a safe storage method as effective as a gasoline tank is needed [1]; at the same time, simple handling and low costs are also very important. 1 kg of hydrogen will occupy a volume of 12.15 m3 and an energy content of 33.5 kWh at STP conditions, whereas for the same amount of energy, the volume that gasoline takes is only 0.0038 m3 . Therefore, to make hydrogen a competitive energy carrier, increasing volumetric density must be ensured. There are considerations for large-scale industrial applications of hydrogen. The requirements that need more attention are good packaging, storing hydrogen, and transferring from the producing point to the application point. Therefore there is a great deal of need for research to ensure that various hydrogen storage materials are secure, trustworthy, and cost-effective. The stored energy in different kinds of fuels is generally stated based on weight (gravimetric energy density or energy density based on mass) or on volume (volumetric energy density or energy density based on volume). According to the chemical reaction, when one mole of hydrogen and half mole of oxygen react, water (in either liquid or gas form) is produced with the release of energy: 1 H2 + O2 ↔ H2 O + H(241.826 kJ). 2

(4.1)

Therefore 241.826 kJ energy will be released if 1 mole of H2 is consumed. The molar mass of hydrogen is 2.02 × 10−3 kg mol−1 , so for pure hydrogen, the gravimetric energy density can be expressed as ρM =

H

M

Fuel Cell Modeling and Simulation https://doi.org/10.1016/B978-0-32-385762-8.00008-7

= 119.716 MJ/kg.

(4.2) Copyright © 2023 Elsevier Inc. All rights reserved.

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At a pressure of 1 atm and temperature of 298.15 K (25 °C) (which is known as STP condition), 1 mole of hydrogen has a volume equal to 24.46 L, and the volumetric energy density of hydrogen can be expressed as ρV =

H

VM

= 9.89 MJ/m3 .

(4.3)

These two energy densities can be measured and presented for other fuels such as propane, methane, and gasoline in a similar way. Hydrogen has the greatest gravimetric energy density (i.e., energy density based on mass) among all fuels. On the other hand, hydrogen has almost the least volumetric energy density. For example, gasoline has a mass and volumetric energy density of 45.7 MJ kg−1 and 34,600 MJ m−3 , respectively. Gasoline has the highest volumetric energy density, which makes this fuel useful, although it has a smaller gravimetric energy density. In the case of practical applications, for example, a light-duty vehicle takes 10 gallons of gasoline to take a 300 mi ride. If hydrogen is used instead to drive the same distance, then a 3495-gallon tank of hydrogen is needed at STP conditions. It is almost not possible to take this volume of hydrogen to run a vehicle. Hence finding some ways to increase the volumetric energy density while maintaining a high gravimetric energy density is a critical problem in hydrogen usage. As previously mentioned in Section 2.1.3, there are five main classes of techniques for hydrogen storage: compression storage techniques, cryogenic storage techniques, chemical hydride storage techniques, metal hydride storage techniques, and carbonbased storage techniques. As the two classes of compression storage techniques and metal hydride storage techniques (employing high-pressure tanks and hydrogen absorbing tanks for hydrogen storage) seem more feasible in comparison to the other classes, in this chapter, we explain these two classes. Additionally, we discuss the modeling and simulation of a metal hydride tank.

4.2. High-pressure tanks The most common methodology of hydrogen storage until now is hydrogen compression. This method is currently widely used, mainly because the process is well understood. In this method, hydrogen is compressed and stored in high-pressure tanks, similar to compressed natural gas tanks. Usually, the compression pressures for hydrogen storage are normally from 20 to 25 MPa. However, for vehicular applications and in situ usage, the pressure is 70 MPa [2], which is an attractive storage due to having a high energy density, low weight, and low cost. For the first time, a pressure tank application for storing hydrogen was proposed in 1880. A wrought iron vessel at 12 MPa was used to store the hydrogen gas. Currently, there are four different types of pressures vessels for compressed hydrogen storage. The

Hydrogen storage systems

Figure 4.1 A hydrogen storage pressure vessel of fourth type [5].

first type is the metallic pressure vessel with a pressure of 20–30 MPa, the mostly used for industrial applications. There is a serious efficiency restriction on hydrogen storage for this type: it can only absorb up to 1 wt.% of hydrogen. In the second type the cylindrical part of the tank is wrapped with fiber resin composite. The next two types are fully composite overwrapped pressure vessels (COPVs), in which either plastic or carbon fibers embedded in the polymer matrix (filament winding) are used to make the composites. These two types differ in mechanical resistance, especially due to the liner design and material. In the third type the pressure vessel has a common metal liner, whereas in the fourth type, the pressure vessel liner is mainly from polymer with an extremely thin metal [3]. For high-pressure storage tanks (Fig. 4.1), the ideal material characteristics include [4] (I) high tensile strength, (II) low density, and (III) no reaction with hydrogen and no its diffusion. This technology, even at very high pressures (700–800 bar), has a weakness in low volumetric density; under the same conditions, the gasoline energy content is even higher. In addition, problems about safety are also a disadvantage according to the possible embrittlement of cylinders. Finally, there are the problems of considerable cost of the (mechanical) compression and the great deal of pressure drop inside the gas cylinder, which is needed when hydrogen is released (e.g., in a hydrogen fuel cell vehicle during the discharging process of the tank). Because of these deficiencies, there are difficulties found in the storage of hydrogen in the case of large-scale compression in automobiles. However, high-pressure hydrogen tanks can be suggested as an initial option for the commercialization of FCEVs.

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4.3. Hydrogen-absorbing tank Generally, in the case of hydrogen economy, storing methods need to be made safer, more efficient, more economical, and more compact. It is expected that hydrogen storage systems have a great capacity (both gravimetric and volumetric) to be responsive for the technological and economical aspects of vehicular applications of hydrogen. For this purpose, a lot of work is currently being done on hydrogen storage in the solid-state materials. A reversible absorption and desorption of hydrogen in the solid-state materials takes place, which makes them a promising option [6]. Solid-state storage systems are divided into two main categories, storage by physisorption and storage by chemisorption [7]. In physisorption or physical storage, hydrogen adsorption occurs by Van der Waals interactions and forces between hydrogen molecules and the surfaces of solids like in carbon-based materials such as fullerenes, activated carbon, carbon nanotubes, fibers, graphene, metal-organic frameworks (MOFs), hollow spheres, and newly added polymers of intrinsic microporosity (PIMs). Then by thermal stimulation or any other sufficient method the hydrogen desorption can happen whenever it is needed. Although these materials look very nice due to the reversibility and fast kinetics characteristics, there are some drawbacks. The low capacity for storing hydrogen in moderate situations, alongside with the requirement of extremely low temperatures for high storage capacity, places terrible restrictions on the usage of these materials in case of practical applications. Note that in the case of physical storage, the main challenges and possibilities lie in the study of the materials. In terms of chemical hydrogen adsorption, a hydride is formed by chemically bonding the hydrogen atoms with a material. Consequently, there is a chemical reaction step in the hydrogen adsorption and desorption process before the hydrogen is stored inside the substance and the following development of chemical bonds. As a result, in the case of chemical adsorption to form a hydride, hydrogen makes strong bonds, and the key problems for the use of hydrides are based on the enhancement of the kinetics and thermodynamics of adsorption/desorption cycles. Chemical hydrogen storage shows one of the best futures among methods for storing hydrogen in the solid state. Although similar materials are applied in the case of both physical adsorption of hydrogen molecules and chemical adsorption of hydrogen atoms at the hydrides interface, these two systems are significantly different from each other. The confusion made by the similar names, “hydrogen storage” has been a difficult topic to understand. Whereas the previous method stores the hydrogen molecules physically inside the large surface area of solid material, in the chemical storage method, hydrogen atoms make atomic bonds. There are a lot of metals and alloys that can store hydrogen by chemisorption and make a solid metal hydride. In most cases the bond between the metal and hydrogen is very strong. There are some metal hydrides that have an even higher volumetric storage

Hydrogen storage systems

ability than liquid hydrogen. Because of this, the metal hydride (MH) receives a great deal of attention for in situ applications of hydrogen storage. A comprehensive comparison of different materials as a matter of both volumetric and gravimetric storage capacities is presented by Züttel [8]. Generally, as it was mentioned above, the volumetric energy density for most metal hydrides is higher compared to liquid hydrogen, but there is still the matter of low gravimetric energy density. In addition, metal hydrides can store hydrogen at low pressures, so they are more attractive in case of safety. Metal hydrides need no complex container to store since they are in solid-state form, and this makes metal hydrides even more desirable. However, both metal hydride development and hydrogen desorption step are chemical reactions in which the breaking or development of hydrogen-metal bonds occurs, and the hydrogen atoms regularly conquer interstitial sites of metal. Therefore investigating the metal hydrides with lightweight and then modifying their reaction kinetic and thermodynamic properties seem to be very interesting. Prachi et al. [9] and Demirba¸s [10] reported that the essential requirements for the hydrogen storage process are the thermodynamic and kinetic conditions. Under this situation, a metal stores the hydrogen gas to the point where equilibrium is touched. Many reaction steps exist that can kinetically delay and slow down a hydrogen storage system from getting to its thermodynamic equilibrium in hydrogen storage within a rational duration. Therefore the reaction rate in a hydride storage system works as a function of temperature and pressure. In case of metal hydrides, hydrogen storage contains some stages in the mechanism and depends on some factors. The essential property for the exterior layer of the metal is the dissociation capability of the hydrogen molecules and allowing fluent motion of hydrogen atoms for hydrogen storage. Considering the surface structures, morphologies, and purities of different metals, each of them has special facilities to dissociate hydrogen molecules. A simple storage model for a metal hydride is shown in Fig. 4.2. The hydrogen adsorption and desorption are the mutually inverse processes in a metal hydride. The storage in hydrides only takes place at a range of pressure and a specific temperature (preferred to be at room temperature), which depends on the thermodynamic nature of the metal hydride and metal. The following reaction indicates how the metal or the metal alloy (M) forms a metal hydride by exposure to hydrogen: x M + H2 ↔ MHx + Q, 2

(4.4)

where Q is the hydride formation heat. Generally, hydrogen desorption is an endothermic process, whereas hydrogen adsorption is an exothermic one. Practically, the adsorption process occurs at high hydrogen pressure, whereas desorption happens at low pressure. A typical pressure–composition–temperature (P–C–T) curve is shown in Fig. 4.3 during hydrogen adsorption and desorption cycle. By increasing the hydrogen pressure

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Figure 4.2 A simplified model of a metal hydride hydrogen storage [11].

Figure 4.3 The absorption and desorption curves at a fixed temperature for a typical MH tank [12].

the metal begins to store hydrogen and finally forms a solid metal–hydrogen solution (α -phase). According to this figure, as soon as the pressure value hits point A on the diagram, the hydride formation of the metal gets started (β -phase). This process occurs at an almost constant hydrogen pressure Pa , and the stored amount of hydrogen in the hydride increases significantly. At point B, the adsorption process is over. The A–B adsorption line shows the characteristics of the effective capacity for hydrogen storage at a certain temperature. Normally, the adsorption plateau pressure increases with increasing temperature and follows the Van ’t Hoff equation ln(P ) =

H

RT

−

S

R

,

(4.5)

Hydrogen storage systems

Figure 4.4 The Van ’t Hoff data for a few MH tanks [13].

where T is the temperature, P is the hydrogen pressure, S and H are the entropy and enthalpy of hydrogen adsorption or desorption, and R is the universal gas constant. As shown in Fig. 4.4, the formation heat can be seen by drawing ln(P ) versus 1/T (Van ’t Hoff diagram) [13]. As shown previously in Fig. 4.3, the desorption and adsorption processes are reversible. It also displays a desorption line with a hydrogen pressure, which is lower and almost constant (Pd ). The desorption and adsorption lines create a hysteresis loop for the hydrogen adsorption and desorption cycle, and this hysteresis effect creates a free energy difference given by 



Pa . Ghyst = RT ln Pd

(4.6)

The hysteresis effect shows a deficiency in the storage cycle because the degradations of materials during hydrogen adsorption/desorption cycles have an irreversible effect. Obviously, a small hysteresis as much as possible should theoretically lead to a decent hydride material. In the case of specifying different properties in metal hydrides, the kinetic rates of hydrogen uptake and release are the most important factors. The rates of these reactions are proportional to the hydrogen sorption details in terms of kinetics, sorption temperature, and pressure. In a microscopic scale the Lennard-Jones potential model can describe this adsorption process, which in fact includes a few steps. A molecule of hydrogen gas faces a very low potential as soon as it gets close to the metal surface, mainly due to atomic adsorption, molecular adsorption, and bulk adsorption. Because of electrostatic attraction or Van der Waals forces, hydrogen is physically absorbed at first

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by the metal surface. At room temperature, this kind of bonding normally does not lead to substantial absorption since it is so weak. The absorbed molecule of hydrogen can be dissociated at effectively high pressure and temperature at the surface of the metal by transmitting an electron between the two elements, and, finally, the hydrogen is chemically absorbed. Because of the energy requirement for dissociation, this process might need activation by thermal energy or a catalyst. The hydrogen atoms travel under the surface sites after the chemical adsorption stage. The diffusion of these atoms through the material happens quickly, and, finally, hydrogen becomes a solution in the metal media. A metal hydride phase with more stability (β -phase) is achieved if the hydrogen accumulation in the α -phase enhances [4]. Metal hydrides can be classified into two subclasses, low- and high-temperature hydrides. This classification is based on the hydrogen absorption and desorption temperature according to Lai et al. [14] and Khafidz et al. [15]. In the case of low-temperature hydrides, hydrogen atoms bond by covalent bonding. As a result, the low-temperature hydride includes material with high molecular weight, which reveals low pressures of hydrogen equilibrium and rapid kinetics because of their low reaction heat. Generally, hydrides made of intermetallic alloy or solid solution are included in the lowtemperature hydrides, which can operate at moderate temperatures. On the other hand, the high-temperature hydrides usually store hydrogen by ionic bonds, so the metal hydride has material with low molecular weight. Because the high-temperature hydrides have relatively higher capacities for hydrogen storage compared with the lowtemperature hydrides, these hydrides are considered a more promising choice though the high temperature puts a restriction on their applications.

4.3.1 1D modeling A good review about the MH tank modeling and simulation is provided by Mohammadshahi et al. [16] in 2015. In 2018, Abdin et al. [17] make a more recent review of the MH tank modeling and simulations. The final outcome of their review is that to choose a model for an MH tank, the complexity of the internal structure of the tank, the adopted way for considering the effective thermal conductivity of the MH bed, and the relation between pressure and composition at different temperatures must be taken into account. They also present a 1D model for an MH tank in their paper [17]. They modeled a long MH cylinder with axial internal symmetry and without any internal structure. They used the following governing equations. a) Conservation of mass Let the hydrogen mass flow rate into the system be ∅˙ per unit volume. For zero hydrogen flow rate (∅˙ = 0), during desorption, the local MH density ρs decreases, causing the local hydrogen gas density ρg to increase, and vice versa for adsorption. Therefore the mass balance equations for the gas and solid for hydrogen in the gas phase (i.e., in

Hydrogen storage systems

the pores of the porous MH tank) are as follows: ε

∂ρg 1 ∂(r ρg V ) ˙ + = ∓m ˙ ± ∅, ∂t r ∂r

(4.7)

and in the solid phase, (1 − ε)

∂ρs = ±m ˙, ∂t

(4.8)

where in the case of double signs, the upper sign indicates absorption, and the lower sign indicates desorption; m˙ is the mass source term of reaction per unit time and per unit volume (i.e., the hydrogen consumption during the adsorption), V is the superficial flow velocity of gas, and ε is the porosity. Assuming ideal gas behavior, which is reasonable PM for the pressures typically encountered in MH tanks [18], the gas density is ρg = Rg HT2 . The mass source term m˙ can be written as follows [19]: dF , dt dF Desorption: m˙ = (1 − ε)(ρs − ρs,in ) . dt Absorption:

m˙ = (1 − ε)(ρsat − ρs )

(4.9) (4.10)

Here ρs is the density of the solid phase (metal or hydride), ρsat is the density of the solid phase when saturated with hydrogen, ρs,in is the density of the empty metal alloys, and dF dt is the rate of reaction. Eq. (4.6) assumes that the density of the MH particles changes linearly with reacted fraction, i.e., it obeys Vegard’s law on average. b) Conservation of energy Assuming local thermal equilibrium, Tg = T = Ts , and a single common energy balance equation can be written as   ∂T 1∂ ∂T ∂T − ρg Cpg v (ρ Cp )eff = keff r ±m ˙ H , ∂t r ∂r ∂r ∂r

(4.11)

where keff is the effective thermal conductivity of the MH bed, H is the enthalpy of phase conversion, which is assumed to be constant, and (ρ Cp )eff is the effective thermal capacity of the MH bed, which can be expressed by a porosity-weighted function of the same quantities for the gas and MH as   (ρ CP )eff = ε(ρ CP )g + (1 − ε) ρ Cp s .

(4.12)

c) Initial and boundary conditions A proper set of boundary conditions must be defined to solve Eqs. (4.7), (4.8), and (4.11). For the present scenario, the applicable conditions are as follows. Initials conditions:

P (r , O) = Pin ,

T (r , o) = Tin .

(4.13)

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Boundary conditions: Cylindrical symmetry imposes an adiabatic boundary condition at r = 0, ∂ T  (4.14)  (0, t) = 0. ∂ r re At the external radius of the MH bed r = R, the continuity of the heat flux between the MH tank and the heat exchange fluid requires − keff

∂ T  (R, t) = hf [T (R, t) − TF ] ,  ∂ r r =R

(4.15)

where hf is the convective heat transfer coefficient. Note that for absorption with rate constant ka , F for calculating the reaction rate is 



F = 1 − exp −ka t , and for desorption,





F = exp −kd t ,

(4.16)

(4.17)

where ka and kd are the rate constants for absorption and desorption, respectively. Abdin et al. [17] solved the presented governing equations by Matlab® -Simulink and compared their results with experimental data. More details about their calculations and results are presented in [17].

4.3.2 CFD simulation In this section, we want to explain the CFD simulation of a Mg2 Ni porous MH tank as an absorbing MH tank. The absorption and desorption of such a tank can be represented by the following reaction: Heat release/absorb

Mg2 Ni + 2H2 ←−−−−−−−−→ Mg2 NiH4 ± 2 · H [kJ/mol H2 ].

(4.18)

To that end, we first introduce the governing equations and then propose a numerical procedure for solving them via ANSYS Fluent software package. Note that before expressing the governing equations, the following assumptions for a common CFD simulation of an MH tank are useful: • Hydrogen is an ideal gas, which is a reasonable assumption due to the modest pressure of an absorbing tank. • The hydrogen gas in the pores and the solid matrix of MH tank are locally in thermal equilibrium [20]. • Radiation can be neglected since it has a minor role in comparison to convection. • Thermophysical properties are assumed to be constant due to the almost constant temperature of the MH tank.

Hydrogen storage systems

The thermal resistance and heat capacity of the canister can be neglected compared to those of the MH bed [21,22]. • The MH is a homogeneousand isotropic porous medium [23]. If we denote the fraction of absorbed hydrogen mass in the tank by X (i.e., X is the ratio of the hydrogen mass absorbed by the solid phase to the solid phase mass when it is out of hydrogen), then: •



Adsorption: Desorption:





PH2 − Pa,eq Ea dX = Ca exp − (Xmax − X ) ; dt RT Pa,eq    PH2 − Pd,eq Ed dX = Cd exp − X. dt RT Pd,eq

(4.19) (4.20)

In these kinetic equations, PH2 is the local hydrogen pressure, Ca and Cd are the rate constants of adsorption and desorption processes, Ea and Ed are the activation energies of adsorption and desorption processes, and Pa,eq and Pd,eq are the equilibrium pressures of adsorption and desorption processes, respectively. To calculate Pa,eq and Pd,eq , the Van ’t Hoff equations can be used: 

Adsorption: Desorption:



Pa,eq Ba ; = 10−5 exp Aa − Pref T   Pd,eq Bd . = 10−5 exp Ad − Pref T

(4.21) (4.22)

Here Pref is the reference pressure equal to 1 bar, T is temperature, and the other variables on the RHSs of the equations are constants, which are presented in Table 4.1 for an Mg2 Ni tank. Based on the calculated rates of absorption or desorption reactions, we can define two main source terms for the conservation of hydrogen mass in the solid phase (Sm ) and the conservation of energy (Se ): dX , dt ρMH (1 − εMH ) dX Se = HMH , MH 2 dt Sm = ρemp

(4.23) (4.24)

where ρemp is the density of the MH tank when it is empty, ρMH is the density if MH tank, εMH is the porosity of MH tank, and MMH is the enthalpy of reaction, as expressed in Eq. (4.18). ANSYS Fluent can solve conservation of mass and energy for a gas flow through a porous medium. In an MH tank the hydrogen gas is absorbed/desorbed by the solid matrix of the porous medium, which is accompanied by the generation or consumption of both mass and energy. Therefore, to simulate the adsorption or desorption

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Table 4.1 The values of parameters in the Van ’t Hoff equations and rate equations [24]. Parameter

Value

Aa in Eq. (4.21) Ba in Eq. (4.21) [K] Ad in Eq. (4.22) Bd in Eq. (4.22) [k] Ca in Eq. (4.19) [s−1 ] Cd in Eq. (4.20) [s−1 ] Ea in Eq. (4.19) [kJ mol−1 ] Ed in Eq. (4.20) [kJ mol−1 ]

26.481 7552.5 26.181 7552.5 175.31 5452.3 52.205 63.468

of an MH tank, it is only required to define a mass source and an energy source in ANSYS Fluent. However, these are not simple constant sources, and using user-defined functions (UDFs) is necessary for defining these sources. To define these sources, as a preliminary step, the rate of reactions must be calculated by Eqs. (4.19) and (4.20) via DEFINE_ADJUST macro. Afterward, two DEFINE_SOURCE macros can be used for calculating the mentioned two sources. An example UDF code for such calculations is provided in Appendix D.

4.4. Summary In this chapter, we introduced various methods for storing hydrogen. Special attention is paid to storing hydrogen in the high-pressure tanks and absorbing tanks. Four types of high-pressure tanks with different storing efficiencies are presented. Two types of absorbing tanks are presented (physically absorbing and chemically absorbing). Afterward, a 1D model for the numerical simulation of a cylindrical MH tank is presented. Finally, CFD simulation of an MH tank (manufactured from porous Mg2 Ni) in the FVM framework of ANSYS Fluent is presented. In this part, the calculation of reactions rates and the mass and energy sources via incorporating UDFs in ANSYS Fluent are explained. The required UDFs are provided in Appendix D.

4.5. Questions and problems Q1. Compare the properties of different materials that can be used for an absorbing tank. Suitable information can be found in [13]. P1. For a metal hydride tank that is a 50-cm-long cylinder of diameter 20 cm, adopt the UDFs in Appendix D and derive the pressure contours during charging by 150 bar hydrogen and discharging by 20 bar.

Hydrogen storage systems

P2. Redo the previous example but assume a tree-shaped fin structure inside the MH tank.

References [1] A. Da Rosa, Fundamentals of Renewable Energy Processes, Elsevier Inc., 2009, Epub ahead of print 2009. https://doi.org/10.1016/B978-0-12-374639-9.X0001-2. [2] Z. Zhang, C. Hu, System design and control strategy of the vehicles using hydrogen energy, International Journal of Hydrogen Energy 39 (24) (2014) 12973–12979. [3] H. Barthelemy, M. Weber, F. Barbier, Hydrogen storage: Recent improvements and industrial perspectives, International Journal of Hydrogen Energy 42 (2017) 7254–7262. [4] L. Schlapbach, A. Züttel, Hydrogen-storage materials for mobile applications, Nature 414 (2001) 353–358. [5] R. von Helmolt, U. Eberle, Fuel cell vehicles: Status 2007, Journal of Power Sources 165 (2007) 833–843. [6] G. Mazzolai, Perspectives and challenges for solid state hydrogen storage in automotive applications, Recent Patents on Materials Science 5 (2012) 137–148. [7] D.C. Elias, R.R. Nair, T.M.G. Mohiuddin, et al., Control of graphene’s properties by reversible hydrogenation: evidence for graphane, Science 323 (2009) 610–613. [8] A. Züttel, A. Remhof, A. Borgschulte, et al., Hydrogen: The future energy carrier, Philosophical Transactions - Royal Society. Mathematical, Physical and Engineering Sciences 368 (2010) 3329–3342. [9] R.P. Prachi, M.W. Mahesh, C.G. Aneesh, A review on solid state hydrogen storage material, Advances in Energy and Power 4 (2016) 11–22. [10] A. Demirba¸s, Fuel properties of hydrogen, liquefied petroleum gas (LPG), and compressed natural gas (CNG) for transportation, Energy Sources 24 (2002) 601–610. [11] J.O. Abe, A.P.I. Popoola, E. Ajenifuja, et al., Hydrogen energy, economy and storage: Review and recommendation, International Journal of Hydrogen Energy 44 (2019) 15072–15086. [12] J.Z. Zhang, J. Li, Y. Li, et al., Hydrogen Generation, Storage and Utilization, 2014, Epub ahead of print 2014. https://doi.org/10.1002/9781118875193. [13] A. Züttel, Materials for hydrogen storage, Materials Today 6 (2003) 24–33. [14] Q. Lai, M. Paskevicius, D.A. Sheppard, et al., Hydrogen storage materials for mobile and stationary applications: current state of the art, ChemSusChem 8 (2015) 2789–2825. [15] N.Z. Khafidz, Z. Yaakob, K.L. Lim, S.N. Timmiati, The kinetics of lightweight solid-state hydrogen storage materials: a review, International Journal of Hydrogen Energy 41 (30) (2016) 13131–13151. [16] S.S. Mohammadshahi, E.M. Gray, C.J. Webb, A review of mathematical modelling of metal-hydride systems for hydrogen storage applications, International Journal of Hydrogen Energy 41 (2016) 3470–3484. [17] Z. Abdin, C.J. Webb, E.M. Gray, One-dimensional metal-hydride tank model and simulation in Matlab-Simulink, International Journal of Hydrogen Energy 43 (2018) 5048–5067. [18] S.S. Mohammadshahi, T. Gould, E.M. Gray, et al., An improved model for metal-hydrogen storage tanks – Part 1: Model development, International Journal of Hydrogen Energy 41 (2016) 3537–3550. [19] P. Marty, J.-F. Fourmigue, P. De Rango, et al., Numerical simulation of heat and mass transfer during the absorption of hydrogen in a magnesium hydride, Energy Conversion and Management 47 (2006) 3632–3643. [20] A. Jemni, S. Ben Nasrallah, J. Lamloumi, Experimental and theoretical study of a metal–hydrogen reactor, International Journal of Hydrogen Energy 24 (1999) 631–644. [21] B.D. MacDonald, A.M. Rowe, Impacts of external heat transfer enhancements on metal hydride storage tanks, International Journal of Hydrogen Energy 31 (2006) 1721–1731. [22] F. Askri, M. Ben Salah, A. Jemni, et al., Optimization of hydrogen storage in metal-hydride tanks, International Journal of Hydrogen Energy 34 (2009) 897–905.

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[23] C.A. Chung, C.-J. Ho, Thermal–fluid behavior of the hydriding and dehydriding processes in a metal hydride hydrogen storage canister, International Journal of Hydrogen Energy 34 (2009) 4351–4364. [24] C.A. Chung, C.-S. Lin, Prediction of hydrogen desorption performance of Mg2 Ni hydride reactors, International Journal of Hydrogen Energy 34 (2009) 9409–9423.

CHAPTER 5

Fuel cell electric vehicles (FCEVs) Contents 5.1. Introduction 5.2. Vehicle dynamics 5.2.1 Resistant and traction forces 5.2.2 Vehicle performance 5.2.3 Vehicle energy consumption 5.3. FCEV configuration and components 5.3.1 PEMFC and battery module 5.3.2 Vehicle control unit (VCU) 5.3.3 Traction motor 5.3.4 PEMFC and hydrogen tank 5.4. Modeling and control of FCEVs 5.4.1 Performance characteristics 5.4.2 Energy consumption characteristics 5.5. Summary 5.6. Problems and questions References

283 284 284 287 288 289 289 291 292 293 294 295 296 299 300 301

5.1. Introduction A significant part of environmental concerns in recent years (such as global warming, air pollution in cities, etc.) stems from the traditional road vehicles, which consume gasoline, gas-oil, etc., in their internal combustion engines (ICEs) for producing traction power. In addition, these ICE vehicles consume a large amount of fossil fuel all over the world, which accelerates the fossil sources depletion and raises the international price of fossil fuels (recall that these fossil resources have definitely a finite volume, and the time will come when they will end). One of the key strategies to control environmental challenges and to mitigate energy restrictions is to widely develop and commercialize clean road vehicles, accompanied by outspreading of renewable infrastructure for electric power generation. To that end, fuel cells are one of the best choices for being employed in clean road vehicles and generating traction power. A clean vehicle that benefits from a fuel cell system is called a fuel cell vehicle (FCV) or fuel cell electric vehicle (FCEV). There are also other types of clean vehicles, such as battery electric vehicles (BEVs) or hybrid electric vehicles (HEVs). In all these clean vehicles, there is at least one electric motor; hence all these three classes of clean vehicles can be considered as different branches of electric vehicles (EVs). In an FCEV, the demanded electric power for the electric motor is supplied by a fuel cell system or a fuel system accompanied by a battery module, whereas in a Fuel Cell Modeling and Simulation https://doi.org/10.1016/B978-0-32-385762-8.00009-9

Copyright © 2023 Elsevier Inc. All rights reserved.

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BEV, it is supplied by a large battery pack, usually comprised of numerous Li-ion cells. HEVs have several types and configurations. In a plugin HEV (also called PHEV) with series configuration, the demanded electric power for the electric motor is supplied by a battery pack and an ICE/electric generator system. However, in a classic HEV with a series configuration, the demanded electric power is provided essentially by an ICE/electric generator system, and the battery module is only for shaving the peak of demanded powers. A fuel cell system generates electric energy rather than storing energy like a battery; in fact, as long as the cell is supplied by fuel, electric power is produced. Therefore, compared with BEVs, FCEVs have the advantage of a longer driving range with no need for tedious charging time of battery. On the other hand, in contrast to an ICE/electric generator system, a fuel cell system provides electric power with less emission and more efficiency. Therefore, compared with HEVs, FCEVs have the advantage of a more limited carbon footprint and reduced fuel consumption. In this chapter, to present the modeling procedure for an FCEV, we explain the key elements of an FCEV powertrain. However, beforehand, we provide a brief description of longitudinal vehicle dynamics, which is necessary for the powertrain modeling, simulation, and control.

5.2. Vehicle dynamics 5.2.1 Resistant and traction forces A vehicle can be broken down into several partitions (power train, body, chassis, suspension, etc.), where each partition consists of several systems (e.g., power train consists of engine, gearbox, differential, and drive-shafts in an ICE vehicle), each system includes several subsystems (e.g., the engine includes subsystems such as fuel injection subsystem, cooling subsystem, engine management subsystem, etc., in such a vehicle), and each subsystem comprises several parts and even subparts. Hence a vehicle is a complex set of elements that can be cascaded in different levels. A comprehensive analysis of the dynamic behavior of such a complex set along and about all three main axes in a 3D Cartesian coordinate system is not a simple task. Here we focus on only one vehicle motion: transitional movement along the road, which is also called the longitudinal dynamics. We neglect other transitional and rotational movements. Therefore, using the Newton second law, we can write the following relation for a vehicle: dV = dt







Ft − δM



Fr

,

(5.1)

where V denotes the vehicle speed, Ft and Fr indicate the resultants of traction forces (also called tractive effort) and resistant forces, respectively, M is the vehicle mass, and δ is the inertia factor (of value a bit larger than unity), which is used to compensate

Fuel cell electric vehicles (FCEVs)

Figure 5.1 Acting forces on a vehicle during uphill climbing.

the effects of neglecting the rotation of power train driveline. The resistant forces acting on a vehicle during uphill climbing are (Fig. 5.1) as follows: a) Drag resistant force stems from both the viscosity of air stream close to the vehicle body and the pressure difference between the front and rear zones of the vehicle and is proportional to the square of vehicle speed: 1 Fw = ρ Af CD (V − Vw )2 , 2

(5.2)

where ρ is the air density, Af is the frontal area of vehicle (i.e., the projected area of the vehicle on a plane normal to the vehicle movement direction), Vw is the wind velocity (consequently, (V − Vw ) is the vehicle velocity relative to the wind), and CD is the aerodynamic drag coefficient, which has a value of about 0.3 for conventional passenger cars. b) Grading resistant force appears as a resistant force during rising uphill and equals Fg = Mg sin α,

(5.3)

where α is the grade angle in rad. For small grade angles, this equation can also be written as Fg = Mg tan α.

(5.4)

c) Rolling resistant force stems from the hysteresis properties of tire material, which leads to an asymmetrical distribution of stress between tire and road, and has a complicated intrinsic. For more details on the physical nature of this force, see [1].

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This force can be formulated as Fr = Mgfr cos α,

(5.5)

where fr is the rolling resistance coefficient. This coefficient depends on numerous tire properties such as stiffness, structure, temperature, inflation pressure, and tread geometry, as well as road specifications such as roughness and wetness. A typical value of the rolling resistance coefficient on dry asphalt or concrete roads for conventional passenger cars is about 0.01. To overcome these forces, the power train of each vehicle powers (i.e., turns) the traction wheels, and at the interface between the tires of the traction wheels and the road, longitudinal traction forces are generated, which move the vehicle ahead. These traction forces stem from the friction phenomenon between the tires and roads. Hence, the value of total traction force will be Ft =

Tw , rd

(5.6)

where Tw denotes the applied torque on the wheels axis, and rd denotes the effective radius of the tire (i.e., the distance between the wheel center and the road). The generated torque by the electric motor is usually intensified in the vehicle transmission system (including gearbox and differential), and then it is delivered to the driven wheels. Thus Tw is proportional to Tem : Tw = Tem ig i0 ηt ,

(5.7)

where ig , i0 , and ηt are the gear ratio, final drive ratio, and transmission efficiency, respectively. The product ig i0 has a value larger than unity, whose specific value depends on the selected gear. By the variation of the electric motor torque in the range of 0 ≤ Tem ≤ Tem,max the traction force will vary in the range of 0 ≤ Ft ≤ Ft,max , where Ft,max =

Tem,max ig i0 ηt . rd

(5.8)

However, due to the frictional nature of the traction force, this force may be also restricted by the friction factor between the tire and road, which is also called the adhesive coefficient between the tire and road. More specifically, for a front-wheeldrive vehicle, Ft,max = μWf ,

(5.9)

Ft,max = μWr ,

(5.10)

for a rear-wheel-drive vehicle, it is

Fuel cell electric vehicles (FCEVs)

and for a 4-wheel-drive vehicle, Ft,max = μ(Wf + Wr ),

(5.11)

where Wf is the normal force acting from the road surface on the front wheels, and Wr is the normal force acting from the road surface on the rear wheels (Fig. 5.1). The resulted maximum traction force from one of the above three relations must be compared with the resulted traction force from Eq. (5.8); the smaller one would be the maximum traction force. The appeared friction factor in the above three equations is not a simple constant factor. In fact, it can significantly vary by the amount of tire deformation and tire slipping on the road. For more details on this complicated factor, see [1]. The values of Wf and Wr can be calculated by writing the balance of moments about the center of rear tire/road interface and center of front tire/road interface, respectively. After some manipulation and assuming that hw = hg (hw is the height of drag resistant force from the road, whereas hg is the height of the center of mass), we will have 



hg dV Lb rd , Wf = Mg cos α − Fw + Fg + Mgfr cos α + M L L hg dt   hg dV La rd , Fw + Fg + Mgfr cos α + M Wr = Mg cos α − L L hg dt

(5.12) (5.13)

where L denotes the distance between the axes (i.e., wheelbase), La is the longitudinal distance between the front axis and the center of mass, and Lb is the longitudinal distance between the rear axis and the center of mass. Note that the term M δ dV dt on the RHSs of the equations depends on the traction force, according to Eq. (5.1). Considering this fact, Eqs. (5.9)–(5.11) can be rewritten respectively as μMg cos α[Lb + fr (hg − rd )]/L , 1 + μhg /L μMg cos α[La + fr (hg − rd )]/L Ft,max = , 1 + μhg /L

Ft,max =

Ft,max = μMg cos α.

(5.14) (5.15) (5.16)

5.2.2 Vehicle performance To evaluate the performance of a vehicle, three main parameters are usually used: maximum speed, gradeability, and the time required for accelerating to a specific speed. We will briefly these three performance indexes. a) Maximum speed: when a vehicle is cruising at a constant speed on a flat road with no wind blowing in the environment, this constant speed has an upper limit, which is called the maximum speed of vehicle. At this limit the traction force will

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be equal to the sum of resistant forces, i.e., at this limit, no extra traction forces can be provided to accelerate and increase the vehicle speed. Therefore we can write Tem,max ig i0 ηt 1 = Mgfr + ρ CD Af V 2 . rd 2

(5.17)

To solve this equation and find V as the maximum speed of vehicle, we must caution that Tem,max in the LHS of the above equation is not constant and can vary with V . More specifically, Tem,max = Tem,max (ωem ), where the electric motor rotational speed ωem is proportional to the vehicle speed via ωem =

Vig i0 60 × rd 2π

(in rpm).

(5.18)

b) Gradeability: the maximum grade of an uphill road that can be climbed by a vehicle at a specific constant speed (e.g., 100 km h−1 ) is recognized as the vehicle gradeability. The angle of such a grade (α ) can be calculated by solving the trigonometric equation Tem,max ig i0 ηt 1 = Mgfr cos α + ρ CD Af V 2 + Mg sin α. (5.19) rd 2 After solving this equation and finding α , the vehicle gradeability tan α can be expressed. c) Acceleration: to evaluate the acceleration capability of a vehicle, the minimum required time for the vehicle to speed up to a specific velocity is reported instead of the maximum average acceleration of the vehicle. This time, called the acceleration time can be calculated by 

ta = 0

Vf



Tem,max ig i0 ηt rd



Mδ − Mgfr − 12 ρ CD Af V 2

dV .

(5.20)

Note that Tem,max in the denominator of the integrant is a function of velocity. Hence calculating the above integral needs numerical integration tools.

5.2.3 Vehicle energy consumption In conventional ICE vehicles the energy consumption attribute is usually reported as the volume of consumed fuel per a specific mileage, e.g., liter of consumed gasoline per 100 km mileage. In EVs, it is more conventional to report the energy consumption attribute as the consumed energy in kWh (and not kJ) per a specific mileage (usually, 100 km). To calculate this attribute for a vehicle, two steps are required: evaluating the energy consumption characteristics of the vehicle power source (e.g., evaluating the fuel economy map of an ICE in an ICE vehicle or evaluating the fuel economy map of a

Fuel cell electric vehicles (FCEVs)

PEMFC system in an FCEV) and adopting a standard pattern for the vehicle movement, called the driving cycle. The former will be discussed in Section 5.3.1. About the latter, driving cycles are simple charts expressing a vehicle velocity versus time. However, these simple charts can effectively impact the energy consumption characteristics of a vehicle (i.e., changing the driving cycle will alter the L/100 km of an ICE vehicle or kWh/100 km of an FCEV). In addition, the emission evaluations of vehicles are also implemented based on these driving cycles; therefore definition and adoption of an international driving cycle is at the center of the automotive industry attention. There are several driving cycles presented by the researchers and energy policymakers, from previous decades until now. A few of them, which are more widely used, are FTP75 (urban and highway), US06, NEDC, and WLTP. There are also some regional driving cycles presented in the literature, such as New York City Cycle (NYCC), Tehran driving cycle [2], etc. Also, there are some driving cycles developed for specific types of vehicles (taxi, bus, etc.). A complete set of numerous driving cycles is presented in [3]. A few general driving cycles are presented in Fig. 5.2.

5.3. FCEV configuration and components 5.3.1 PEMFC and battery module In an FCEV, a PEMFC system is used as a power source to generate electric power. This electric power supplies an electric motor to produce mechanical power, and the produced mechanical power is employed to drive the wheels via a transmission system. However, since the demanded power by the vehicle (which is a function of vehicle properties, driver expectations, and road situations) has a quite dynamic behavior (i.e., it experiences many sudden changes in usual driving), and the batteries have a better dynamic response in comparison to PEMFCs, usually, a battery module is also employed in an FCEV power train (Fig. 5.3). Here the role of this battery module supplying the dynamic part of the demanded power, whereas the role of PEMFC is supplying the static (i.e., average) part of the demanded power. By this division of tasks, when the demanded power is more than the PEMFC optimum power (Fig. 5.4), the battery module will be discharged and produce the remained part of demanded power. On the other hand, when the demanded power is less than the PEMFC optimum power, the battery module will be charged by the extra power from the PEMFC. Therefore, with a proper design and control strategy, the battery will never need to be charged from outside of the vehicle. Another advantage of employing a battery module in an FCEV power train is the capability of harvesting and storing electric energy from the mechanical inertia of vehicle during the braking process, which is called regenerative braking or electric braking. Since the output voltage of PEMFC and battery module may differ, an electronic inter-

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Figure 5.2 Famous driving cycles: a) FTP75 urban, b) FTP75 highway, c) NEDC, d) WLTP.

face between these two systems is required to provide electric power with a stabilized voltage for the electric motor.

Fuel cell electric vehicles (FCEVs)

Figure 5.2 (continued)

Figure 5.3 Power train configuration for an FCEV.

5.3.2 Vehicle control unit (VCU) A key element of an FCEV power train is the vehicle controller, which receives and analyzes signals from the vehicle and power train components (mainly PEMFC and battery module) and, based on its control strategy, decides and commands the electric motor to produce a specific torque and rotational speed to satisfy the driver expectation (either in the propulsion mode or in the braking mode). Another key element, also controlled by VCU, is the electronic interface. By controlling this interface the fraction of electric

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Figure 5.4 Characteristic curves for a typical PEMFC of a fuel cell system.

power required for the electric motor operation, supplied by either PEMFC or battery module, can be adjusted. This fraction strongly depends on the vehicle situation and the designed control strategy. A control strategy for such an FCEV is presented in [1]. The main objectives in this control strategy are: a) providing the desired traction power expected by driver, b) PEMFC operation in its optimum range (Fig. 5.4), and c) maintaining the battery state of charge (SOC) in a prescribed mid interval. When PEMFC system operates in its optimum range, its produced power will be larger than a minimum, which is called the minimum power of fuel cell system (Pfc−min ), and smaller than a maximum, which is recognized as the rated power of the fuel cell system (Pfc−rated ). Out of this interval, the PEMFC efficiency will be poor. The battery module also must operate in a midinterval of SOC (e.g., from 30% to 70%) to provide better net efficiency. Besides, when the SOC of the battery module is too low, it cannot provide a sufficient power in the peak of demanded powers; on the other hand, when the SOC of the battery module is too high, it cannot store a considerable amount of electric energy during regenerative braking. The details of conditions and decisions in this control strategy are presented in Fig. 5.5.

5.3.3 Traction motor Today’s advanced electric motors for EV application have considerably high power density with quite dynamic response. An ideal electric motor must be able to produce constant power at all speeds. However, since this is not possible at near-zero speeds (because motor torque cannot be infinity in practical), there should be a range of speeds 0 ≤ ω ≤ ωb for which the torque is constant; ωb is called the base speed. It is established that smaller ωb is more suitable, and it can lead to either lower motor power or a trans-

Fuel cell electric vehicles (FCEVs)

Figure 5.5 The control strategy for FCEV proposed by Ehsani et al. [1].

mission system with smaller number of gears [1]. Therefore the maximum torque of an advanced electric motor should have a characteristic such as that presented in Fig. 5.6. According to this figure, at each rotational speed the maximum torque of the electric motor Tem,max can be written as

Tem,max =

⎧ ⎨

1000Pr

⎩

1000Pr

ωb ω

×

60 2π

for 0 ≤ ω ≤ ωb ,

×

60 2π

for ωb < ω.

(5.21)

The torque obtained from this equation will be in Nm, whereas the rotational speeds are in rpm, and the rated power of the electric motor is in kW. The equation can be used for the performance calculation of an FCEV in Eqs. (5.17), (5.19), and (5.20).

5.3.4 PEMFC and hydrogen tank Although employing the non-PEM type of fuel cells in electric vehicles is reported (such as using a small SOFC stack by Nissan company), PEMFCs are the best type of

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Figure 5.6 Ideal maximum torque versus speed for a traction electric motor.

fuel cells for FCEV applications. High efficiency, low operating temperature, a better dynamic response in comparison to the dynamic response of other fuel cells, and high power density are a few advantages of these attractive fuel cells. However, PEMFCs employ noble metals such as Pt in their CLs, which are quite sensitive to CO. Even a few ppm of CO can poison the CLs. Hence providing and storing pure hydrogen with a sufficient amount of energy in FCEV is a crucial task, which is currently done by high-pressure storage tanks (previously, in situ hydrogen production via a reformer was also used in FCEVs, which is currently abundant due to its poor achievable purity of hydrogen and also emission production during reformation). As discussed in Chapter 2, Section 2.1.3, there are other methods for hydrogen storage, such as metal-hydride storage. However, these methods are either inefficient or at the research stage, and they are not commercialized. However, the hydrogen storage in high-pressure tanks has a few challenges such as ensuring safety, great required power for compressing hydrogen, and low energy density, which makes research and development for proposing new storage methods inevitable. For example, if hydrogen is stored in a high-pressure tank with 700 bar, then the energy per liter of hydrogen will be less than 2.0 kWh, equivalent to about 0.2 L of gasoline. In addition, it is estimated that about 25% of hydrogen energy must be consumed to increase its pressure to such high pressure.

5.4. Modeling and control of FCEVs In this section, we explain 1D modeling of a PEMFC by a simple example. Consider a FCEV with the features presented in Table 5.1. We want to obtain the performance and energy consumption characteristics of this vehicle.

Fuel cell electric vehicles (FCEVs)

Table 5.1 Features of the investigated FCEV. Parameter

Value

Total mass, M Frontal area, Af Drag coefficient, CD Rolling resistant coefficient, fr Effective radius of tires, rd Gear ratio × final drive ratio, ig i0 Efficiency of transmission system, ηt Inertia factor, δ Electric motor rated power Base speed of electric motor Maximum speed of electric motor PEMFC rated power Battery module rated power Battery module energy capacity

1200 kg 1.8 m2 0.3 0.01 0.22 m 3.0 0.95 1.1 75 kW 1250 rpm 5000 rpm 40 kW 35 kW 1 kWh

5.4.1 Performance characteristics To evaluate the maximum speed of vehicle, we first integrate Eq. (5.21) with Eq. (5.18) as follows: Tem,max =

⎧ ⎪ ⎨

1000Pr

⎪ ⎩

r   Vi1000P g i0 60 r × 2π

ωb

×

60 2π

= 573.25

×

60 2π

=

5500 V

for 0 ≤ ω ≤ 1250, for 1250 < ω.

(5.22)

d

Assuming that the vehicle maximum speed happens for 1250 < ω, Eq. (5.17) can be rewritten as 5500 V

r

5500 d V

ig i0 ηt

1 2

= Mgfr + ρ CD Af V 2

→

× 2.5 × 0.95 1 = 1200 × 9.81 × 0.01 + × 1.1 × 0.3 × 1.8 × V 2 . 0.22 2

(5.23)

This equation, which is an algebraic third-degree equation, can be easily solved by numerical solving tools such as MATLAB® software or WolframAlpha website. The results are Vmax = 56.22 m s−1

or 202.4 km h−1 .

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Now we use Eq. (5.19) to evaluate the vehicle gradeability at the constant speed of 100 km h−1 . Using a similar assumption for maximum speed, we have 5500

ig i0 ηt

V



rd 5500



(100/3.6)

1 2

= Mgfr cos α + ρ CD Af V 2 + Mg sin α

→

× 3 × 0.95

= 1200 × 9.81 × 0.01 × cos α   1 100 2 + 1200 × 9.81 × sin α. + × 1.1 × 0.3 × 1.8 × 2 3.6

0.22

(5.24)

By solving this trigonometric equation via a numerical solver we get α = 0.1898 rad = 10.875 deg.

Therefore the gradeability of the vehicle will be tan α = 19.21%. The third performance characteristic we want to calculate is the acceleration time. To that end, we must calculate the integral presented in Eq. (5.20). However, since Tem,max has two relations, the integral must be divided into two parts, a part for 0 ≤ V ≤ Vb and a part for Vb < V ≤ Vf , where Vb = ridgωi0b × 260π = 9.60 m s−1 and Vf = 100 3.6 = 27.78 m s−1 . Thus 

ta =

Vf



0

 =

Vb



0

 +

Vf

Vb

Tem,max ig i0 ηt rd

573.25ig i0 ηt rd

  5500 





Mδ − Mgfr − 12 ρ CD Af V 2

Mδ − Mgfr − 12 ρ CD Af V 2

ig i0 η t

V

rd



dV

dV

Mδ

dV .

(5.25)

− Mgfr − 12 ρ CD Af V 2

Using MATLAB for calculating the above integral, ta = xx + yy = zz s.

5.4.2 Energy consumption characteristics To evaluate the energy consumption of the proposed FCEV, a driving cycle must be selected. Here we employed WLTP. Afterward, the driving cycle must be discretized to N intervals, numbered 1, 2, . . . , i, . . . , N, where the interval i can be represented as ti−1 ≤ t ≤ ti (the vehicle velocities at two edges of each interval are known according to the driving cycle data set). In each interval the mean velocity of the vehicle and the

Fuel cell electric vehicles (FCEVs)

mean acceleration of the vehicle can be calculated by Vi + Vi−1 , 2 Vi − Vi−1 ai = . ti − ti−1

Vi =

(5.26) (5.27)

By these two values the traction power (i.e., the power produced by the electric motor for the vehicle propulsion) in each interval can be calculated via 

P t ,i = V i F t ,i = V i



1 2 Mgfr cos α + ρ CD Af V i + Mg sin α + M δ ai . 2

(5.28)

Note that when the vehicle brakes, ai is negative, and consequently, Pt,i may be negative. This negative power shows the available power, which can be stored in the battery module. However, in practice, only a fraction of this power can be stored, where this fraction is called the regenerative fraction. The required electric power for producing such a power will be P s, i =

P t ,i ηt ηem

,

(5.29)

where ηem is the efficiency of the electric motor. This required electric power is supplied by either PEMFC or battery module or both of them, depending on the amount of Ps,i and the SOC of the battery module according to the adopted control strategy. Here we employ the presented control strategy in Fig. 5.5. In addition, we assume that the minimum power Pfc,min for PEMFC in the optimum power region is 10 kW (Pfc,rated equals 40 kWh). Besides, we assume that the desired SOC interval for the battery module is 30% < SOC < 70%, i.e., Emin = 0.3 × 1 = 0.3 kWh and Emax = 0.7 × 1 = 0.7 kWh. Also, we assume that the initial battery SOC is 50%, i.e., E0 = 0.5 × 1 = 0.5 kWh. By this information, in each time interval, the PEMFC power Pfc,i , the battery module power Pbat,i , and the battery energy variation Ebat,i can be calculated as follows. a) For the propulsion mode (P⎧ s,i > 0): ⎪ Pfc,i = Pfc,rated , ⎪ ⎪ ⎨ P =P −P bat,i s, i fc ,rated , a-1) If Ps,i > Pfc,rated , then where PEMFC is on, and ⎪ Ebat,i = −Pbat,i ti , ⎪ ⎪ ⎩ E =E bat,i bat,i−1 + Ebat,i , the battery is discharging (Pbat,i > 0), i.e.,⎧we have the hybrid traction mode. ⎪ Pfc,i = Pfc,rated , ⎪ ⎪ ⎨ P =P −P bat,i s, i fc ,rated , where a-2) If Ps,i < Pfc,min and Ebat,i−1 < Emin , then ⎪  E = − P bat,i bat,i ti , ⎪ ⎪ ⎩ E =E bat,i bat,i−1 + Ebat,i , PEMFC is on, and the battery is charging (Pbat,i < 0).

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⎧ ⎪ Pfc,i = 0, ⎪ ⎪ ⎨ P =P , bat,i s, i a-3) If Ps,i < Pfc,min and Ebat,i−1 > Emin , then where ⎪  E = −Pbat,i ti , bat,i ⎪ ⎪ ⎩ E =E bat,i bat,i−1 + Ebat,i , PEMFC is off, and the battery is discharging (Pbat⎧,i > 0). ⎪ Pfc,i = Ps,i , ⎪ ⎪ ⎨ P = 0, bat,i a-4) If Pfc,min ≤ Ps,i < Pfc,rated and Ebat,i−1 > Emax , then where ⎪  Ebat,i = 0, ⎪ ⎪ ⎩ E =E bat,i bat,i−1 , only PEMFC is on (Pbat,i = 0). ⎧ ⎪ Pfc,i = Pfc,rated, ⎪ ⎪ ⎨ P =P −P bat,i s, i fc ,rated , a-5) If Pfc,min ≤ Ps,i < Pfc,rated and Ebat,i−1 ≤ Emax , then ⎪  E = − P bat,i bat,i ti , ⎪ ⎪ ⎩ E =E bat,i bat,i−1 + Ebat,i , where PEMFC is on, and the battery is charging (Pbat,i < 0). b) For the braking mode (Ps,i < 0): ⎧ ⎪ ⎪ Pfc,i = 0,   ⎪ ⎨ P = − P  × η 2 ,   (ηt em ) bat,i s, i b-1) If Ps,i × ηem  < Pem,rated and Ebat,i−1 < Emax , then ⎪ Ebat,i = −Pbat,i ti , ⎪ ⎪ ⎩ E =E bat,i bat,i−1 + Ebat,i , where only regenerative braking is active (Pmech−br⎧,i = 0). ⎪ Pfc,i = 0, ⎪ ⎪ ⎨ P = 0,   bat,i where b-2) If Ps,i × ηem  < Pem,rated and Ebat,i−1 ≥ Emax , then ⎪  Ebat,i = 0, ⎪ ⎪ ⎩ E =E bat,i−1 ,   bat,i only mechanical braking is active (Pmech−br ,i = P⎧s,i  × (ηt ηem )). ⎪ Pfc,i = 0, ⎪ ⎪ ⎨ P = −P   bat,i em,rated × ηem , b-3) If Ps,i × ηem  > Pem,rated and Ebat,i−1 < Emax , then ⎪  E = − Pbat,i ti , bat,i ⎪ ⎪ ⎩ E =E bat,i bat,i−1 + Ebat,i ,

where both regenerative braking and mechanical braking are active   (Pmech−br ,i = Ps,i  × (ηt ηem ) − Pem,rated /ηt ). ⎧ ⎪ Pfc,i = 0, ⎪ ⎪ ⎨ P = 0,   bat,i b-4) If Ps,i × ηem  > Pem,rated and Ebat,i−1 ≥ Emax , then where ⎪ Ebat,i = 0, ⎪ ⎪ ⎩ E =E bat,i−1 ,   bat,i   only mechanical braking is active (Pmech−br ,i = Ps,i × (ηt ηem )). This procedure, including five propulsion modes and four braking modes, can be implemented easily for a discretized driving cycle. We implement it for the WLTP driving cycle. The results are presented in Fig. 5.7. The hydrogen consumption of such

Fuel cell electric vehicles (FCEVs)

FCEV in the mentioned driving cycle can be calculated by mH2 =

N 

m˙ H2 ti =

i=1

N  i=1

Pfc,i   ti , ηfc,i hrxn 

(5.30)

where hrxn is the enthalpy of the PEMFC overall reaction (in kJ/kg H2 ), which is equal to −2147 kJ/kg H2 at 80 °C and 1 atm. The fuel cell efficiency ηfc is a function of PEMFC current density. Hence, according to Fig. 5.3, it can be related to the PEMFC power. Adopting the presented ηfc in Fig. 5.3, assuming that ηfc,max = 0.5, and using Eq. (5.30) the mass of consumed hydrogen can be calculated, which is equal to 0.55 kg. This hydrogen consumption is for one driving cycle, in which the vehicle travels a distance of S=

N 

V i ti = 23262.39 m.

(5.31)

i=1

Therefore the hydrogen consumption of this vehicle for 100 km traveling will be HC = 0.55 × 100,000/23269.39 = 2.32 kg H2 per 100 km. Details of the presented model for evaluating the performance characteristics and the energy consumption characteristics for the proposed vehicle in Table 5.1 are provided in Appendix E. This simple model can be further developed. For example, the electric motor efficiency can be defined via a map in the model (i.e., the variation of this efficiency can be taken into account). This 1D model can also be used to develop a predictive controller for an FCEV.

5.5. Summary In this chapter, we started from longitudinal vehicle dynamics. We introduced three resistant forces and the way to calculate the vehicle traction force. Afterward, three main vehicle performance parameters, namely maximum speed, gradeability, and acceleration time are introduced, and their calculations are explained. Driving cycle as a standard tool for taking into account the driving pattern effects is described, and, subsequently, the way to extract energy consumption characteristics based on a driving cycle is explained. All these discussions about vehicle dynamics were general (i.e., for any vehicle and not only for FCEV). At the next section, we explained an FCEV configuration and its components such as FCEV, battery module, electric motor, VCU, and hydrogen tank. In the last section of this chapter, a 1D model for an FCEV, which can evaluate both the performance characteristics and the energy characteristics of FCEV is presented via an example modeling problem. The implemented code for this modeling is also provided in Appendix E.

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Figure 5.7 Results of 1D modeling of the proposed FCEV: a) vehicle speed, b) PEMFC power, c) battery power, and d) battery energy variation.

5.6. Problems and questions Questions 1. Find three correlations for the rolling resistant coefficient for three types of vehicles: passenger car, bus, and truck. 2. Can you explain the physics of rolling resistance on both hard and soft roads? 3. Find the drag coefficient for three specific vehicles from the three types of passenger: car, bus, and truck. 4. Explain the role of electronic interface in an FCEV power train. 5. Find and explain three famous control strategies for HEVs. Can they be adopted for FCEVs? Why?

Fuel cell electric vehicles (FCEVs)

Problems 1. Derive Eqs. (5.12) and (5.13). 2. Redo the example presented in the last section for NEDC driving cycle. 3. What are the effects of presented parameters in Table 5.1 on the performance and energy consumption of an FCEV? Employ the code to provide discussion.

References [1] M. Ehsani, Y. Gao, S. Longo, K. Ebrahimi, Modern Electric, Hybrid Electric, and Fuel Cell Vehicles, 3rd ed., CRC Press, Boca Raton, 2018. [2] A. Fotouhi, M. Montazeri-Gh, Tehran driving cycle development using the k-means clustering method, Scientia Iranica 20 (2013) 286–293. [3] T.J. Barlow, S. Latham, I.S. McCrae, P.G. Boulter, A reference book of driving cycles for use in the measurement of road vehicle emissions, TRL Publ Proj Rep, 2009.

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CHAPTER 6

Fuel cell power plants Contents 6.1. Applications 6.1.1 Residential sectors 6.1.2 Power plants 6.1.3 Automotive 6.2. SOFC power plant components 6.2.1 Main reactor 6.2.2 Materials 6.2.2.1 Anode 6.2.2.2 Cathode 6.2.2.3 Electrolyte 6.2.2.4 Interconnection 6.2.3 Reformer 6.2.4 Voltage regulator 6.2.5 Thermal management components 6.2.5.1 Fuel preheater or HX1 6.2.5.2 Steam generator or HX2 6.2.5.3 Reformer, Rf 6.2.5.4 Air preheater, HX3, and air heater, HX4 6.2.5.5 Fuel heater, HX5 6.2.5.6 SOFC anode, An, and cathode, Ca 6.2.5.7 Combustor or the afterburner or AB 6.3. Fuel 6.3.1 External reforming 6.3.2 Internal reforming 6.3.3 Gasification 6.3.3.1 Entrained flow 6.3.3.2 Moving bed 6.3.3.3 Fluidized bed 6.4. Summary 6.5. Problems References

303 304 305 308 310 311 314 314 315 316 319 319 320 321 322 322 323 323 324 324 324 324 326 328 330 331 332 334 335 335 335

6.1. Applications SOFCs are produced in various shapes and sizes for different purposes. Small blocks are used for automotive and residential sectors, whereas large-scale blocks power the industries. Their ability to consume a wide range of fossil fuels enabled them to be incorporated in many different locations. The main applications of SOFCs are categoFuel Cell Modeling and Simulation https://doi.org/10.1016/B978-0-32-385762-8.00010-5

Copyright © 2023 Elsevier Inc. All rights reserved.

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rized into a) residential sector, b) industry and power stations, c) automotive industry, and d) other applications.

6.1.1 Residential sectors In Chapter 1, we discussed the roadmap of different countries toward the application of fuel cells. As discussed in the Japanese roadmap, SOFCs were the chosen candidate for the residential sector. The most significant advantage of such a selection is that SOFCs can consume a variety of fuels, including natural gas, diesel, and other petroleum. This fact has made the usage of SOFCs quite attractive since they can be used in many countries and different places. Another advantage of the application of SOFCs in the residential sector is that these devices are usually set up for cogeneration. Hence they provide both electricity and heat, which increases the overall efficiency of the device. They are also quiet systems and operate silently. Therefore you can install them almost anywhere, which is very important in building design. In many countries, natural gas is delivered to buildings by pipelines. In other countries, fossil fuels are available in other shapes, such as oil, petroleum, or coal. Hopefully, SOFCs are able to consume a variety of fuels by means of reformers. Reformers are devices that convert hydrogen-rich fuels to pure hydrogen. For example, natural gas is mainly composed of CH4 . A steam reformer converts it to H2 and H2 O according to the following reaction: −  CH4 + H2 O −  −− − − 3 H2 + CO.

(6.1)

This reaction requires steam; hence it is commonly known as steam reforming. As we can see, steam reforming produces carbon monoxide. If the carbon monoxide is purged in the atmosphere, then the process is not environmentally friendly, and the obtained hydrogen is called gray hydrogen. If the produced carbon monoxide is captured, then hydrogen is called blue hydrogen. Note that we have green hydrogen produced by direct electrolysis of water that intrinsically produced no carbon emission. Fig. 6.1 shows a commercial SOFC unit designed for residential buildings. You can find 3 kW and 5 kW modules specially designed for a single home. The unit provides electricity and heat for the building, which means that the whole efficiency is very high. However, using SOFC for electricity generation has its own drawbacks. Since SOFCs work at high temperatures, their startup time is time-consuming, which means that they need a long time to warm up for operation. Consequently, they should run for a long time once they are working, and we cannot stop them at night and start them the next day. Therefore the unit must work in partial loads at night and full loads during the day. In some countries the government buys extra electricity from the residential buildings. However, reverse power flow is not allowed in other countries, such as Japan [15]. If the reverse power flow is allowed, then the owners of the residential buildings can sell

Fuel cell power plants

Figure 6.1 A residential SOFC power block [4].

the extra produced electricity to the government. However, if it is not allowed, then they should make special plans to maximize the use of their power unit. The improvement of technology can add attractive features to the residential power systems. For example, SOFC power plants can be integrated and equipped with the internet of things (IoT). As an example, Osaka Gas has been working on the development of different residential SOFC blocks in cooperation with Kyocera, Toyota Motor, and Aisin Seiki [18]. They have been offering their products with IoT technology since 2014. The communication servers are provided by Osaka Gas, as is shown in Fig. 6.2. With IoT technology, the customers are able to monitor their power generator state using an application. The owners can monitor and control the consumed gas, load, safety parameters, and many other things. Moreover, they can remotely run the floor warming system or fill a bathtub with warm water some minutes before they reach their homes.

6.1.2 Power plants The electricity that comes from power plants is delivered either on- or off-grid. The generated power is directly fed to the grid in an on-grid system and is transferred to the end users. However, off-grid generations are more suitable for many different applications, including distant areas and charging electric cars. The off-grid power is becoming more and more popular and is attracting attention in many applications because of the following reasons:

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Figure 6.2 Concept of IoT used in SOFC power units.

In a global grid a considerable amount of power is lost in transportation. However, in an off-grid or local grid, the energy loss is minimized since the power plant is built close to the end consumer. • Many people live in rural areas with no access to grid power. Extending the grid in these areas is less economical than installing a local power plant. • Building a new on-grid power plant has its own proponents and opponents. Usually, many communities do not welcome constructing new transmission lines due to their impact on the environment, economic issues, political problems, and many other factors. Therefore the best alternative is the construction of local power plants. • Constructing local microgrids instead of global ones affects power security. For example, in the case of natural disasters, the power plants may become out of control or damaged. In such cases, small local power plants will have fewer outcomes than the conventional large-scale ones. Many different companies are working on SOFC for stationary applications. Most of them are in the research stage, and few of them have commercial products. For instance, Bloom Energy in California is the first US-based company that produces a commercial product named Bloom Box. The product (shown in Fig. 6.3) has an efficiency of over 50% without cogeneration. The company offers boxes with 50, 200, and 250 kW power, which can be connected in parallel to increase the power. Many renowned companies such as Walmart, Google, IBM, NetFlix, and many other benefit Bloom Box to reduce their carbon emission. Some companies have combined Bloom Box with other carbon reduction methods to reduce carbon emission as much as possible. As an example, Walmart uses biogas as the main feed. This combination leads to a much better •

Fuel cell power plants

Figure 6.3 Bloom Box from Bloom Energy [6]. Table 6.1 Specification of Bloom Box for some selected customers [6]. Customer

Capacity (kW)

Year

Location

Adobe

400 1200 4000 1000 500 500 500 400 1000 1500 500

2012 2010 2016 2013 2010 2009 2010 2008 2012 2017 2015

San Francisco (CA) San Jose (CA) Cupertino (CA) Maiden (NC) Southern CA Financial San Jose (CA) Oakland Mountain View (CA) Torrance (CA) CA (4 plants) Irvine (CA)

Apple Bank of America eBay FedEx Google Honda Ikea Johnson & Johnson Kaiser Permanente Morgan Stanley Panasonic Walmart WGL Yahoo

4300 750 750 +30 installations 2600 1000

7 projects in CA 2016 2014 from 2009 2014

New York City (NY) Lake Forest (CA) California Santa Clara County (CA) Sunnyvale (CA)

carbon reduction process. Currently, Walmart produces more than 60% of its energy from Bloom Box [8]. Table 6.1 shows the application of Bloom Box for some selected companies. The data are taken from Bloom Energy’s official site [6]. The site contains detailed information about the installed packages. Also, more companies are available on the official site.

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6.1.3 Automotive The power unit suitable for transportation should meet some different criteria. For example: • The volume of the whole unit must be as small as possible. The unit should be installed inside the car under the front hood. Hence its size has its own limitations. • In addition to the size, the weight of the cell is also very important. The reason is that a portion of produced energy is wasted for carrying the extra weight. Thus the lighter the unit, the longer the range of the vehicle. • The unit must have as high efficiency as possible. The higher efficiency is translated to a longer range with a certain amount of fuel. • The startup time of the cell must be very low compared to conventional fossil-fueled cars. For these reasons, the best fuel cell choice would be PEMFC. In this regard, some statements must be more discussed to compare PEMFCs with SOFCs. It is clear that SOFC provides higher efficiency compared to PEMFC. Moreover, we can use many different fuels for running a SOFC since it contains internal reformers. Therefore a big problem of PEMFCs, hydrogen storage, is diminished when using SOFCs. However, a SOFC complete system generally requires more space than a PEMFC system. By system we mean all the accessories required to provide fuel and oxygen, regulate the output power, balance the temperature, and other subsystems. Moreover, the startup time of SOFCs is adequate for passenger cars or other transportation means. These factors have led the automotive industries to focus on the PEMFCs despite the great advantages of SOFCs. However, SOFCs are considered to be used as auxiliary power units (APUs) or range extenders in some applications. Most notably, they are specially designed and used as APUs for anti-idling laws to reduce diesel consumption during idling. In many cases the vehicles are run at idle, which means that their engines are working while the vehicle is not moving. For example, police officers, firefighters, and smalland heavy-duty trucks run idle for a long time. Idling contributes to global emission but has no useful outcome. It is reported [16] that passenger cars, light-, medium-, and heavy-duty trucks consume more than 6 billion gallons of diesel and petroleum. Idling not only increases the air emission but also affects the engine wearing and the need for oil and associated filter changing. This adds to the maintenance cost for truck owners and frequent engine overhauls. Consequently, many countries and regions have passed different laws for reducing idling time. For instance, the US focuses on research and development of technologies, economic incentives, and education. Different states have passed different laws against idling. Some of them are more strict than others. According to Hawaii Administrative Rules, no idling is permitted “while the motor vehicle is stationary at a loading zone, parking or servicing area, route terminal, or other

Fuel cell power plants

Figure 6.4 The general concept of AVL APU.

off-street areas” [16]. Other countries also have passed similar laws for idling reduction. For instance, in Europe, vehicles are equipped with start-stop facilities to reduce idling. One fuel cell-based technology, basically designed for idling reduction, is the APU designed by AVL. In this technology a SOFC provides the required power and heat for the cabin in idle times. Especially when the trucks are at rest overnight, the cabin requires electricity for TV, cooking, refrigerators, light, and other similar stuff. Moreover, truck owners need heat when traveling in cold places. Without an APU, the required power and heat are provided by engine idling, which means that the truck engine is on overnight to supply the owner’s requirement without any movement. Adding an APU to the truck reduces the fuel consumption by up to 70%. So what is an APU? An APU is a SOFC complete system specially designed for consuming diesel as its main fuel. As shown in Fig. 6.4, the SOFC is located at the heart of the package. The most sophisticated part of the APU is the reformer developed to reform diesel to H2 and CO. Reforming natural gas is more common in the industry, but diesel reformer requires more attention and is, of course, more challenging. Diesel reforming requires heat supplied from the recirculation of anode gas. The catalysts of the reformer break down the diesel hydrocarbon chains into hydrogen and carbon monoxide. These gases are fed into the anode of the SOFC as the main fuel. A blower blows air into the stack at the cathode side through a heat exchanger, as shown in the figure. The SOFC as the main reactor of the system uses H2 and CO as the main fuel and the heated air to convert them into electricity and heat. It is quite evident that the off-gas of anode still has a certain amount of fuel and carbon monoxide. These materials are further burned in the off-gas burner to produce more heat. Therefore the output contains only H2 O and CO2 , and also the efficiency of the system is increased by such CHP concept.

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Figure 6.5 AVL SOFC APU [2]. Table 6.2 The characteristics of AVL APU [2]. Item

Electrical power El. efficiency Thermal power Weight Volume Noise Fuel

Value

Unit

3 > 30 10 75 80 45 road diesel (< 15 ppm S)

kW % kW kg L dB (A) -

The AVL APU is shown in Fig. 6.5, and its characteristics are tabulated in Table 6.2. As we can see, the unit has 3 kW electrical power with an efficiency of more than 30%. At the same time the thermal power of the APU reaches 10 kW, which can be used for different purposes. This means that the overall efficiency of the unit is higher than 30%. One of the advantages of AVL APU is that it has a compact design and occupies a limited amount of space. As another APU characteristic, we can see that it consumes normal diesel. The level of sulfur must be less than 15 ppm. If there is a higher amount of sulfur, then the cell becomes poisoned. Sulfur poisoning occurs at cathode materials, one of the most important degradation factors in SOFCs. Therefore, if the sulfur contaminant is too high, then it must be refined before sending it to the reformer.

6.2. SOFC power plant components So far, we have mostly been discussing the cells or cell stacks when talking about SOFCs. However, a SOFC power unit contains different components for proper operation. For example, the hydrogen-rich hydrocarbons must be cracked or reformed into hydrogen

Fuel cell power plants

and carbon monoxide before entering the anode. Light hydrocarbons such as methane can be internally reformed at the anode, but other heavy hydrocarbons must be prereformed to methane. Therefore a reformer is crucial when feeding heavy hydrocarbons. As another example, the fuel and oxygen do not react completely inside the SOFC. In the best practical cases, the efficiency would be about 80%, which means that the exhaust gases contain H2 , CO, and O2 . These gases are either explosive or toxic. Hence they should be burnt in an after-burner to produce heat, water, and carbon dioxide, which is not toxic. The heat of the after-burner would be used to prewarm the inlet gases. Because if the inlet gases are not warm, then they will cause thermal shocks and eventually destroy the SOFC cells. This scenario shows that an SOFC plant requires an after-burner and some heat exchangers to warm the incoming fuels and air. In addition to the above examples, other components are necessary. Different SOFC types require different components. For instance, if the feeding fuel is methane, then the prereformers are not necessary. In general the basic components of a SOFC power plant are as follows: • A fuel processor, as described. • A desulphurizer unit to remove the sulfur out of the input gases. The input gases must be sulfur-free, whether it is anode or cathode; at least, it must contain less than 0.1 ppm sulfur. • Fuel cell power module, which is the central heart of the plant. • Different process gas heat exchangers. • Power conditioning equipment for DC–AC converter is also required for converting the generated power to useful grid-scale power.

6.2.1 Main reactor When talking about SOFC technology, we normally mean the cell stack. The smallest component of a stack is called cell, in which the main electrochemical reactions occur. A single cell usually provides a very small power of about tens of watts. To gain higher power, we need to internally connect the cells in parallel or series. The result is called stack, which is also known as the main reactor. There are different types of cells and stacks, each of which has its pros and cons. Different manufacturers have proposed different shapes for developing their products. Here we give a brief introduction to these technologies. As explained, a single SOFC cell has a very low voltage, which practically makes it useless. However, to increase the voltage and, consequently, the whole system power, cells must be connected in series. The result is called stack. There are two different types of stack designs, a) planar and b) tubular. The planar stacks have a very compact shape and can provide higher powers compared to the tubular shapes. However, the tubular shapes are mechanically more stable and require fewer packing. In addition to these two types, a new combined design is proposed by Osaka

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Figure 6.6 A typical planar cell stack [13].

Gas [13], which is known as flat-tube type. Her we explain these three configurations in more detail. Planar cells are manufactured in the same manner as is conventional for construction of other FCs such as PEMFCs. In this configuration the cells are made of bipolar plates as shown in Fig. 6.6. The fuel enters from one side of the plate and the air from its other side. There are various variants of the design. For example, the bipolar plates may be constructed using circular or rectangular shapes. In a circular shape the fuel is fed from the center of the disc and moves radially to the edges. In the rectangular types the fuel enters from the edges, as is shown in the figure. In general, the construction of the planar cells is simpler than of other types. Moreover, this type is more compact compared to the other types. Therefore planar cells are more common and are manufactured by many manufacturers. These cells can be easily converted to a stack configuration, as is shown in the figure. Several cells are put together to form a pile of cells connected in series. Therefore the voltage of the stack would be the sum of the voltages of all the individual cells. At both ends of the stack, end-plates close the stack. One of the drawbacks of the planar type is that it requires advanced packing. Compared to the tubular types, suitable packing that can operate at elevated temperatures is more costly and has higher technology. Tubular cells were first introduced by Siemens Westinghouse. The configuration is shown in Fig. 6.7 and consists of a porous doped MaMnO3 , which is closed at one end. The electrolyte is YSZ, and the anode is made of Ni – YSZ. These materials, as will be explained later, are thermally compatible. This means that all of them have more or less the same thermal expansion coefficient. Thus they do not encounter mechanical stresses when they are getting hot. In addition to the tubular cell, interconnectors made of doped lanthanum chromite are used to connect the cells in series. Several tubular cells can be connected together to increase the voltage and eventually the power of the stack.

Fuel cell power plants

Figure 6.7 A typical planar cell stack [13].

Figure 6.8 A typical planar cell stack [13].

The oxygen flows from the closed end of the tubes and moves around the cells. This configuration requires the least packing for isolating the air and fuel. This advantage has led to a more stable operation of the system, and the tubular SOFCs can operate for a long time without any major problem. However, the power density of tubular cells is slightly lower than that of planar types. Several designs have been used to increase the power density of the tubular cells. For example, they have been arranged in a flat shape, or smaller tubes have been used in some designs. Flat-tube type is not actually a different type. These cells are tubular cells arranged in a rectangular shape. As discussed in the previous item, the tubular cells show very low power density. To increase the power density of the cells, they are arranged in a flat shape. The configuration is shown in Fig. 6.8. Although the main concept is the same, the power density of flat-tube-type cells is greater than the standard tubular ones. Another advantage is that they can be easily connected to each other to increase the voltage and power of the whole SOFC.

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6.2.2 Materials The main components of SOFC cells are anode, cathode, electrolyte, and cell interconnects. The selection of proper material for each component is tricky [14]. This is very important not only because of their electrochemical behavior but also for their thermal expansion consistency. The cell temperature of a SOFC reaches about 1000 ◦ C and reaches room temperature while it is at rest. Therefore all the components must have more or less the same expansion coefficient. Otherwise, they encounter mechanical stresses during the stop and start operations and eventually break down.

6.2.2.1 Anode The selection of anode materials requires specific attention. The following characteristics must be considered for anode materials: • The material used in the construction of anode must show an excellent catalytic behavior. Since different fuels are used for SOFCs, the used materials should be separately selected. • The chemical stability of the materials is also important. In contrast to many other FCs, the fuel of SOFC is not very pure since different types of fuel are fed to the cells. For example, almost pure hydrogen is fed to the PEMFCs, or at least its impurity is not very high. The feeding fuel of SOFCs has many different impurities. Therefore the materials of the anode should be stable when it is in contact with impurities. • The electrodes must be porous to let the gaseous fuel and the resulting species move in and out. The porosity also increases the active surface area of the reaction site. • The thermal expansion coefficient of the electrodes is also important. The selected materials of the anode should show almost the same expansion as the other components to minimize the mechanical fatigue and stress. At the anode, the oxidation of fuel occurs, as well as the reforming process if fuel is other than hydrogen. Therefore the anode must show catalytic behavior in addition to thermal stability. There are different materials available for the anode side. Table 6.3 summarizes some of the frequently used ones. As we can see, nickel-based anodes are divided into nickel-oxide and NiO/YSZ composite ones. In SOFCs with the electrolyte made of YSZ (Yttria-stabilized zirconia), this combination shows better thermal expansion compatibility with the electrolyte and gives a good electrochemical efficiency [5]. The choice of nickel as the main material for anode has several advantages, including: • It has a good catalytic capability for converting hydrogen into electrons and ions. • It also has good electrical conductivity; hence it is also used as the main current collector for the anode. • Nickel also is highly active for steam reforming of methane. • It has a reasonable price compared with other precious metals.

Fuel cell power plants

Table 6.3 Anode materials [10]. Green nickel oxide Composition

Name

NiO-AFL NiO-AS-F NiO-AS-M NiO-AS-C

NiO NiO NiO NiO Stabilized zirconia Composition

Name

8YSZ-C 5YSZ-C 3YSZ-C

(Y2 O3 )0.08 (ZrO2 )0.92 (Y2 O3 )0.05 (ZrO2 )0.95 (Y2 O3 )0.03 (ZrO2 )0.97 Lanthanum strontium titanate Composition

Name

LST

La1 – xSrxTiO3 NiO/ YSZ composite powders Composition

Name

NiO/ YSZ-AFL NiO/ YSZ-AS

NiO : 8YSZ = 57 : 43 NiO : 8YSZ = 60 : 40

Name

Spinel powders Composition

NCF NCC

Ni0.5 Cu0.5 Fe2 O4 Ni0.5 Cu0.5 Co2 O4

Name

APS-A

SSA (m2 /g) 8 ∼ 10 5∼7 3∼5 1∼3 SSA (m2 /g) 4∼6 4∼6 4∼6 SSA (m2 /g) 5 ∼ 10 SSA (m2 /g) 8 ∼ 10 5∼7 SSA (m2 /g) 5 ∼ 10 5 ∼ 10

Sintered granule for air plasma spray coating Composition PSD (μm) All Anode D50 = 20 ∼ 40

Other materials tabulated in Table 6.3 are also available for anode side. In addition to these materials, lots of various compositions are under examination. For example, Cubased, Ru-based, and other metallic and nonmetallic materials are available in lab-scale and research papers. Some of these materials are also available in the industrial units.

6.2.2.2 Cathode The material used for the construction of cathodes must have various characteristics. For instance: • The used material should show a high electrochemical activity to convert oxygen molecules into oxygen ions. It is well known that the most electrical polarization of the whole cell is due to the cathodic reactions. Hence the used material must perform a high catalytic behavior.

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In addition to the high catalytic behavior, the material used to construct the cathodes must show high electronic conductivity to reduce electronic losses. We know that by increasing the temperature the resistance of materials increases. Hence the cathode materials working at elevated temperatures must show reasonable electric conductivity to minimize the loss. • The cathodes are porous with lots of microstructures to provide a suitable pass for the diffusion of gaseous oxygen. The porosity of the electrode also increases the available active area and increases the activity of the electrode. However, it reduces the electronic conductivity. Therefore a balance must be considered between the increase in active surface area and electronic conductivity. • The materials should have chemical stability depending on the operating temperature and fuel. As explained earlier, different SOFCs work at different temperatures and may use different fuels. Therefore the used material should show good chemical stability for each usage. • As for the anodes, the cathodes also encounter the same temperature difference when they are working and when they are stopped. Therefore the thermal expansion of cathodes must match the other components. Otherwise, they will bear lots of mechanical stresses and will break down. Table 6.4 lists the most commonly used materials used in cathodes. As we can see, lanthanum is practically a great candidate. Lanthanum manganite LaMnO3 provides a good choice since it shows almost all the above-mentioned characteristics. It is stable, has adequate activity, and has a reasonable thermal expansion, which matches YSZ electrode. To increase the performance of lanthanum, it is doped with different materials. For instance, calcium Ca and strontium Sr are frequently used as dopants. Although manganite shows good performance, it is not the only candidate. Other lanthanum compounds, such as strontium ferrite, cobalt ferrite, and strontium cobalt ferrite, are used, which are tabulated in Table 6.4. In addition to lanthanum, other materials such as barium and samarium are suitable for cathodes. These materials are used instead of lanthanum in different compounds. As shown in Table 6.4, different compounds with different characteristics are used. •

6.2.2.3 Electrolyte The electrolyte in FCs is quite an important part. A closer examination of electrolytes shows that their materials must pose the following characteristics: • It must be a good ion conductor; otherwise, the ionic losses become excessive, and the efficiency decreases. • The electrolyte must be an electronic insulator to prevent the internal short circuit.

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Table 6.4 Cathode materials [11]. Name

LSM-73-N LSM-73-F LSM-73-C LSM-82-N LSM-82-F LSM-82-C Name

LSCF-6428-N LSCF-6428-F LSCF-6428-C Name

LSC-64 Name

LSF-82 Name

LNF-64 Name

BSCF-5582 Name

SSC-55

•

•

Lanthanum strontium manganite Composition

(La0.7 Sr0.3 )0.95 MnO3 (La0.7 Sr0.3 )0.95 MnO3 (La0.7 Sr0.3 )0.95 MnO3 (La0.8 Sr0.2 )0.98 MnO3 (La0.8 Sr0.2 )0.98 MnO3 (La0.8 Sr0.2 )0.98 MnO3 Lanthanum strontium cobalt ferrite Composition

(La0.6 Sr0.4 )0.97 Co0.2 Fe0.8 O3 (La0.6 Sr0.4 )0.97 Co0.2 Fe0.8 O3 (La0.6 Sr0.4 )0.97 Co0.2 Fe0.8 O3 Lanthanum strontium cobaltite Composition

La0.6 Sr0.4 CoO3 Lanthanum strontium ferrite Composition

La0.8 Sr0.2 FeO3 Lanthanum nickel ferrite Composition

LaNi0.6 Fe0.4 O3 Barium strontium cobalt ferrite Composition

Ba0.5 Sr0.5 Co0.8 Fe0.2 O3 Samarium strontium cobaltite Composition

Sm0.5 Sr0.5 CoO3

SSA (m2 /g) 10 ∼ 15 5 ∼ 10 1∼5 10 ∼ 15 5 ∼ 10 1∼5 SSA (m2 /g) 10 ∼ 15 5 ∼ 10 1∼5 SSA (m2 /g) 5 ∼ 10 SSA (m2 /g) 5 ∼ 10 SSA (m2 /g) 5 ∼ 10 SSA (m2 /g) 5 ∼ 10 SSA (m2 /g) 5 ∼ 10

The thermal expansion coefficient of the electrolyte must be consistent with the other components. Otherwise, when the cell is in operation, the mechanical stresses will lead to the breakdown of the system. The stability of the electrolyte is vital because it must meet the following stability criteria: • Thermal stability must be as proper as the other components. The electrolyte must have the same thermal stability as the other components meet. For instance, in SOFCs the electrolyte must operate at elevated temperature levels. Therefore it must be stable at that range if it is going to operate for a reasonable amount of hours.

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Table 6.5 Electrolyte materials based on Zirconia [12]. Scandia-stabilized zirconia Composition

Name

10 Sc0.5 Ce0.5 GdSZ 9.5 Sc0.5 Gd0.5 YbSZ

(Sc2 O3 )0.1 (CeO2 )0.005 (Gd2 O3 )0.005 (ZrO2 )0.89 (Sc2 O3 )0.095 (Gd2 O3 )0.005 (Yb2 O3 )0.005 (ZrO2 )0.895 Ytterbia/Scandia-stabilized zirconia Composition

Name

6Yb4ScSZ

(Yb2 O3 )0.06 (Sc2 O3 )0.04 (ZrO2 )0.9

Name

Yttria-stabilized zirconia Composition

8YSZ

SSA (m2 /g) 10 ∼ 15 10 ∼ 15 SSA (m2 /g) 10 ∼ 15 SSA (m2 /g) 10 ∼ 15

(Y2 O3 )0.08 (ZrO2 )0.92

Table 6.6 Electrolyte materials based on Ceria [12]. Name

GDC-10-N GDC-10-F GDC-20-F GDC-20-N GYBC-LTS GDC-LTS Name

SDC-20-N SDC-20-F SYBC-LTS

Gd-doped ceria Composition

Gd0.1 Ce0.9 O1.95 Gd0.1 Ce0.9 O1.95 Gd0.2 Ce0.8 O1.9 Gd0.2 Ce0.8 O1.9 Gd0.135 Yb0.015 Bi0.02 Ce0.83 O1.915 GDC+dopant Sm-doped ceria Composition

Sm0.2 Ce0.8 O1.9 Sm0.2 Ce0.8 O1.9 Sm0.16 Yb0.02 Bi0.02 Ce0.8 O1.9

SSA (m2 /g) 10 ∼ 15 5 ∼ 10 10 ∼ 15 5 ∼ 10 10 ∼ 15 10 ∼ 15 SSA (m2 /g) 10 ∼ 15 5 ∼ 10 10 ∼ 15

• In contrast to the anode or cathode material, the electrolyte is subjected to both

oxidations from the cathode side and reduction from the anode side. Therefore the used material should chemically withstand both processes. These characteristics have made it difficult to choose the proper material. Many different materials have been used as the base for constructing the electrolyte. Tables 6.5–6.7 summarize some of the more conventionally used materials. As we see, they are divided into yttrium-, ceria-, and perovskite-based ones. Yttrium-stabilized ZrO2 , also known as YSZ, is the most commonly used electrolyte. Compared to the ceria and perovskite electrodes, it has a lower conductivity but a longer life and stability, which has made it the best choice at the present time. Unfortunately, the YSZ achieves its high conductivity at temperature levels of about 1000 ◦ C and loses its conductivity when it is getting colder. Thus, to use it at lower

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Table 6.7 Electrolyte materials based on Perovskite [12] Lanthanum strontium gallium magnesium oxide Composition SSA (m2 /g) LSGM-9182 La0.9 Sr0.1 Ga0.8 Mg0.2 O2.85 3∼6 La0.8 Sr0.2 Ga0.8 Mg0.2 O2.8 3∼6 LSGM-8282 La0.8 Sr0.2 Ga0.8 Mg0.18 Zn0.02 O2.8 3∼6 LSGM-8282-LTS Name

temperatures, the thickness of the electrolyte must be reduced. Another approach is to dope it with other materials such as Sc. These choices are shown in Table 6.5. In comparison to YSZ, the ceria and perovskite materials show greater conductivity, which is very attractive. For instance, perovskite shows an excellent ionic conductivity at temperatures about 650 ◦ C. Various used materials based on ceria are shown in Table 6.6. Also, Table 6.7 shows different choices for perovskite materials.

6.2.2.4 Interconnection The interconnection provides the required electrical pass for conducting the electrons from cell to cell. Like the other components of the SOFCs, these materials must also meet several characteristics. Most of them are similar to the characteristics of the other components. We can name the following items: • They must be excellent conductors of the electron to reduce the electronic losses as much as possible. • They should show good thermal stability at elevated temperatures. • They must also have a compatible thermal expansion coefficient compared to the other components to eliminate the mechanical stresses. • The used materials should have acceptable mechanical strength. • Chemical stability is also important since they are connected to the other components. Doped lanthanum chromite is widely used at the interconnects. To increase its stability and performance at different temperature levels, other materials such as calcium and cobalt are also used as dopants. Other materials such as spinel are also used for making interconnects. Table 6.8 lists some of the more commonly used ones.

6.2.3 Reformer Many different fuels are used in SOFCs. This is one of the advantages of these technologies over their competitors, especially PEMFCs. This is because SOFCs work at very high temperatures, and the fuels can be reformed to produce hydrogen. In other low-temperature FCs, such as AFC, PAFC, PEMFC, and so on, the fuel must be pure hydrogen. For this reason, they have to be equipped with hydrogen storage tanks, which is a big problem. In the SOFC case, almost any fuel can be used, including gaseous, liquid, and solid fuels. However, the fuel must be reformed to produce hydrogen.

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Table 6.8 Interconnect materials [12]. Name

MCF MC-11 CMF Name

LCC LCCC

Spinel Composition

MnCo1.9 Fe0.1 O4 Mn1.5 Co1.5 O4 CuMn1.9 Fe0.1 O4 Lanthanum chromite Composition

La0.7 Ca0.3 CrO3 La0.8 Ca0.2 Cr0.9 Co0.1O3

SSA (m2 /g) 5 ∼ 10 5 ∼ 10 5 ∼ 10 SSA (m2 /g) 5 ∼ 10 5 ∼ 10

Sintered granule for air plasma spray coating Composition PSD (μm) APS-IC All interconnect D50 = 20 ∼ 40 Name

Reforming the fuel requires an additional process. There are two different major reformers: External reformer, where the fuel is processed is a separate equipment, and the resulting hydrogen is fed into the FC. Internal reformer, where the fuel is directly injected into the FC, and the fuel internally is broken into hydrogen and CO. In the case of an external reformer, the produced hydrogen can be delivered to any FC. It does not matter if it is a SOFC, a PEMFC, or any other technology. External reformers have different types and technologies (to be discussed later) and are manufactured on large scales. These reformers can accept a wide range of hydrocarbons and even noncarbonaceous materials to produce hydrogen. The purity of the hydrogen depends on the input feed and the reformer technology. More details are given in the next section, which covers the fuels. The internal reformer is a unique quality of SOFCs. This quality is attributed to the high operating temperature of the cell. Some catalysts must be coated on an anode side to make an internal reformer. Usually, the catalyst is made from Ni, which is stable at that temperature. The high temperature also increases the reaction rates, which is favorable for the reforming process. In the following section about fuels, we give more detail about the internal reformers.

6.2.4 Voltage regulator SOFC is a DC generator, and hence its power is not suitable for consumption. The generated power must be regulated to a more appropriate source. In power generator SOFCs the produced electricity is usually fed to the grid. Hence the DC power must be converted to the 3-phase AC. The general block diagram of this conversion is shown in Fig. 6.9. As we can see, the DC power must first be converted into another DC source

Fuel cell power plants

Figure 6.9 Schematic of voltage regulator of a SOFC.

with appropriate voltage. Then it will be converted into the AC power by means of a DC/AC inverter. The AC power is then filtered to eliminate the noise and fed to the grid. Since voltage regulation requires two levels of regulations, we have to care about the following issues: • The conversion cost is high. Note that the cost of energy is very important. Any process that adds to the cost of energy should be treated with care. • The converter draws the current ripple from the FC. Therefore the converter must be designed appropriately to cope well with the FC. • The FC shows a slow dynamics, whereas the converters usually have a fast dynamics. This fact may result in an imbalance between the converters and SOFC. In choosing the DC/DC converter, we should pay attention to the following issues: • It should have very high efficiency. Otherwise, lots of energy would be wasted, and as discussed, the cost of the energy increases. • It should also have high reliability to operate effectively with the SOFC. • It must have a low ripple current. • It must be quite consistent with SOFC since this is the interface between the power regulator and SOFC. Many factories produce DC/DC converters. All of them are trying to produce converters that have higher efficiency, lower ripple current, and good response with SOFC. All these characteristics are important and affect the overall cost of energy.

6.2.5 Thermal management components A SOFC power plant requires many different heat exchangers. Yoshida and Iwai [19] presented a clear discussion about the usage of various heat exchangers. Here we give a brief discussion. More information can be found in the same reference and also other related publications. In general, we can divide the heat exchangers into two categories, low- and hightemperature heat exchangers. The low-temperature heat exchangers operate at about 600 ◦ C, whereas the high-temperature ones work at the SOFC operating temperatures (around about 1000 ◦ C). These heat exchangers are schematically shown in Fig. 6.10. As illustrated, the main key components that play important roles in thermal management of the SOFC operation are the following:

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Fuel preheater or HX1 heats up the feed to reach the levels at which the reforming process can take place. Steam generator or HX2 is used to generate steam at a desirable temperature for reforming. Reformer, Rf, may be indirect or direct. The one shown in the figure is an indirect reformer. Air preheater, HX3, is a preheater for warming up the air. The warmed air is then fed to HX4 to reach the SOFC temperature level. Air heater, HX4, takes the preheated air and increases its temperature up to the SOFC operating temperature. Fuel heater, HX5, heats up the syngas generated by the reformer to reach them to the levels of the SOFC operational temperature. SOFC anode, An, is the main reactor in which the electron is produced, and heat is released. SOFC cathode, Ca, is the oxygen electrode in which the oxygen is reduced to oxygen ions. Combustor or the afterburner or AB burns all the residual fuels for providing the necessary heat for heat exchangers. Since these parts are important, we will discuss them with more detail.

6.2.5.1 Fuel preheater or HX1 When the SOFC uses an internal reformer, the fuel is not pure hydrogen, and hence it should be reformed to produce H2 or CO. The types of fuel are discussed in the next section, but to become clear, note that any hydrogen or carbon-rich fuel can be fed to the SOFC. The fuel is then reformed to H2 , CO, CO2 , and H2 O, as is shown in the figure. The reforming process requires heat at elevated temperatures. Therefore the feeding fuel must be preheated and reach the reforming temperature, which is around 600 ◦ C. This happens in HX1, where the fuel temperature increases by means of the excess heat of the SOFC.

6.2.5.2 Steam generator or HX2 The reforming process requires steam at the same temperature as the fuel enters the reformer. HX2 is responsible for both steam generation and warming up the steam to the desired level. Therefore HX2 consists of two parts, the steam generator and heating part. Since the steam and fuel should be at the same temperature, the generated steam can be mixed by fuel to form a mixture. Then the mixture can be heated up to the reforming temperature. However, they can be separately heated, as is shown in the figure.

Fuel cell power plants

Figure 6.10 Thermally important components.

6.2.5.3 Reformer, Rf In an internal reforming process, the fuel and steam are converted to the products shown in Fig. 6.10. Reforming is a strong endothermic process and consumes about 20% of the system heat. In an indirect internal reforming, the products are in lower temperatures than those needed to enter the SOFC anode. Hence, as the figure shows, these products will be fed to other heat exchangers to increase their temperature further.

6.2.5.4 Air preheater, HX3, and air heater, HX4 For a smooth operation, air also must be heated to the same temperature as SOFC operates. This will be done using HX3 and HX4. The fresh air is preheated in HX3

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to a moderate temperature and then is superheated to the desired value in HX4. The control of air temperature at the cathode inlet is very important.

6.2.5.5 Fuel heater, HX5 As explained, the products of the reformer are in low temperatures that are not suitable for injection into the fuel cell. They must become superheated and reach the operational temperature of the SOFC. Therefore HX5 is used to raise the fuel temperature by means of the burned gases in the combustor. The temperature of the burned gases is high enough for this purpose.

6.2.5.6 SOFC anode, An, and cathode, Ca These channels are the main reactor in which the feed is oxidized by oxygen to produce electricity. In addition to the electricity, heat is produced as a side product, which is important in the system thermal behavior. One of the important issues in SOFCs is that the electrolyte of these cells is made of ceramic. Hence any temperature gradient at the anode and cathode sides results in thermal stresses of the electrolyte. This fact requires: • A proper design for heat exchangers at the anode and cathode side so that both sides become in the same temperature. • A proper design for the cell dimensions and operational conditions so that the temperature becomes uniform across the electrolyte. • Increasing startup and shutdown time to eliminate fast transient effects. The fast transient causes a nonuniform temperature gradient. Usually, the startup and shutdown of the system is in the order of 10 hours.

6.2.5.7 Combustor or the afterburner or AB Obviously, not all the input fuel and oxidants are consumed inside the SOFC. As is shown in Fig. 6.10, the unused fuel and oxygen are fed to a combustor, also known as afterburner, to burn the SOFC exhaust gases. The generated heat of the combustor is fed to the above-mentioned heat exchangers.

6.3. Fuel In PEMFCs, carbon monoxide is considered a poisoning material because it reacts with platinum and reduces its functionality. In PEMFCs the carbon monoxide combines with platinum and degrades its performance. Therefore the air and fuel must be carbon monoxide-free. This fact prevents the usage of many hydrocarbons from providing hydrogen for PEMFCs because hydrocarbon reforming produces CO. In contrast to PEMFCs, carbon monoxide is considered a fuel for SOFCs. According to the main reactions of SOFC, oxygen reacts with the electrons from the external

Fuel cell power plants

circuit and becomes oxygen ion via the following reaction: 1 O2 + 2 e− −−→ O2− . 2

(6.2)

The oxygen ions move toward the anode via the solid-state membrane, where it combines with the fuel. If the fuel is hydrogen, then the following reaction takes place: H2 + O2− −−→ H2 O + 2 e− .

(6.3)

The overall cell reaction then becomes H2 +

1 O2 −−→ H2 O + Electricity + Heat. 2

(6.4)

If instead of hydrogen, carbon monoxide is used as the fuel, then we have CO + O2− −−→ CO2 + 2 e− ,

(6.5)

and the overall cell reaction then becomes CO +

1 O2 −−→ CO2 + Electricity + Heat. 2

(6.6)

In this regard, it is clear that carbon monoxide can be directly used as a fuel in SOFCs. This is a great advantage of SOFCs because we can reform almost any hydrocarbon and use it as a fuel. The reforming process of hydrocarbons produces hydrogen gas and carbon monoxide. Therefore the result is a mixture of H2 and CO, which can be used as a fuel in SOFC. The anodic reaction then becomes α H2 + β CO + (α + β) O2− −−→ α H2 O + β CO2 + 2 (α + β) e− ,

(6.7)

and the cathodic reaction becomes 1 (α + β) O2 + 2 (α + β)e− −−→ (α + β) O2− . 2

(6.8)

Consequently, the overall cell reaction is 1 (α + β) O2 + α H2 + β CO −−→ α H2 O + β CO2 + Electricity + Heat. 2

(6.9)

In theory, all the hydrocarbons can be reformed to make hydrogen and CO. We can do this via an external reformer and feed the hydrogen to any available fuel cell technology. However, the main difficulty of external reforming is that the produced hydrogen must be stored and transferred to the power stations. Usually, the reformers are built on a large scale and cannot be integrated with small FCs.

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As one of the beneficial aspects of SOFCs, the hydrocarbons can be directly fed into the SOFC anode. In this case the anode contains a proper catalyst layer to reform the fuel into hydrogen and carbon monoxide directly. Hopefully, the produced carbon monoxide can also be used as fuel and is not required to be converted into hydrogen. This advantage is quite attractive since the storage and transportation of hydrogen are a big problem. However, in SOFCs the fuel can be a hydrogen-rich material such as methane, methanol, ethanol, biogas, etc. The storage of these materials is quite easy! The conventional fuels used in SOFCs are either hydrocarbon or carbon-free fuels. The fuels can be in the form of gas, liquid, and solid. Gaseous fuels. Many different gaseous fuels are used in SOFCs. Perhaps, methane is the most used gaseous fuel, but other heavier hydrocarbons can also be used. Methane is obtained from fossil reservoirs, but biosources are also important. These hydrocarbons are reformed to produce H2 and CO as described. As explained before, carbon monoxide is used as fuel for SOFCs. However, it is not a hydrocarbon, but is considered a carbonaceous fuel. Usually, CO is not used alone, and as it was discussed, accompanies hydrogen generated in reformers. Liquid fuels. Liquid hydrocarbons include methanol, ethanol, and dimethyl-ether. Dimethyl-ether is in the gas phase at normal conditions, but it is stored in high-pressure tanks in liquid form. The advantage of these fuels is that they are in liquid shape and can be easily stored and shipped. Nonhydrocarbon liquid fuels are also used. For instance, ammonia, NH3 , and hydrazine, N2 H4 , are nonhydrocarbon fuels used for this purpose [3]. These fuels are stored in liquid form and contain high energy density. Another advantage of these fuels is that they do not produce any carbon dioxide or coke. Finally, these fuels produce hydrogen when they flow over Ni/YSZ catalyst. Therefore they can be used directly in SOFCs with internal reformers. Solid fuels. Solid fuels include different coals, bitumen, biowastes, etc. [9]. Solid fuels are all hydrogen-rich materials, which can be used to produce hydrogen. This conversion happens with different technologies, including gasification, anaerobic digestion, fermentation, and liquefaction. The syngas is then fed into the SOFC.

6.3.1 External reforming Methane reforming is a well-developed technology to convert natural gas into pure hydrogen. There are different types of methane reformers, including autothermal reforming (ATR), steam methane reforming (SMR), and partial oxidation. In SMR the natural gas or methane is mixed with steam and is fed over a catalyst (mostly nickel) at a very high temperature. The present reaction occurs at the catalyst surface: CH4 + H2 O (steam) −−→ CO + 3 H2 (Endothermic).

(6.10)

Fuel cell power plants

Figure 6.11 An SMR plant [7].

The produced CO is further processed to react more with water and generate more hydrogen. The reaction is CO + H2 O (steam) −−→ CO2 + H2 (Exothermic).

(6.11)

Reaction (6.11) is very important because it not only increases the amount of hydrogen but also converts the carbon monoxide into a nontoxic material CO2 . In addition to methane, SMR is able to convert many different light hydrocarbons into hydrogen, including methanol, ethanol, and biogas. Therefore it would be a good candidate for combining with different fuel cells. However, SMR is efficient on a large scale. Unfortunately, small SMRs are not efficient enough to be used in small FC units. Fig. 6.11 shows an SMR build in Germany. Therefore their usage is limited to large reforming plants, and the produced hydrogen must be stored and further transferred to FC power plants. In ATR, oxygen also requires reforming. In fact, the main difference between SMR and ATR is that in the latter technique, methane is oxidized directly. ATR may use steam or carbon dioxide for reforming. If steam is fed in the chamber, then the reaction would be 4 CH4 + O2 + 2 H2 O −−→ 10 H2 + 4 CO,

(6.12)

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and if carbon dioxide is fed, then we have 2 CH4 + O2 + CO2 −−→ 3 H2 + 3 CO + H2 O.

(6.13)

ATR is an exothermic reaction because it is actually the combustion reaction of methane. From Eqs. (6.12) and (6.13) it is clear that using steam produces a higher H2 /CO ratio. This ratio is 1:1 for CO2 and 2.5:1 using steam [17]. The exothermic nature of ATR increases the reaction temperature. The normal temperature of an ATR reformer is something between 950 and 1100 ◦ C.

6.3.2 Internal reforming Steam reforming is an endothermic process that requires heat at temperature levels between 750 and 900 ◦ C. Compared with SOFC operational temperature, we see that the cell itself can provide the required heat at an acceptable temperature. Practically, SOFCs can provide 40–70% of the heat needed for the reforming of methane. Consequently, we can construct an internal reformer (IR) inside the SOFC to make the following benefits: • An IR directly uses the produced heat of SOFC. Thus it reduces heat loss and increases the efficiency of the whole system. • Elimination of an external reformer leads to a much simpler configuration. • By means of an IR we do not need to store hydrogen as the main fuel. An IRequipped SOFC directly uses natural gas or methane, whose storage is much easier. • Since a significant amount of waste heat is used for reforming, we need less cooling for SOFC. Hence the cooling systems are smaller and less costly. IR is divided into two major categories: Indirect internal reforming or IIR, in which a separate reformer is fabricated adjacent to the anode. Direct internal reforming or DIR, in which the fuel is directly reformed on the catalyst layer of the anode. As is clear in an IIR-SOFC, the reformer is a separate device at which the reforming is taken place. Such a configuration schematically is shown in Fig. 6.12a. The reformer contains a catalyst layer over which the fuel is converted into synthesis gas. The produced gas is then fed to the anode, as illustrated. Depending on the shape and flow direction, there are lots of configurations available both in practice and research. The reformer may have tubular, planar, and monolithic shapes. These reformers may also have cross-flow, coflow, counterflow, or forced periodic reversal of flow [1]. All these configurations have their own pros and cons and are under consideration. In an IIR-SOFC the anode catalyst layer is separated from the reformer catalyst layer. Hence different catalysts can be used and optimized for different purposes. For instance,

Fuel cell power plants

Figure 6.12 Different types of internal reforming.

the anode catalysts can be studied for maximizing the cell reaction performance, whereas the catalysts of the reformer are optimized for maximizing the reforming reactions. However, in an IIR-SOFC a mismatch between the required heat for reforming and the heat generated by SOFC may exist. Also, the working temperature of SOFC may not be optimized for steam reforming. The actual reason for this problem is that the reaction kinetics of steam reforming is much faster than the kinetics of the fuel cell reactions. Obviously, the IIR must be designed and optimized to take the best advantage out of the system. In a DIR-SOFC the fuel directly flows over the anode surface as shown in Fig. 6.12b. At the anode surface, reforming takes place, and methane is converted into synthesis gas. The gas then contributes to the electrochemical reactions as is normal in SOFCs. According to this concept, a DIR-SOFC contains anode that has good catalytic performance for reforming methane. It is known that nickel cermet anodes have good potential for steam reforming. This cermet requires no additive since its activity on methane reforming is sufficient. Conventionally, Ni/ZrO [1] is used as the main anode material for many SOFCs. The resulting anode poses a good performance on both methane reforming and working with ZrO2 electrolyte. DIR has some disadvantages, including: • When reforming occurs, coke is generated as a side product. Coke will cover the surface of the anode and reduce its efficiency. • Since reforming is a highly endothermic reaction, it absorbs a lot of thermal energy resulting in anode cooling down. This arises the following problems:

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• It decreases the cell temperature and consequently reduces the reforming effi-

ciency. • It results in a temperature gradient inside the cell, which eventually results in lower efficiency. • The temperature gradient causes mechanical stresses inside the cell.

6.3.3 Gasification Since a SOFC can consume any syngas as its primary fuel, any device that produces syngas can be integrated with it. For instance, all types of gasifiers are practically suitable for this integration. This fact extends the application of SOFC even more since we can use any source of hydrogen and carbon monoxide to generate syngas and feed it to the cell. The consumption of syngas is promising because we can combine the SOFCs with a wide range of waste materials such as biofuels and even urban waste. Such a combination not only provides a sustainable energy source but also eliminates other important environment-related issues. Technically, gasifiers are available in many different forms, sizes, and operational conditions. The design of a gasifier greatly depends on the type of fuel and the required products. There are different commercially available gasifiers that work with coal, petcoke, and other solid biomass. There are many gasifiers dedicated to converting urban wastes into hydrogen and are combined with different thermal cycles, including internal combustion engines. As explained, the design of a gasifier depends on the purpose of the device. In other words, depending on our needs, we can adjust the gasifier to produce different compositions, so the content of syngas differs from design to design. The operational conditions of gasifiers can be adjusted to optimize syngas production and composition. In the gasification process, the cold gas efficiency is a key parameter. According to Fig. 6.13, the gasifying process requires steam and air. The product of the gasifier is syngas, which is desirable. However, a portion of the feed is converted to ash, tar, and soot, which are unfavorable. As schematically illustrated, only syngas is fed to the SOFC as a fuel. This gas is fed to the SOFC, which contains an internal reformer and produces electricity. Other products are considered waste and must be removed. Consequently, the cold gas efficiency is defined as the ratio of the product energy content to the input feed energy content: ηcold−gas =

Eproducts , Efeed

(6.14)

where E is the energy content, and η is the efficiency. The energy content can be evaluated from the product of mass flow rate and the calorific value. Therefore Eq. (6.14)

Fuel cell power plants

Figure 6.13 Integration of a gasifier and a SOFC.

can be rewritten as m˙ G HG , (6.15) m˙ F HF where H is the lower calorific value, and m˙ is the mass flow rate. The indexes G and F refer to syngas and feed, respectively. The cold gas efficiency is important since it determines the final price of the energy produced by SOFC. The higher the value of ηcold−gas , the higher the energy efficiency and lower electricity price. There are three different types of gasifiers: entrained flow, moving bed, and fluidized bed. ηcold−gas =

6.3.3.1 Entrained flow This technology has been mostly incorporated by GE Energy, Shell, E-Gas, and Siemens. It is one of the most mature operational systems. They work at very high temperatures and use oxygen and steam to produce a low methane content syngas. The cold efficiency of entrained flow gasifiers is about 80%. Fig. 6.14 shows the schematic diagram of an entrained flow gasifier. The fine solid material such as coal, biomass, etc., oxidant (such as air or oxygen), and steam are fed into the main vessel. As shown in the figure, they produce a dense cloud of entrained particles moving at a very high temperature and pressure. Since the flow is extremely turbulent, the reaction rate is very high, and it has a very high carbon conversion efficiency. The syngas is composed mainly of H2 , CO, and a small amount of hydrocarbons such as methane. Since the operating temperature is high, the resulting ash is converted into an inert slag, but the generated syngas is very clean and tar-free. Hence the entrained-flow gasifier is used to produce syngas almost from any feedstock. In general, the following characteristics can be named for an entrained gasifier: • This technology accepts a variety of solid feeds. In that view, we can use it for converting almost any solid waste.

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Figure 6.14 Schematic of an entrained-flow gasifier.

• • • • • • • •

It requires a large amount of oxidant The oxidant may come from the air, or oxygen can be directly fed to the gasifier. In most commercial plants, oxygen is the choice. One of the characteristics of the entrained flow gasifier is that the temperature within the reactor is almost uniform. We can see this in Fig. 6.13. The operation of the gasifier is slagging. It has a short reactor residence time. Entrained flow gasifier has a high carbon conversion, but it shows a low cold gas efficiency. Hence the percentage of syngas is lower compared to other technologies. Since it has a high level of sensible heat in its produced gas, heat recovery is required to improve efficiency. It is known as an environment-friendly technology.

6.3.3.2 Moving bed Moving-bed gasifiers operate at moderate temperature levels. It requires oxygen and stem and produces low methane-content syngas. The cold gas efficiency is higher than the entrained flow gasifiers and may reach up to 90%. Fig. 6.15 presents a schematic operation of a moving-bed gasifier. They operate at moderate pressure (25 to 30 atm), as shown in the figure. The feed of the gasifier is in the form of large particles. The oxidant is flowing upward in a countercurrent direction injected from the bottom of the main vessel.

Fuel cell power plants

Figure 6.15 Schematic of a moving-bed gasifier.

The gasification takes place at three different zones. For this purpose, the reactor consists of the following zones: Drying zone is at the top of the vessel, where the feed is dried and cools down the produced gas before leaving the reactor. Carbonization zone is located at the middle of the reactor. The feed is further heated and devolatilized as it descends in this area. Gasification zone is located below the carbonization zone. The devolatized particles are gasified with steam and carbon dioxide in this zone. Combustion zone locates at the bottom of the vessel, where the oxygen reacts with the remaining char and produces heat. Moving-bed gasifiers share the following characteristics: • The design of moving-bed gasifiers is simple and easy to design. • Although it has a simple design, its efficiency is high. • Compared to other technologies, the amount of oxidant is lower. • Since the coarse feed particles are fed into the gasifier, the feed preparation is relatively less complex. • Since the produced gas loses its temperature at the drying zone, the product temperature is low; therefore, we do not need expensive heat exchangers to reduce the syngas temperature. • As for entrained-flow gasifiers, different feeds can be used for gasification in moving-bed gasifiers. An advantage of moving-bed gasifiers is that the feed may contain a high moisture content.

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Figure 6.16 Schematic of a fluidized-bed gasifier. • •

Moving-bed gasifiers have a very high cold-gas efficiency. Compared to other technologies, the methane content of this technology is higher.

6.3.3.3 Fluidized bed As the name shows, in a fluidized-bed gasifier the solid particles are suspended in an oxygen-rich gas to make a fluidized bed. In these gasifiers, the feed particles are mixed with the particles that are undergoing gasification. For a sustainable operation, the size of the particles must be very small (less than 6 mm). Fluidized-bed gasification proceeds with similar characteristics as those of moving-bed, but it contains lower methane content of about 2–3%. Fig. 6.16 shows a schematic diagram of a fluidized-bed gasifier. In this design, feed particles enter the vessel from its side, and the oxidant is blown from the bottom of the reactor to fluidize the bed. As shown in the figure, the sustained nature of the bed generates a uniform temperature inside it. The operating temperature of the fluidized-bed reactors is at moderate levels, and the carbon conversion reaches about 90–95%. In these reactors, some char particles leave the vessel along with the syngas. Hence they must be recovered and recycled back to the vessel. Fluidized-bed gasifiers display the following characteristics: • The main characteristic of a fluidized bed is its high rate of heat transfer. The fluidized-bed gasifier benefits the same advantage. • This design can gasify a wide range of solid fuels.

Fuel cell power plants

It requires moderate oxygen and steam. The temperature of the gasifier is at a moderately high level. One of the benefits of the reactor is that the temperature is uniform throughout the gasifier. The cold gas efficiency of the fluidized bed gasifier is higher than that of entrainedbed gasifiers, but it has lower carbon conversion.

• • •

6.4. Summary Fuel cells are promising candidates for energy production in the future, where hydrogen plays the most important role as an energy carrier. SOFCs are among the best choices among different FC technologies since they are mature and operate at high temperatures. The present chapter studied different aspects of SOFCs as stationary power plants. Note that as a power plant, different components are crucial for a smooth and sustainable operation. These components were studied in more detail. Obviously, the chapter is just an introduction to the components and is used to start an opening to different aspects and knowledge of component design. Each component requires special attention and has lots of theoretical and experimental issues.

6.5. Problems 1. 2. 3. 4.

Investigate the usage of SOFC as a power plant module. What would be the advantage of lowering the operational temperature of a SOFC? Investigate the effect of temperature gradient inside the SOFC cell. How the temperature gradient in a DIR-SOFC is prevented?

References [1] P. Aguiar, D. Chadwick, L. Kershenbaum, Modelling of an indirect internal reforming solid oxide fuel cell, Chemical Engineering Science 57 (10) (2002) 1665–1677. [2] AVL, Idle reduction, https://www.avl.com/documents/10138/885889/AVL+Fuel+Cell+ Engineering+and+Testing, 2021. (Accessed 18 December 2021). [3] Massimiliano Cimenti, Josephine M. Hill, Direct utilization of liquid fuels in SOFC for portable applications: challenges for the selection of alternative anodes, Energies 2 (2) (2009) 377–410. [4] Miura Co., Miura Co launches fuel cell product in Japan with Ceres Power technology, https://fuelcellsworks.com/news/miura-co-launches-fuel-cell-product-in-japan-with-cerespower-technology, 2021. (Accessed 3 December 2021). [5] Sudhanshu Dwivedi, Solid oxide fuel cell: Materials for anode, cathode and electrolyte, International Journal of Hydrogen Energy 45 (44) (2020) 23988–24013. [6] Bloom Energy, Bloom energy case studies, https://www.bloomenergy.com, 2021. (Accessed 4 December 2021). [7] Air Liquide Engineering and Construction. Steam methane reforming plant, Germany, https:// www.engineering-airliquide.com/project-delivery-services-references/steam-methane-reformingplant-germany, 2022. (Accessed 10 January 2022). [8] Marta Gandiglio, Andrea Lanzini, Massimo Santarelli, Large stationary solid oxide fuel cell (SOFC) power plants, in: Modeling, Design, Construction, and Operation of Power Generators with Solid Oxide Fuel Cells, Springer, 2018, pp. 233–261.

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[9] N.V. Gnanapragasam, M.A. Rosen, A review of hydrogen production using coal, biomass and other solid fuels, Biofuels 8 (6) (2017) 725–745, https://doi.org/10.1080/17597269.2017.1302662. [10] KCERACELL, Anode materials, http://www.kceracell.com/anode.html, 2021. (Accessed 21 December 2021). [11] KCERACELL, Cathode materials, http://www.kceracell.com/cathode.html, 2021. (Accessed 21 December 2021). [12] KCERACELL, Electrolyte materials, http://www.kceracell.com/electrolyte.html, 2021. (Accessed 21 December 2021). [13] OsakaGas, About the solid oxide fuel cell, https://www.osakagas.co.jp/en/rd/fuelcell/sofc/sofc/ system.html, 2022. (Accessed 20 January 2022). [14] Subhash C. Singhal, Solid oxide fuel cells for power generation, Wiley Interdisciplinary Reviews: Energy and Environment 3 (2) (2014) 179–194. [15] Tetsuya Wakui, Ryohei Yokoyama, Effect of increasing number of residential SOFC cogeneration systems involved in power interchange operation in housing complex on energy saving, Journal of Fuel Cell Science and Technology 8 (4) (2011). [16] Wikipedia, Idle reduction, https://en.wikipedia.org/wiki/Idle_reduction, 2021. (Accessed 11 December 2021). [17] Wikipedia, Methane reformer, https://en.wikipedia.org/wiki/Methane_reformer, 2022. (Accessed 10 January 2022). [18] Masakazu Yoda, Shuichi Inoue, Yuya Takuwa, Kenichirou Yasuhara, Minoru Suzuki, Development and commercialization of new residential SOFC CHP system, ECS Transactions 78 (1) (2017) 125. [19] Hideo Yoshida, Hiroshi Iwai, Thermal management in solid oxide fuel cell systems, in: Proceedings of 5th International Conference on Enhanced Compact and Ultra Compact Heat Exchangers: Science, Engineering and Technology, Hoboken, NJ, USA, 2005.

CHAPTER 7

Combined heat and power systems Contents 7.1. CHP and fuel cells 7.1.1 The produced heat of an FC 7.1.2 Exergy 7.1.3 Thermoeconomics 7.2. General procedure for CHP designs 7.3. SOFC-based CHP system 7.3.1 Combined SOFC and gas turbine 7.3.2 Application for space warming 7.3.3 Combined SOFC and desalination 7.3.4 Application in supply utilities 7.3.5 Application for high-temperature batteries 7.4. PEMFC-based CHP system 7.4.1 The main concerns of a PEMFC-based CHP system 7.4.1.1 Efficiency 7.4.1.2 Temperature stability 7.4.1.3 Impact on the environment 7.4.1.4 Safety 7.4.1.5 Economy 7.4.2 Organic Rankine cycles 7.4.3 Calculation of CHP cycle 7.4.3.1 Fuel cell relations 7.4.3.2 The thermal cycle or ORC relations 7.4.3.3 Cooling tower relations 7.4.3.4 Heat exchanger relations 7.4.3.5 Heat transfer coefficient 7.4.3.6 Exergy 7.4.3.7 Exergy balance on the whole system 7.4.3.8 Thermoeconomics 7.4.3.9 Cost functions 7.4.4 Worked example 7.5. Summary 7.6. Problems References

337 338 344 347 349 350 350 351 352 354 354 355 355 356 356 356 357 357 357 361 362 363 366 366 369 370 371 372 372 373 383 383 384

7.1. CHP and fuel cells Increasing the overall efficiency of a power plant is possible through combined heat and power, also known as the CHP concept. In this technique the dissipated heat of the power plant is used for other purposes. For example, the heat generated from an Fuel Cell Modeling and Simulation https://doi.org/10.1016/B978-0-32-385762-8.00011-7

Copyright © 2023 Elsevier Inc. All rights reserved.

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electric generator for an industrial sector can heat up the offices and other buildings. In other cases the heat is transferred to other industrial processes. It can be used for vaporizing water for making drinking water out of seawater, as a preheater for boilers, and for many other industrial purposes. Fuel cells (FCs), as power-generating devices, are also suitable for CHP purposes. There are different high-temperature FCs, such as solid-oxide (SOFC), molten carbonate (MCSF), phosphoric acid (PAFC), and alkaline (AFC), which are considered in many different CHP cycles. Other low-temperature FCs such as direct methanol (DMFC) or polymeric electrolyte membrane (PEMFC) are not very common in CHP cycles since they provide a low-temperature heat source of obviously low quality. Some researchers proposed to use the low-temperature FCs in CHP cycles to increase the efficiency resulting in a reduction in payback return time. The present chapter focuses on the CHP cycles and on how an FC can be combined with other heat-requiring systems. The importance of CHP for high-temperature FCs is quite evident. In fact, there are lots of theoretical papers in the open literature on the usage of high-temperature FCs in CHP cycles, and more importantly, there are many systems in the market. However, for the low-temperature FCs, we will discuss the CHP method and give some CHP examples.

7.1.1 The produced heat of an FC Regardless of the FC type and the electrochemical reactions that take place, we know that the overall reactions cause the fed fuel to be oxidized by oxygen to produce electricity. The typical reaction of any fuel cell is expressed by Eq. (1.4), in which the fuel F is oxidized by oxygen O2 and produces FO. It is quite evident that the electrochemical reactions are not reversible and produce heat as a byproduct. In that point of view, we can rewrite the overall reaction as F+

1 O2 −−→ FO + heat. 2

(7.1)

Fig. 7.1 shows the energy balance for an FC, in which the input power is converted to either electricity or heat. Obviously, as discussed before, we use electric power as the main product of the FC. However, the produced heat can be used in combination with another process to benefit from such a valuable source of energy. Here we have to find the amount of the produced heat to make a good design. Hopefully, we can have a good estimation of the amount of thermal waste power by considering the efficiency of the FC. We know that the efficiency of any FC is described as the ratio of the electric output power to the input chemical power: η=

Pe , Pin

(7.2)

Combined heat and power systems

Figure 7.1 Conservation of energy on an FC.

where P means power. Also, from the fundamentals of electrochemistry we know that the efficiency of the FC is also described by η=

E , Eth

(7.3)

where E is the cell operational voltage, and Eth is the thermoneutral potential of the cell (described by Eq. (1.35)). Example 7.1. Calculate the thermoneutral potential of a SOFC that works at 1000 ◦ C. Answer. Thermoneutral potential should be calculated at the operating conditions. Hence, at 1000 ◦ C, for a SOFC, we must obtain the enthalpy at the same temperature. Since the overall reaction of SOFCs is the same as that of PEMFCs, we can use the results of Table 1.3. Thus we have H = −242524 J mol−1 .

Consequently, from the definition of thermoneutral potential we can write Eth =

−H

nF

=

242524 = 1.257 V. 2 × 96485

We see that by increasing the temperature the thermoneutral potential decreases. Example 7.2. The characteristic or I–V curve of the SOFC of Example 7.1 is given in the following table: Calculate and plot the efficiency of the FC. i (A cm−2 ) E (V)

0

100

200

300

400

500

600

700

800

900

1000

0.66

0.6

0.56

0.51

0.48

0.44

0.39

0.34

0.30

0.23

0.10

Answer. The efficiency of the cell is obtained using Eq. (7.3). For the present FC, the thermoneutral potential was calculated in the previous example. Therefore the efficiency of the cell is simply obtained by dividing its potential by 1.257. The results are plotted in Fig. 7.2.

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Figure 7.2 Efficiency of the cell.

From Fig. 7.1 we see that the energy balance results in Pin = Pe + Pth .

(7.4)

Although Eq. (7.4) gives an energy balance for the FC, it is not a good tool for obtaining the thermal energy dissipated by the cell because in this equation, we do not have any information about the input power or Pin . The input power is the result of electrochemical reactions taking place at the electrodes and is unknown. The only known parameter in Eq. (7.4) is Pe since we measure the voltage and current of the FC during its operation. Fortunately, we can calculate the thermal dissipated power of the cell by combining Eq. (7.4) with Eq. (7.2). The result yields 1 Pth = Pe ( − 1).

(7.5)

η

Eq. (7.5) helps us calculating the thermal heat of any FC. This heat may be used in any other heat-requiring process. Example 7.3. The I–V curve of a high-temperature PEM is given in the following table: The FC is made of 15 × 15 cm2 plates and generates 5 kW electricity at its i (A cm−2 ) E (V)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.9

0.7

0.63

0.57

0.5

0.43

0.38

0.34

0.3

0.18

0

maximum power point. For this FC: • Calculate its power curve and plot it on the I–V curve. • How many cells does this FC contain? • Calculate the efficiency of the cell if it operates at 150 ◦ C. Answer. High-temperature PEMs (HT-PEM) are considered hydrogen FCs. The only difference between these cells and PEMs is that in an HT-PEM the temperature can be increased above 100 ◦ C.

Combined heat and power systems

Figure 7.3 Efficiency of the cell. •

The power of the cell is obtained using Pe = E × i .

•

Since the cell has different values for voltage and current, it produces different electrical power at different current densities. Fig. 7.3 shows the power curve versus current density. The calculation shows that the maximum power of the cell is reached at i = 1.6 A cm−2 and is equal to Pmax = 0.48 W. Since the area of each cell is 15 × 15 = 225 cm2 , each cell can produce Pcell = 225 × 0.48 = 108 W. For a 5 kW stack, we need n=

•

Ptot 5000 = = 46.3, Pcell 108

which must be rounded to 47 cells. Since the overall reaction of HT-PEMs is the same as that of a normal PEM, we can use the results of Table 1.3 at 150 ◦ C. At this temperature, we have H = −242941 kJ mol−1 ,

and from this value we can calculate the thermoneutral potential: Eth =

−H

nF

=

242941 = 1.259 V. 2 × 96485

Finally, the efficiency of the cell at different current densities is obtained by dividing its voltage by Eth = 1.259. The result is plotted in Fig. 7.4.

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Figure 7.4 Efficiency of the cell.

Figure 7.5 The generated power of the cell.

Example 7.4. For the HT-PEM of Example 7.3, calculate the amount of the generated heat and discuss. Answer. The generated heat is obtained using Eq. (7.5). On the other hand, the efficiency of the cell is a function of its voltage. Hence we have to calculate the generated heat for each operating voltage according to the cell power and efficiency. Fig. 7.5 shows the result of the calculation. As we can see, the generated heat power increases by increasing the current density, but the input electric power is not monotonic. The electric power has a maximum depending on the FC and its characteristic curve. By having Pe and Pth we can calculate the input power. This value is also plotted in Fig. 7.5, and as we can see, it has a linear variation versus current density. This example shows that the thermal power increases by increasing the current density. Also, the input power has a linear shape attributed to the linear increase in input

Combined heat and power systems

fuel and oxygen. Therefore, if we want to design the FC for operating on the point of maximum power, then we will get a lot of waste heat. The cell efficiency at this point would be about 25%. Its quite evident that this point is not an appropriate case for stack design. This was also clear from the fact that at the maximum power point the cell experiences concentration polarization, which is not very good. In practice the stack is designed according to the efficiency. For example, we can choose the efficiency of 50% for design purposes. At this point, 50% of the input energy is converted to electricity, and obviously, we will have the same amount of heat power. The problem with this selection is that the number of cells in the stack increases because the higher the efficiency, the lower the current densities. Example 7.5. Assume that the stack is going to be made according to 50% efficiency. Calculate the number of the cells in the stack and compare it with the results of Example 7.3. Answer. According to Fig. 7.4, the cell has an efficiency of 50% at i = 0.4. Hence, for a 5 kW stack with 15 × 15 cells, we must have n=

5000 = 55.5, 0.4 × 15 × 15

which means that we need 56 cells. Comparing the result with Example 7.3, we need nine more cells. The above examples step-by-step illustrate the importance of CHP calculations. As we see, the heat and electric power are completely coupled, and various designs dramatically affect the efficiency and heat. Moreover, we cannot design upon the highest efficiency of the cell because the higher the efficiency, the larger the stack, which in turn makes the investment higher. Consequently, we have to make the best design with • a proper efficiency, • the least number of cells or the lowest possible investment, • a proper design for the usage of the waste heat (note that we cannot avoid it), and • the best economic design. Fig. 7.6 shows the general concept of CHP with FCs. In this concept, the generated heat of the FC is fed into another process that requires heat. The combined process uses the heat to generate useful products and also unavoidable waste. The combination of the two processes must be optimized to maximize the overall useful outputs. One of the major problems of the combined cycle is evaluating the outputs and comparing them with the input energy because the outputs of the FC and the combined process may be different. The output of the FC is its produced electric power. However, the other process may have a completely different output. For example, the other process may be a desalination unit where the heat is used to produce drinking water out of the seawater.

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Figure 7.6 General concept of CHP in FCs.

Then how can we compare the drinking water and electricity to calculate the overall performance and efficiency of the combined cycle? The answer to the above question is thermoeconomics. Thermoeconomics is a science of converting the thermodynamic processes to economic language. In other words, any energy or material flow is given a proper cost. By this concept fuel, thermal and electric energies, or any other useful output of the processes can be translated into money and thus are comparable. Since the value of the energy of different forms is better understood in exergy analysis, economic relations are usually coupled with exergy flows. Therefore, in some references, thermoeconomic analysis is also known as exergoeconomic analysis. To understand the concept, we first introduce the exergy and focus on its relations and then discuss the thermoeconomic concept.

7.1.2 Exergy Exergy analysis is important since it involves both the quantity and quality of the energy process. In this analysis the quality of energy is as important as or even more important than its quantity. One of the advantages of exergy analysis is its generality, meaning that it is carried on the whole components and whole types of energy. Its other advantage is that it directly connects the energy process to the economic factors. In other words, since we calculate the energy destruction and losses, we can calculate the total loss of energy and convert it to cost. Therefore the exergy of the process can be a good optimization goal for minimizing the net loss. When two systems are not in thermodynamic equilibrium, they can produce useful work, which is not possible when they are in equilibrium [31]. Now if one of the systems is larger than the other one, then we call it the environment and the smaller one the system. Exergy is defined as the maximum available work that the system can deliver until it reaches the equilibrium state with the environment [15]. In a closed system the exergy is conserved, but in reality the exergy may be destroyed due to friction [31,35]. By this definition exergy is, in fact, the amount of energy that can be used as the useful work. As we can deduce from the definition, exergy depends on the state of both the system and the environment. We have to define a reference state for obtaining the exergy since

Combined heat and power systems

Figure 7.7 The flow of mass and energy for an FC.

the amount of the work depends on the difference between the states of the system and environment [4]. It is pretty reasonable that the state of the environment is the reference point [3]. The discussion above shows that the exergy of a system is not a state variable because when a system is in equilibrium with its environment, it can produce no work. This means that its exergy is zero. However, if the same system is located in another environment, then it still can produce useful work. For this reason, exergy is more known as a pseudostate variable [3]. The main advantage of exergy analysis is that this method enables us to calculate the amount of input energy converted to useful work. Or better yet, we can calculate the irreversibility of the system by means of exergy analysis. For exergy analysis, it is important to be aware of the following definitions. Surrounding By definition surrounding is all the things except the system under consideration. This may include lots of things that are not even in thermal contact with the main system. Environment The environment is defined as a part of surroundings whose state variables are uniform and never change. By this definition the processes of the environment are always reversible. Therefore the irreversibility happens inside the system and at its boundaries [3]. Dead state The system can produce work as long as it is not in equilibrium with its environment. By producing work the state of the system moves toward the environment. When it reaches the environment state, it reaches an equilibrium state and cannot produce any further work. This state is called the dead state or sometimes reference state. Note that the FC works as a control volume since we have a flow of mass in and out of the cell. Fig. 7.7 shows the situation for an FC located in an environment with Tamb and Pamb . As stated before, for exergy analysis, we assume that these values remain unchanged. Moreover, as an electrochemical system in which chemical species are reacting with each other, the exergy of the system is divided into physical and chemical parts. Therefore we have to use a proper relation for defining the exergy of the system.

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Table 7.1 Standard chemical exergies of different materials for Tamb = 298.15 ◦ C and Pamb = 1.0 atm. Chemical exergy [kJ kmol−1 ] 236,100

Material

H2 O2 N2 H2 O(l)

3970 720 900

The following equation shows the total exergy of a process [3,22,37] for an FC: e = eph + ech .

(7.6)

The physical exergy has the following components: eph = (h − hamb ) − Tamb (s − samb ) +

c2 gz + , 2000 1000

(7.7)

where Tamb , hamb , and samb are the temperature, enthalpy, and entropy of the environment, respectively. For an ideal gas, this equation is reduced to the equation 

eph = cp Tamb



T T P k−1 − 1 − ln + ln( ) k , Tamb Tamb Pamb

(7.8)

where k is the specific heat ratio of the gas. The chemical exergy is obtained using the equation e = ch

n  i=1

xi eich

+ Ru Tamb

n 

xi eich ,

(7.9)

i=1

where Ru is the universal gas constant, and xi is the partial molar fraction of each component of the mixture. The values of exergy for chemical species are tabulated and available if reference handbooks. Example 7.6. For a hydrogen FC, obtain the exergy value for different species. Answer. For hydrogen fuel cells such as SOFC, PEM, or HT-PEM, the overall reaction is as follows: 1 H2 + O2 −−→ H2 O. 2 However, most of the time, oxygen is taken from the atmosphere. Therefore it also contains other materials, particularly nitrogen. For such a system, we need to obtain the exergy value for all the involved species. These values are tabulated in Table 7.1.

Combined heat and power systems

7.1.3 Thermoeconomics Enhancing the efficiency of the system is always favorable unless it violates economic issues. By increasing the energy consumption in developing countries the new system designs must follow the economic calculations. Combining the first and second laws of thermodynamics with economic concepts gives a powerful tool for systematic analysis of the energetic systems, including power plants. The resulting science is called Thermoeconomics [1]. Many different methodologies have been developed during the past decades to relate the exergy parameters to economic factors. All of them are nothing other than the correct application of the second law of thermodynamic [6,26]. Thermoeconomics is also known as the biophysical economics, in which the thermodynamic concepts are studied with economic concerns [28]. This term is attributed to Myron Tribus, who is an American engineer [9,13,30], and was developed by Nicholas Georgescu-Roegen, who was a statistic and economist in the 1980s [12]. Thermoeconomic analysis deals with the fact that the efficiency of the system should not be considered with the thermodynamic concerns, but it should be optimized to increase the economic factors [7,8]. Thus in this field of study the thermodynamic relations are combined with economic relations [28] to relate the energy, work, and heat to the overall cost of the plant [29]. By this definition thermoeconomics can be considered as an interdisciplinary field [36]. In economic analysis, we have to define the plant costs during its operating life. The costs have different parts, including plant investment and construction, operating and maintenance, fuel, and wearing of the parts. These costs must be included in the overall analysis [3]. The cost function, also known as the total annual cost (TAC), is the final goal of the analysis and is considered as the final objective function of thermoeconomic analysis. This function in the most general form is written as TAC = Cinv + Cop ,

(7.10)

which contains two parts, initial investment cost and annual operation cost. We discuss these parts in detail. The initial investment cost is the product of the cost of all the system components and capital recovery factor (CRF). This factor is in fact an indication of the total investments for one year and is expressed by the equation Cinv = CRF ×



Call components .

(7.11)

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The capital recovery factor depends on the interest rate i and the expected operating years of the plant, n. The relation is defined by the equation CRF =

i . 1 − (i + 1)−n

(7.12)

The annual operation cost As is shown in the following equation, the annual costs are divided into the fuel cost and the maintenance parts: Cop = Cfuel + Cmain .

(7.13)

Usually, the cost of hydrogen pump is negligible in comparison to the FC itself and also to other components. Hence we neglect the power of the hydrogen pump since it is not comparable with the overall power generation. Therefore the fuel cost solely depends on the hydrogen cost [33]: Cfuel = cH2

WFC ηcell

,

(7.14)

where cH2 is the cost of hydrogen per one kWh power generation, ηcell is the efficiency of the system, and WFC is the cost of electricity, which can be obtained using the equation [33] WFC = P × 24 × 365 × Z ,

(7.15)

where P is the power of the fuel cell in kW, and Z is the capacity factor of the plant. The capacity factor is the ratio of the actually produced energy of the plant to the maximum calculated theoretical energy. The capacity factor is usually calculated over a whole year. In addition, the operation and maintenance costs are calculated using the equation Cmain = 0.07Cinv .

(7.16)

From Eqs. (7.14) and (7.16) the total operation cost can be calculated using Eq. (7.13). Then the operational cost is added to the investment cost or Eq. (7.11) to find the TAC from equation (7.10). This function is considered as the main objective function that must be minimized. A thermoeconomic optimized design is the one in which the TAC is minimized, translated to the fastest return rate. One of the benefits of such calculation is that thermoeconomic analysis gives a single value to all the different products. In other words, all the products and losses are translated to the annual cost, and if TAC is minimized, then we will have the most benefit of the whole combined cycle.

Combined heat and power systems

7.2. General procedure for CHP designs As discussed before, combining the heat of an FC with another process has lots of possibilities. For high-temperature FCs, wasted heat can be used in many different ways. For instance, the SOFCs and MCFCs work at temperatures higher than 900 K. This temperature is sufficient for evaporating water and generating superheat vapor. Therefore we can use the vapor as a product for generation of additional electricity or use it in desalination process. We can also use the FC heat for space warming, gasification, preheating process, and any other heat requiring process. The FCs that work at moderately high temperatures, such as AFC, HT-PEM, or PAFC, usually operate at temperature levels above 400 K. Although the temperature of these cells is not as high as SOFC, they can still produce superheat vapor. The obtained vapor is not at suitable levels for generating power at steam power cycles. However, they can be used at organic Rankine cycles (ORCs) to produce electricity. ORCs are quite similar to Rankine cycles, but their working fluid is not water. In an ORC the working fluid is chosen such that the hot temperature produces enough pressure for driving a steam turbine. In many applications, ammonia is the best choice and has been used in many different designs. In addition to power generation, the temperature of these FCs is suitable for other applications such as desalination and space warming. The application of CHP for high- and moderate-temperature FCs is known and practically used in many designs. In fact, at the present time, it would be very strange if these FCs were not designed for cogeneration. By proper design the efficiency of an FC may be increased up to 80 or 85%, whereas without cogeneration, the normal efficiency is about 50%. This increase in efficiency cannot be ignored. In contrast to the high- and medium-temperature FCs, the application of CHP is rarely considered in low-temperature FCs such as PEMFCs or DMFCs. The normal operating temperature of these FCs is about 320–370 K. At this level the water remains in its liquid form at atmospheric pressure. Therefore we do not have steam or vapor in these FCs. At first glance, the low level of temperature suggests that low-temperature FCs are not suitable for CHP. However, the investigations reveal that we can still benefit from cogeneration in such FCs. For example, the cooling water of FCs can be used for space heating. Also, we can feed the generated heat to ORCs operating at low temperatures. For such temperature levels, many different coolants, including R134a, R143, and many other refrigerants, can be used as the primary working fluid at Rankine cycles. Also, by lowering the pressure the temperature would be sufficient for desalination purposes. All the above-mentioned scenarios are possible, provided that the economic factors are satisfied. In other words, if a CHP cycle is economically profitable, it will be welcomed by the customers. Thus for having a reasonable design, we must follow the following steps;

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1. Draw a primary sketch for the CHP plant. 2. Perform an exergy analysis and obtain the exergy of all the involved processes. Also, the exergy analysis gives a detailed and quantitative estimation of exergy destruction in each plant component. 3. Use proper correlations to convert the exergy to the cost and calculate the TAC. 4. Use optimization algorithms to minimize the TAC as the main cost function of the optimization algorithm. These steps give a general guideline for any cogeneration design, including the cycles involving FCs.

7.3. SOFC-based CHP system SOFCs are considered as one of the best choices for CHP applications due to the following reasons. Firstly, they work at the highest possible temperature among the other FCs. Secondly, they are usually manufactured with high powers since they are considered for large-scale power generation. Thirdly, they are currently under mass production, and their production costs are becoming lower and lower. Consequently, using SOFCs in CHP applications is quite welcome. Typical value for a SOFC efficiency is something between 50 and 60%. The rest of the input energy is almost converted into heat, which can be used for any desirable CHP purposes. Since the temperature of SOFCs is very high, the thermal heat source is suitable for a variety of applications. For instance, it is suitable for steam generation, space heating, used in different thermal cycles, and any other heat required system. In the following subsection, we explain different applications of SOFCs in CHP systems by some examples. The examples show the capability of SOFCs in different CHP cycles. Obviously, there are many other ideas available in the open literature, and the applications of SOFCs are not limited to these examples.

7.3.1 Combined SOFC and gas turbine The high-temperature rejected heat of SOFCs is suitable for exploiting in thermal power plants. According to the nature of the power plant, the rejected heat can be used in different ways. For instance, in a steam power plant the waste heat is used to evaporate water to produce superheated steam. The steam is subjected to a steam turbine to generate additional electricity. The combination of SOFC and steam turbines is quite straightforward. Theoretically, this combination can be used with any Rankine cycle or even with Stirling cycles [24]. Although we can use the excess heat in such thermal power cycles, different scenarios are studied theoretically. The research is still progressing to increase the efficiency of the combined cycle. In contrast to the Rankine cycles, we have to use a different strategy in combination with a gas turbine. Fig. 7.8 shows a combination of a SOFC and a gas turbine. In this

Combined heat and power systems

Figure 7.8 Illustration of CHP for SOFC and gas turbine.

combination the exhaust gases of the gas turbine or expander flow through the heat exchangers to warm up the inlet fuel and air, which is fed to the SOFC by the fuel and air compressors. The fuel and air react in the SOFC with an internal reformer to produce heat and electricity. The excess fuel and air are very hot due to the high temperature of the SOFC. Then these hot flows are fed in the combustion chamber. The combustion chamber burns the excess fuel and completes the gas turbine cycle. As we can see, in this combination the rejected heat of both SOFC and the gas turbine is used. This means that the efficiency of the system becomes as high as possible.

7.3.2 Application for space warming The high temperature of SOFCs is used in space warmings. This is the easiest possible CHP cycle. However, we can combine more complex cycles to increase the efficiency of the whole assembly as much as possible. For instance, [20] showed that the CHP system suggested in Fig. 7.8 can be further improved by exploiting more energy from the waste heat. The concept of their work is schematically shown in Fig. 7.9. In this figure the whole cycle has remained the same, but as we can see, the excess heat of the gas turbine is used to make warm water used for space heating or drinking hot water. Such a combination would be very attractive for large residential or even industrial buildings.

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Figure 7.9 Improvement of CHP system of Fig. 7.8 by adding an HRSG for space warming.

7.3.3 Combined SOFC and desalination The desalination sectors produce fresh water through different technologies. According to the used technologies, these sectors are divided into many different types. One of the types is called thermal desalination since it uses thermal energy to convert the seawater into drinking water. In thermal sectors the water is heated up and converted into steam. Then the steam is cooled down to produce fresh water. Since thermal desalination requires heat, they can be integrated with SOFCs too. The rejected heat of a SOFC is quite suitable for producing steam out of seawater. The produced steam can then be fed into desalination units. Fig. 7.9 is one of the designs proposed by Meratizaman et al. [19]. This design is, in fact, the extension of their previous CHP works discussed in the previous section. Compared with Fig. 7.9, the heat used for producing utility warm water is used for steam generation. The steam is then fed in a MED desalination unit, where the seawater is converted into desalinated water pumped into the corresponding tank and the brine, which is returned to the sea by the brine pump.

Combined heat and power systems

Figure 7.10 Combination of desalination with the CHP system of Fig. 7.8.

Other desalination systems can also be integrated with SOFCs due to their highquality characteristics. It is not necessary to use a gas turbine or any other extra facilities. The SOFC itself is sufficient for producing superheated steam, which would be suitable for desalination. The same combinations can also be done by means of molten-carbonate fuel cells (MCFCs). Just note that the operating temperature of an MCFC is lower than those of SOFCs. A wide range of high-temperature FCs can be integrated into the desalination units when it comes to water desalination. Since desalination requires superheated steam, even the medium high-temperature FCs such as AFCs and HT-PEMs can also be used. The operating temperature of these FCs is high enough for the evaporation of water. Therefore they are also good candidates for desalination.

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7.3.4 Application in supply utilities It is well known that the industries with high energy consumption produce a considerable amount of CO2 . Therefore these sectors are to be optimized to reduce the total CO2 emission. Many different scenarios have been considered, studied, and even applied to minimize carbon emissions. One of the solutions is the integration of hightemperature FCs with supply utilities. Since the operating temperature of these devices is high, they can be used in these sectors. Usually, in an industrial unit, the boilers are responsible for heat generation, which is used to produce superheated steam with different degrees of pressure and temperature. The steam is fed to different units of the factory according to the required pressure and temperature of a unit. In some other industries, a part of the consuming electricity comes from the installed generators. One of the widely used generators is the gas turbine. The exhaust of the gas turbine is used to evaporate steam through HRSGs to increase the efficiency of the total site. These CHP plants have been studied by many researchers and published in many papers. It is important here that SOFCs can also be integrated with supply utilities. Whether the site contains gas turbines or not, the high temperature of SOFCs can help the boilers produce superheated steam. Many different scenarios are defined and developed by different scientists in the field. As an example of such works, we can name the works of Fakour et al. [10,11]. In these studies the combination of SOFCs with supply utility is considered. They combined the SOFCs with the site supply utilities in different ways to see which scenario is more affordable, which one produces less carbon, and which case is the most economical. To achieve these goals, they made thermoeconomical calculations based on the same exergy analysis discussed here.

7.3.5 Application for high-temperature batteries Some different battery systems work at high temperatures and stop working when cooled down. As an example, sodium-sulfur or NaS batteries invented by the Ford Company in 1966 [5]. In these batteries the molten sulfur works as the positive electrode, and the molten sodium as the negative electrode. A solid beta alumina ceramic electrolyte separates these liquid electrodes. Since sodium and sulfur are solid at room temperature, the battery should be kept at a high temperature to melt their electrodes. The normal operating temperature of the cells is about 300 ◦ C. As another example of such batteries, we can name ZEBRA batteries [27]. These batteries are rechargeable or, better to say, are secondary cells used for large-scale energy storage. The materials used in these batteries are very different but typically are made of sodium and nickel. The ZEBRA batteries (also known as sodium-nickel-chloride) work at very high temperatures since the materials can generate electricity while they are melted. Therefore they must be kept at 260–350 ◦ C.

Combined heat and power systems

Figure 7.11 Application of CHP for SOFC and ZEBRA batteries.

The above examples show that these batteries can be combined with SOFCs in a win-win situation. In other words, SOFC supports the heat required for keeping the batteries in their operational state, whereas the batteries store the SOFC electricity for utility shaping. The concept of the design is proposed by Antonucci et al. [2] and is schematically shown in Fig. 7.11. As we can see, the power produced by SOFC is stored in the ZEBRA energy storage, and the required heat is fed to the battery pack through the heat exchanger shown by HX1. The battery provides the demand and stores the additional energy produced by SOFC. Note that here the concept of CHP is important, and the energy dispatching is not studied; the way the produced electricity is fed to the grid or stored is out of the scopes of the present book.

7.4. PEMFC-based CHP system As a high-temperature FC, the concept of CHP is quite natural for SOFCs, MCFCs, or any other high-temperature FC. However, for low-temperature devices such as PEMFCs, using CHP is not quite common. PEMFCs usually work at around 80 ◦ C, which is not high enough for CHP purposes. However, this level of temperature may be used in organic Rankine cycles (ORC) to produce additional electrical power. It is worth mentioning that ORC cycles have been used for electricity generation out of temperature sources as low as room temperature such as ocean thermal energy conversion (OTEC) systems, where the hot source is the ocean surface water with temperature as low as 25 ◦ C.

7.4.1 The main concerns of a PEMFC-based CHP system An ORC works with the cycle in the same way as a normal Rankine cycle, but the only difference is that in an ORC the working fluid is an organic fluid that evaporates at much lower temperature levels. Selecting an appropriate working fluid is crucial since it affects the efficiency and environment. There are many different organic substances used in ORCs, each of which is suitable for a special purpose. When selecting an organic material, we must answer the following questions: • Can the working fluid generate enough power? Or better said, is the efficiency of the cycle appropriate?

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Is the fluid stable in the operating temperature? Is the substance environment friendly? Is the substance safe enough? Is the used material economically affordable? Therefore these parameters must be taken into consideration before any further study is carried out. • • • •

7.4.1.1 Efficiency Selecting a proper working fluid greatly affects the cycle performance. Maizza and Maizza [18] suggested that in addition to chemical stability, the working fluid must have a high latent heat and high density but a low specific heat. In that case the working fluid absorbs larger amount of energy at evaporator resulting in smaller component size. In contrast, Yamamoto et al. [34] claims that the fluids with lower latent heat are superior since they produce a higher superheat pressure.

7.4.1.2 Temperature stability In addition to the evaporation, the condensation characteristics are also important. The condensation temperature must occur below 300 K so that the absorbed heat can be delivered to the atmosphere. Therefore the fluids such as methane, which have a very low condensation temperature, are not suitable for the current purpose. The freezing point of the working fluid is also important. The freezing temperature should be far lower than the operating conditions to prevent the fluid from freezing. Moreover, the working pressure must be in a reasonable range. A very high pressure or very low vacuum reduces the system reliability and operating life.

7.4.1.3 Impact on the environment For the temperature levels about the working temperature of a PEMFC, the common refrigerants including R−134 a, R−141 b, R−1311, R−7146, and R−125 can be used. These refrigerants have been used in refrigeration cycles for many years and have shown good performance and stability. They can be stored in stainless steel tanks and pipes for many years without being affected by the material of the container and show negligible corrosion. For environmental considerations, the refrigerants are revised every year, and each year some of them are put away from usage. For instance, R−11, R−12, R−113, R−114, and R−115 were commonly used before but are not used in refrigerant cycles anymore. They are believed to have harmful impact on ozone layer. Also, R−21, R−22, R−123, R−124, R−141 b, and R−142 b are going to be removed out of the list of refrigerants in near future (maybe up to 2030). The main problems of refrigerants are the following: 1. ozone depletion potential,

Combined heat and power systems

2. global warming potential, and 3. atmospheric lifetime. These problems lead to irreversible impacts on the environment and must be seriously considered.

7.4.1.4 Safety The refrigerants must be safe in different factors. They must be inflammable, nontoxic, and noncorrosive. In practice, most alkanes are flammable. Some of them are toxic, and others are corrosive. Therefore the ideal case is not always reachable. Consequently, for any application, we must select the best material among different types. From a safety point of view, some substances are unsafe although they are safe within an acceptable threshold. For example, R-601 is flammable only when it is in contact with a proper flame. Otherwise, it is not considered a flammable substance. Other alkanes are self-flammable when they are over 200 ◦ C. The classifications of refrigerants regarding safety are discussed in ASHRAE handbook. The safety factor then must be checked before using a refrigerant in practice. Note that the flammability is not the only issue. Other parameters must also be considered according to the guidelines of the handbook.

7.4.1.5 Economy Working with refrigerants as the main working fluid in an ORC requires economic consideration. These fluids usually are expensive and in some cases are not extensively available. Hence their usage in an ORC increases the cycle cost. This fact is more important if we know that to generate quite a lot of power, a great amount of working fluid is needed. Also, in maintenance periods, we need to recharge the cycle with the same material. Thus, in addition to the safety and other mentioned issues, the cost of the working fluid is a great issue. In some cases, making an ORC is not a big deal itself, but the economic considerations make the whole project impossible.

7.4.2 Organic Rankine cycles The classic Rankine cycles are available in any thermodynamic book. However, when different organic fluids are used as the main working fluid, different conditions must be studied. Here we briefly check the main characteristics of a classic Rankine cycle and then we will study different ORCs. For studying a Rankine cycle, it is customary to work on its T–S diagram. The T–S diagram gives much useful information about the cycle. For instance, according to Fig. 7.12, we have three different fluid types.

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Figure 7.12 Different working fluids.

Wet fluid is a fluid in which the slope of saturation vapor curve is negative, or in other words, we have ds/dT < 0. Many different materials such as water have such a characteristic. Dry fluid is a fluid in which ds/dT > 0 (such as pentene). Isentropic fluid is a fluid in which ds/dT = 0 (such as R11). Wet fluids need to reach the superheat region to produce proper power in a Rankine cycle. However, the dry fluids can be used even if they do not reach the superheat state. Fig. 7.13 shows the main components of a standard Rankine cycle. In this cycle the water evaporates inside the evaporator and runs a steam turbine. After passing the turbine, the water loses energy and falls in the liquid-vapor region. Then the water mixture passes through a condenser and becomes a saturated liquid. The saturated liquid is pumped to a higher pressure to become ready for getting heated again and evaporates once again. An ORC has the same configuration, and the only difference is that in ORC an organic fluid exists instead of water. Since in an atmospheric pressure, water evaporates at 100 ◦ C, the boiler or evaporator temperature must be much higher than that. However, when the evaporator temperature is not that high, we can use organic materials that can be boiled in much lower temperature. For example, the refrigerants have a large range of evaporation temperature. Some of them evaporates at room temperature, and others even at lower levels. According to the T–S diagrams shown in Fig. 7.14, we can distinguish two different cycle categories, bell-shaped, shown in Fig. 7.14a, and overhanging, shown in

Combined heat and power systems

Figure 7.13 Schematic of Rankine cycles.

Figure 7.14 Thermodynamic behavior of Rankine cycles for different working fluids.

Fig. 7.14b. For simplicity, we use B-ORC to refer to the bell-shaped types and OORC for overhanging ones. Consider the T–S diagram of a B-ORC shown in Fig. 7.14a. The working fluid leaves the condenser at states T3 and P3 . Then it pumps to a designed pressure by a pump and reaches Pmax = P4 = P1 . From point 4 to 1, the working fluid is evaporated by being heated in a boiler or evaporator. Then it enters a turbine and generates power. The turbine reduces the pressure of the working fluid to Pmin = P2 . Point 2 is located in the saturated region, in which both the liquid and gas phases of the working fluid exist. It is customary to consider an isentropic process for describing the turbine. We can also assume a more realistic situation in which the turbine increases the entropy of the working fluid. In such a case, instead of point 2s , the working fluid reaches point 2 as shown in Fig. 7.14a. From point 2 to 3 the working fluid looses its temperature through the condenser and starts the cycle again.

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Figure 7.15 Details of a GPHE [32].

The O-ORCs also follow the same cycle as B-ORCs. However, as we see in Fig. 7.14b, when the working fluid passes the turbine, points 2s and 2 fall into the superheat region. The difference shows that these two different working fluids require different attention and cannot be used interchangeably. The main components of an ORC are shown in Fig. 7.13. Just like a normal Rankine cycle, it includes a pump, a turbine, and two heat exchangers, one for condensing the fluid when it leaves the turbine and the other for evaporating it after the pump. Since the ORCs work in low temperature levels, the gasket-plate heat exchangers (GPHEs) are the best choice for the condensers and evaporators. These heat exchangers are used in low-pressure applications where the temperature differences are not very high. The choice of GPHEs requires special attention. Since the arrangement of the plates, their size and other characteristics make the design more complex. Unlike a shell and tube heat exchanger, there is no classic way for selection and designing of GPHEs. The main advantage of GPHEs is that they have a large exchanging area, which is capable to transfer proper heat in very small temperature difference. A typical GPHE is shown in Fig. 7.15.

Combined heat and power systems

Figure 7.16 Schematic of PEMFC hybrid cycle.

7.4.3 Calculation of CHP cycle Due to the low temperature of PEMFCs, we can use an ORC to produce electrical energy. This can happen by combining the ORC shown in Fig. 7.13 with the FC shown in Fig. 7.16. The cooling water of the FC is used as the hot source to evaporate the organic working fluid to produce superheated vapor. We also need a cold source to condense the working fluid in the condenser for this to happen. As we can see, we can use a cooling tower (CT) to produce the required cooling for the whole plant. The above configuration means that we have three different subsystems to evaluate: 1. FC subsystem in which the net electric power and the rejected heat must be obtained. 2. CT subsystem in which the net cooling load required for the operation of the ORC must be determined. 3. ORC subsystem is the connection link between the above two subsystems. The input power of the ORC cycle comes from the FC subsystem, and the CT subsystem determines the cooling power. Therefore the ORC subsystem must be designed such that the whole system produces as much electricity as possible. For each subsystem, we need to develop a model and simulate the subsystem. Then the connection between the subsystems is made by equalizing the heat flow according to Fig. 7.16. In the present subsections, we separately consider each subsystem and present the involved equations. In addition to the thermodynamic relations of each cycle, we try to give the related equations for the calculation of each component of the cells. Then we present the exergy and thermoeconomic analysis of the whole system. Finally, we try to give different examples to show the present analysis results on an actual sample.

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7.4.3.1 Fuel cell relations A fuel cell is made of a stack of N cells in series. Therefore the voltage of each individual cells and the net stack voltage are related to each other by the equation Vcell =

V , N

(7.17)

where, obviously, Vcell is the voltage of a single cell, and V is the voltage of the whole stack. The cell voltage is also useful for obtaining cell efficiency. Since, as mentioned in Chapter 1, the efficiency of a low-temperature fuel cell can be obtained using its thermoneutral potential Vcell ηcell = . (7.18) 1.481 The advantage of working with cell voltage rather than with stack voltage is that we can use thermodynamic relations for the calculation of the former. The relations were discussed in detail in Chapter 1. Here we repeat the final result: 









RT i RT i RT iL,c − − Vcell = ET ,P − ln ln ln αc F i◦,c αa F i◦,a nF iL,c − i   RT iL,a − iRi , ln − nF iL,a − i



(7.19)

where i is the current density of the cell, F is the Faraday constant, αa and αc are, respectively, the anodic and cathodic transfer current densities, and i◦,a and i◦,c are the anodic and cathodic exchange current densities. Also, the reversible open circuit voltage ET ,p of the cell, which is a function of temperature and pressure, is obtained using the equation 0.5 ET ,p = 1.482 − 0.000845T + 0.0000431T ln(PH2 PO ), 2

(7.20)

where PH2 and PO2 are the hydrogen and oxygen partial pressures, respectively. Having the current and voltage of the whole stack, the power of the fuel cell is obtained using the equation P = VI .

(7.21)

According to Faraday’s law, the consumed hydrogen is proportional to the stack current [14]; thus the molar flux of hydrogen can be found according to the equation n˙ =

I . 2F

(7.22)

The efficiency of the fuel cell defined by Eq. (7.18) indicates the amount of input energy converted to useful electrical energy. The rest of the input energy to the fuel

Combined heat and power systems

Table 7.2 Some common working fluids. Refrigerant

Chemical name

Cycle type

R-32 R-125 R-134a R-143a R-152a R-290 R-600 R-600a R-717 R-718 R-1270

Difluoromethane Pentafluoroethane 1,1,1,2-Tetrafluoroethane 1,1,1-Trifluoroethane 1,1-Difluoroethane Propane n-Butane Isobutane Ammonia Water Propylene (propene)

b b b b b b o o b b b

Tc (K) 351.26 339.17 374.21 345.86 386.41 369.83 425.13 407.81 405.40 647.10 365.57

Pc (MPa) 5.78 3.62 4.06 3.76 4.52 4.25 3.80 3.63 11.33 22.06 4.66

cell stack is lost by converting to heat. Therefore the lost energy is calculated by the equation ˙ FC = N × n˙ × HHV × (1 − ηcell ), Q

(7.23)

where HHV is the higher heating value of the fuel cell reaction. By this value we can find the water flow rate at which water is generated according to the fuel cell reactions: m˙ FC =

˙ FC Q . TFC cp,FC

(7.24)

To drive the water out, we need to use a pump that consumes a portion of the useful electrical energy. As a result, the net energy available will become less than the produced energy. Consequently, the efficiency of the fuel cell is ηFC =

˙ pump,FC P−W , N × n˙ × HHV

(7.25)

˙ pump,FC is the power of the water where P is the power of the fuel cell stack, and W pump, both in watts or kilowatts.

7.4.3.2 The thermal cycle or ORC relations The relations of TEC system depend on the working fluid, as explained before. Since the shape of the T–s diagram of the working fluid greatly affects the efficiency and calculations. As discussed before, selection of the working fluid depends on the critical temperature. In fact, the critical point temperature of the working fluid must be less than the higher temperature of the cycle. The properties of some common working fluids are given in Table 7.2.

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1. The relations of wet cycles The temperature of point 1 in Fig. 7.14a is known, since the temperature of the evaporator is known. Thus the pressure is also known since the pressure and temperature are related in the saturation state: T1 = Tevp , P1 = Psat@T1 .

(7.26) (7.27)

By having two distinct state variables the state of the fluid is known, and all other state variables are also known: h1 = hg@T1 , s1 = sg@T1 .

(7.28) (7.29)

The state of point 2 is also known, since its temperature is equal to the temperature of the condenser, and the pressure is also related to its temperature: T2 = Tcnd ,

(7.30)

P2 = Psat@T2 .

(7.31)

We can assume that the turbine undergoes an ideal process. Hence, from point 1 to point 2 we have an isentropic process: s2,s = s1 ,

(7.32)

h2,s = h(T2 , s2,s ).

(7.33)

Now we can include the real efficiency of the turbine to its isentropic model to obtain the real turbine process: h2 = h1 − ηturb,s (h1 − h2,s ), s2 = s(T2 , h2 ).

(7.34) (7.35)

The state of point 3 can also be obtained using its pressure: P3 = P2 ,

(7.36)

h 3 = h f@ T 2 , v3 = vf@T2 .

(7.37) (7.38)

Now the work of the pump can be found as w˙ pump,WF = v3 (P1 − P2 ).

(7.39)

Combined heat and power systems

Then having the specific work of the pump and the state variables at point 3, we can find the thermodynamic values at point 4: h4 = h3 + w˙ pump,WF ,

(7.40)

T4 = T (P1 , h4 ).

(7.41)

The above relations are used to obtain the exchanged heat between the FC component and the TEC part. This heat is transferred from FC to TEC in the evaporator: q˙ evp = h1 − h4 .

(7.42)

On the other hand, from the temperature difference at the outlet and inlet of the evaporator we can write the following relation for the FC cycle: ˙ evp = m Q ˙ FC cp,evp Tevp .

(7.43)

Therefore we can also obtain the water flow rate at TEC: m˙ WF =

˙ evp Q . q˙ evp

(7.44)

Finally, by obtaining the cycle flow rate we can calculate the condenser net heat and the net power generation of the turbine. Moreover, the power consumption of the recirculating pump can also be determined: ˙ cnd = m Q ˙ WF (h3 − h2 ),

(7.45)

˙ turb = m W ˙ WF (h1 − h2 ),

(7.46)

˙ pump,WF = m W ˙ WF w˙ pump,WF .

(7.47)

2. The relations of dry cycles The main difference between the wet and dry cycles is illustrated in Fig. 7.14. In a dry cycle, point 2 is located in the superheated zone, and we must refer to Eq. (7.48) instead of Eq. (7.33): h2,s = h(P2 , s2,s ).

(7.48)

Then the actual state variables are obtained using Eqs. (7.34) and (7.35) and the temperature from T2 = T (P2 , h2 ). The rest of the calculations are the same as for the wet cycles.

(7.49)

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7.4.3.3 Cooling tower relations The operation of the whole system depends on cold water produced in cooling tower. The flow G of air at the CT is calculated using the equation [21] hCT,in − hCT,out =

G [(hair,out − hair,in ) − (ωair,out − ωair,in )hmu ], m˙ CT

(7.50)

where hmu is the specific enthalpy of the water. The required work of the circulating pump must be calculated after the design of the heat exchangers. The overall efficiency of the PTEC plant is ηPTEC =

˙ turb + W ˙ FC − W ˙ pump,WF − W ˙ pump,FC ) (W , (70 × n˙ × HHV )

(7.51)

˙ turb is the turbine power, W ˙ FC is the power generated by the FC, W ˙ pump,WF is where W ˙ pemp,FC is the power required by the FC the power required by the working fluid, and W pump.

7.4.3.4 Heat exchanger relations The most relations used for the calculations of shell and plate heat exchangers are not very accurate. The flow pattern in these heat exchangers is almost counterflow but still has many complex patterns. The relations that exist today are suitable for having a rough estimation of the required surface area. These relations are divided into two categories, calculation of pressure drop and calculation of heat transfer coefficient [16,25]. 1. Calculation of heat exchanging surface In shell and plate heat exchangers the wavy structure of the surface increases the actual surface of the plates comparing if they were flat. The ratio of the increased surface, called the developed length, to the flat or projected length is called the enlargement factor and denoted φ . According to Fig. 7.17, we can write φ=

developed length . projected length

(7.52)

The factor φ depends on the corrugation pitch, corrugation depth, and plate pitch as shown in Fig. 7.17. The enlargement factor is almost something between 1.15 and 1.25 depending on the design. Typically, for early design stages, it is common to use φ = 1.17 as a reference value. Eq. (7.52) can be rewritten as φ=

A1 , A1p

(7.53)

where A1 is the actual heat transfer area, and A1p is the projected area defined by the manufacturer.

Combined heat and power systems

Figure 7.17 Geometrical characteristics of the heat exchangers.

The area A1p may be estimated from Fig. 7.17a according to the equation A1p = Lp Lw .

(7.54)

According to the figure, Lp and Lw can be estimated using other geometrical parameters as follows: Lp  Lv − D p ,

(7.55)

L w  Lh + D p .

(7.56)

The value of φ is also used for the calculation of the effective flow path. In these heat exchangers the flow channel is the path by which the two successive plates are connected. A gasket is placed between each plate to prevent leakage. Since the flow channel is not a simple pipe, its surface area is calculated using the mean flow channel gap shown by b in Fig. 7.17b. From the figure it is clear that b = P − t,

(7.57)

where P is the pitch of the plate, and t is the thickness of the plates. Note that the plate pitch P should not be mistaken with the wave pitch Pc . The mean flow channel gap b is used for the calculation of flow rate and its Reynolds number, which is important in the calculation of heat transfer coefficients. Therefore it is an

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Fuel Cell Modeling and Simulation

important parameter, but, unfortunately, it is not usually available in the catalogs or brochures. Thus its calculation is not very accurate. For instance, to calculate P, we can refer to Lc determined by the manufacturers: P=

Lc , Nt

(7.58)

where Nt is the total number of the plates. For fluid calculations, it is assumed that the channels have circular cross-section with an effective diameter. This is the same notion, which we also know as the hydraulic diameter: deff =

4 × Channel wet area 4Ac = . Channel wet perimeter Pw

(7.59)

According to the hydraulic diameter concept, the effective diameter is related to b via Eq. (7.59): deff =

4bLw 2b = , 2(b + Lw φ) φ

(7.60)

knowing that b  Lw. 2. Pressure drop The Reynolds number depends on the mass flow rate Gc and the equivalent diameter of the channel, deff : Re =

Gc deff μ

.

(7.61)

The mass flow rate is calculated using the equation Gc =

m˙ , Ncp bLw

(7.62)

where Ncp is the number of channels in each pass, which is calculated by the equation Nt − 1 , (7.63) Ncp = 2Np where Nt is the total number of the plates, and Np is the number of passes. The total pressure drop in the heat exchangers is due to the channels Pch and the connecting ports Pp . The channel pressure drop is calculated using the equation Pch = 4fch

Lch Np Gc2 μb −0.17 , ( ) deff 2ρ μw

(7.64)

Combined heat and power systems

where fch is the friction factor, kp , (7.65) Rez and kp and z are some coefficients that must be determined from the geometrical characteristics of the plates, such as the Reynolds number, and its chevron angle β . In some references such as [16] and [25], these parameters are tabulated. The pressure drop due to the connecting ports of plates is roughly estimated as 1.4 times the velocity equivalent head: fch =

Pp = 1.4Np

Gp2 , 2ρ

(7.66)

where m˙ . (7.67) π dp2 4 In the last equation, m˙ is the total mass flow rate of the port, and dp is the diameter of the port. As mentioned before, the total pressure drop is the sum of the channel frictional drop and the port pressure drop: Gp =

Ptot = Pch + Pp .

(7.68)

7.4.3.5 Heat transfer coefficient The heat transfer coefficient in shell-and-plate heat exchangers depends on the material and also is a function of the transferred heat between the plates via their connections. Therefore the fluid flow between the plates must also be considered. For this reason, we have to use the equivalent or effective hydraulic diameter deff , since they are not usually round. The references for heat transfer phenomena suggest the following equation for obtaining the Nusselt number: Nu = Ch (

deff G μ

)y (

Cp μ 1 μb 0.17 μb )3 ( ) = Ch Rey Pr 0.33 ( )0.17 , k μw μw

(7.69)

where deff is the hydraulic diameter obtained by Eq. (7.59), Ch and y are obtained from [16] and [25] by having the Reynolds number and the chevron angle, and Pr is the Prandtl number. In these heat exchangers the transition from laminar to turbulent flow happens in low Reynolds numbers. Therefore the heat transfer rate becomes larger, which is quite favorable.

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Fuel Cell Modeling and Simulation

The overall heat transfer coefficient is 1 1 1 t = + + , U hh hc kw

(7.70)

where t is the thickness, kw is the conductive heat transfer coefficient of the plates, and hc and hh are the convective heat transfer coefficient obtained from the Nusselt number or via the equation deff h , (7.71) k where k is the conductive heat transfer coefficient. ˙ Based on the The final goal of these calculations is obtaining the net heat transfer Q. obtained data, the net transferred heat is calculated using the equation Nu =

˙ = UAe Tlm , Q

(7.72)

where Ae is the heat transferred area, and Tlm =

T1 − T2 . T1 ln T2

(7.73)

7.4.3.6 Exergy In a PEM fuel cell, we deal with a control volume, rather than with a system. Therefore we need to develop our equations for a control volume. The exergy balance for a control volume is obtained using the combination of the first law of thermodynamic with the entropy generation (the second law) [17]. In control volume, the exergy is transferred by three different mechanisms: work, heat, and mass transfer. These mechanisms are illustrated by Fig. 7.18. For the rate of exergy destruction, the following is used: E˙ D =

 j

(1 −

  Tamb ˙ ˙ cv + )Qj − W m˙ in ein − m˙ out eout , Tj out in

(7.74)

˙ j is the heat transfer rate at the control where E˙ D is the rate of exergy destruction, Q ˙ cv is the total rate of transferred work volume boundaries, Tj is the temperature, and W between the control volume and the environment. For a complete exergy analysis, the balance of exergy must be carried on over each individual component. For our CHP system, we explained in detail this analysis. Fuel cell The physical exergy of each component is obtained using Eqs. (7.7) and (7.8) and the chemical part from Eq. (7.9) together with the values of Table 7.1. By using

Combined heat and power systems

Figure 7.18 Exergy analysis of a control volume.

these values the total exergy destruction of the PEM fuel cell is obtained by Eq. (7.6): ˙ FC , E˙ D,FC = m˙ H2 eH2 + m˙ air eair − m˙ H2 O eH2 O + m˙ FC (eFC,in − eFC,out ) − W

(7.75)

˙ FC is the fuel cell output power. where W Condenser and cooling tower The exergy destruction rate in the condenser and cooling tower is

E˙ D,CT = G(eair,in − eair,out ) + m˙ mu (emu − eair,out ) + m˙ CT (ecnd,in − ecnd,out ).

(7.76)

Turbine The exergy balance for the turbine yields ˙ turb . E˙ D,turb = m˙ WF (e1 − e2 ) − W

(7.77)

Pumps For pumps, the exergy destruction is calculated using the equation ˙ pump . E˙ D,pump = m˙ WF (e3 − e4 ) + W

(7.78)

7.4.3.7 Exergy balance on the whole system The only mass input to this system is hydrogen from the anode side and oxygen from the cathode side. Therefore the inlet exergy to the control volume is defined by the equation E˙ in = E˙ air + E˙ H2 ,

(7.79)

and the exergetic efficiency can be calculated using the equation

=1−

E˙ D,tot , E˙ in

where E˙ D,tot is the net exergy destruction for all the components.

(7.80)

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Fuel Cell Modeling and Simulation

7.4.3.8 Thermoeconomics To calculate the final cost of the PTEC combined plant, we can use the equation CPTEC =

TAC . WPTEC

(7.81)

This function is also considered as the objective function of the whole system since the net income is directly related to the net generation of the system. In this equation, WPTEC is the net annual generation of the cycle.

7.4.3.9 Cost functions The costs of different parts are the most challenging steps toward the economic analysis. In a real world, there are many different manufacturers that produce the plant components. The cost of these components depends on the brands and the make. Therefore, for a real design, we need to obtain the real cost for each component. However, from many references and handbooks (such as [23]) we can have a rough but reasonable estimation. Usually, the costs of the components are tabulated or given in the form of figures such as those shown in Figs. 7.19–7.21. According to such figures, the cost of each component can be obtained [23]. Fuel cell The cost of PEM fuel cells is a function of their power as expressed by the equation CFC = 500P .

(7.82)

Evaporator and condenser In the present system, as discussed, we have used the shell-and-plate heat exchangers both for the evaporators and condensers. Hence the cost evaluation of both of them follows the same procedure. The only difference between the evaporation and condensation is the value of the overall heat transfer coefficient U in Eq. (7.72). Therefore the evaporator and condenser require different heat transfer areas and have different costs. The cost of evaporator and condenser is obtained from Fig. 7.19 and is calculated using the equation Cevp/cnd = −0.0009A2cnd + 13.123Acnd + 590.78.

(7.83)

Cooling tower In the present proposed plant the cold water is provided by a cooling tower. The cost of cooling tower is shown in Fig. 7.21 and expressed by the equation 0.6947 . CCT = 257978V˙ CT

(7.84)

Pump As mentioned, the used pumps are not comparable in cost to the other components. Hence we can neglect their costs.

Combined heat and power systems

Figure 7.19 Cost evaporator and condenser.

Figure 7.20 Cost of turbine.

7.4.4 Worked example To clarify the formulations and methods presented here, we focus on some examples. Example 7.7. Use the formulations of a CHP cycle and obtain the efficiency of the proposed cycle and compare your results with the cycle proposed by Xie et al. [33]. Answer. Xie et al. [33] used a 10 kW PEMFC working at 55 ◦ C. In that cycle, ammonia or R-717 was used as the main working fluid with properties given in Table 7.2. The basic parameters used in this example are tabulated in Table 7.3. These data are consistent with the data used by Xie et al. [33]. The main difference of the present study

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Fuel Cell Modeling and Simulation

Figure 7.21 Cost of cooling tower.

with the original work is that we have used a TEC subsystem for providing the cooling water. Then the whole system is also improved by using an optimization algorithm so that the best performance of the whole system is achieved. The results of the simulations are tabulated in Table 7.4. Table 7.3 Typical parameters of the process. Parameter

Property

Value

P V N Pair PH2

Nominal fuel cell output power Fuel cell output voltage Total number of cells Fuel cell inlet air pressure Fuel cell inlet hydrogen pressure Fuel cell coolant TEC working fluid Cooling tower fluid

10 kW 48 V 70 3 atm 5 atm Water Ammonia Water

Table 7.4 The efficiency of the hybrid cycle. Subsystem

PEMFC TEC Total

[33]

Efficiency (%) Present modified cycle

40 4.4 45.3

46 9 51.5

Combined heat and power systems

Figure 7.22 Exergy destruction in different components. Table 7.5 The base values for design variables. Parameter

Value

Unit

N I TFC,out

70 208 55

A ◦C

The results show that although the cooling system is added to the system, the overall cycle can achieve higher net efficiency. Example 7.8. Discuss the share of exergy destruction in different components. Answer. Fig. 7.22 shows the results of exergy analysis on all the components. As we can see, 93% of the total exergy destruction of the system happens in the PEM fuel cell. This means that if we are going to make any optimization, it would be much better to concentrate on the fuel cell itself. Optimization of other components does not have any significant meaning. Also shown in the figure, the condenser and cooling tower are in the second place by 4% exergy destruction followed by the turbine having 2% share. Other components including pumps and evaporator share only 1%. Since the main exergy destruction happens in the fuel cell, it would be reasonable to focus on this component and optimize its parameters. In the present example, we choose the number of the cell stacks N, the working current I, and the outlet temperature of the cell TFC,out as the main design variables to optimize the net output electricity cost as the main objective function. In other words, we seek the best values for N, I, and TFC,out to reach the highest TAC. The base parameters are tabulated in Table 7.5 and used as designed variables for optimization. We discuss the results of optimization in the following examples. Example 7.9. Using the design variables, try to optimize the cycle.

375

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Fuel Cell Modeling and Simulation

Figure 7.23 Exergy destruction in different components for optimized cycle and ammonia as working fluid.

Answer. The optimization means minimizing the exergy destruction in the whole system. Therefore we use the design variables shown in Table 7.5 and use a genetic algorithm (GA) code for optimization of the whole system. Note that any optimization scheme can be used here. For ammonia, the optimization results in N = 207,

I = 89 A,

and

T = 69 ◦ C.

The results are shown in Fig. 7.23. As we can see, the share of exergy destruction in the fuel cell reduces to 90%. This shows that the cycle has lower exergy destruction, which means that the efficiency of the whole system is increased. It is known that the working fluid influences the cycle efficiency. The importance of working fluid was discussed in this section. Here we will simulate the whole system with different working fluids to investigate their effect on exergy of the system. Example 7.10. Recalculate the combined cycle using different refrigerants. In your investigation, compare the results of the base and optimized cycles. Answer. The whole system is simulated using the data of Table 7.5. Then for each refrigerant, the cycle is optimized individually. We see that for different refrigerants, we obtain different values for the design variables. The results are tabulated in Table 7.6. The results show that the type of refrigerant strongly affects the working parameter. In particular, the fuel cell output temperature is a function of the refrigerant. Therefore, when designing a CHP cycle, these factors must be kept in mind. Example 7.11. Discuss the share of exergy destruction for different working fluids. Answer. Figs. 7.24–7.32 show the share of exergy destruction for each component of the system. From the figures we can deduce that the working fluid also affects the exergy destruction share.

Combined heat and power systems

Figure 7.24 Share of exergy destruction of each component for R-32.

Figure 7.25 Share of exergy destruction of each component for R-125.

Figure 7.26 Share of exergy destruction of each component for R-134a.

Figure 7.27 Share of exergy destruction of each component for R-143a.

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Fuel Cell Modeling and Simulation

Table 7.6 Optimized values. Working fluid

R-32 R-125 R-134a R-143a R-152a R-290 R-600 R-600a R-717 R-1270

Parameter

N 242 198 244 225 241 205 200 237 207 237

I (A) 95 96 91 81 96 89 100 92 89 86

TFC,out (◦ C) 65 64 68 74 68 69 63 67 69 71

Figure 7.28 Share of exergy destruction of each component for R-152a.

Figure 7.29 Share of exergy destruction of each component for R-290.

One of the interesting results of the exergy analysis is cost evaluation. As discussed, exergy analysis relates the energy costs to operation and maintenance, which makes a powerful tool for decision making. In the following example, we will show how this analysis can help us to reduce the overall cost of a power plant.

Combined heat and power systems

Figure 7.30 Share of exergy destruction of each component for R-600.

Figure 7.31 Share of exergy destruction of each component for R-600a.

Figure 7.32 Share of exergy destruction of each component for R-1270.

Example 7.12. Discuss the share of cost of each component of the CHP cycles for different working fluids. Answer. Figs. 7.33–7.40 show the share of costs for each component of the system. Again, to have a better understanding, we have used the optimization methods to optimize the base cycle.

379

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Fuel Cell Modeling and Simulation

Figure 7.33 Share of cost of each component in the base and optimized cycles for R-32.

Figure 7.34 Share of cost of each component in the base and optimized cycles for R-125.

Figure 7.35 Share of cost of each component in the base and optimized cycles for R-134a.

Example 7.13. Calculate the total annual cost (TAC) for the base and optimized cycles obtained in Example 7.13. Discuss the results and show how optimization will affect TAC.

Combined heat and power systems

Figure 7.36 Share of cost of each component in the base and optimized cycles for R-143a.

Figure 7.37 Share of cost of each component in the base and optimized cycles for R-290.

Figure 7.38 Share of cost of each component in the base and optimized cycles for R-600.

Answer. TAC of the base and optimized cycles are summarized in Table 7.7. The results show that the TAC of the optimized cycles are dramatically higher in an optimized cycle. Therefore we can deduce that CHP can be best practiced when the optimization of economic parameters is considered.

381

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Fuel Cell Modeling and Simulation

Figure 7.39 Share of cost of each component in the base and optimized cycles for R-600a.

Figure 7.40 Share of cost of each component in the base and optimized cycles for R-1270.

Table 7.7 Total annual cost of the cycles. Working fluid

TAC for base cycles ($)

TAC for optimized cycles ($)

R-32 R-125 R-134a R-143a R-152a R-290 R-600 R-600a R-717 R-1270

21361 17671 20724 17157 21562 17101 18697 20356 17142 18963

10589 10570 10570 10576 10576 10578 10570 10568 10587 10584

Combined heat and power systems

7.5. Summary In this chapter, we discussed the application of CHP in increasing the efficiency of different FC technologies. For high-temperature FCs, the CHP is quite natural and has been studied and applied as discussed. However, we see that CHP can also be used in combination with low-temperature FCs such as PEMFCs. Therefore it is a good practice in thinking of CHP when dealing with any FC technology. For becoming more familiar with CHP, we first introduced its concept. Then we discussed different scenarios in which CHP can be applied to SOFC as one of the best high-temperature choices. Finally, we showed that PEMFCs can also be used in CHP cycles to produce higher efficiency. In all the discussed cases, we must keep in mind that increasing the efficiency of the system is not interesting unless it has economic attraction. For this reason, we focused on the thermoeconomic or exergoeconomic concepts to emphasize that the efficiency increase must be done by considering the economic concerns. Otherwise, the design would be meaningless.

7.6. Problems 1. Calculate and plot the thermoneutral voltage of hydrogen FCs with respect to time and plot the results on a chart. 2. Using the data for SOFC of Example 7.2, calculate the efficiency of the FC at different temperatures and plot all the results on the same chart. Use T = 800, 900, 1000, 1100 ◦ C as the operating conditions. 3. Example 7.3 deals with the calculation of generated heat at 150 ◦ C. Repeat the example for other temperatures such as 125, 175, and 200 ◦ C. 4. Calculate the generated heat at different temperatures for the FC of the previous problem. Plot the results on the same chart and discuss. Is there an optimum value for operating temperature at which the highest possible electricity is produced? 5. Investigate the concepts of a. exergy, b. exergy balance, and c. exergy destruction. 6. Investigate some real examples of CHP with different FC technologies. 7. Eq. (7.82) gives a rough estimation for PEMFC price. The new achievements in this industry result in lower prices. Obtain a finer formula for the price of PEMFCs by gathering recent prices and drawing the trends. 8. For a 100 kW PEMFC working at 80 ◦ C, calculate the rejected heat if the cell efficiency is 60%. Then calculate the required surface area of a suitable heat exchanger and obtain its price according to the given relations. Note that the problem is openended, and you must assume some values according to the conventional data.

383

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Fuel Cell Modeling and Simulation

9. Assume that a 20 kW ORC is coupled with the FC of Problem-8. Design a proper cooling tower and calculate its price. Again, the problem is considered open-ended, and you must choose some details from your own experiences and the conventional values.

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[21] P.K. Nag, Power Plant Engineering, Tata McGraw-Hill, 2008. [22] Pierre Perrot, A to Z of Thermodynamics, Oxford University Press on Demand, 1998. [23] Max Stone Peters, Klaus D. Timmerhaus, Ronald Emmett West, et al., Plant Design and Economics for Chemical Engineers, vol. 4, McGraw-Hill, New York, 2003. [24] Rokni Masoud, Thermodynamic analysis of SOFC (solid oxide fuel cell)-Stirling hybrid plants using alternative fuels, Energy 61 (2013) 87–97, https://doi.org/10.1016/j.energy.2013.06.001. [25] E.A.D. Saunders, Heat Exchangers, John Wiley and Sons Inc., New York, NY, 1988. [26] Luis Serra, César Torres Cuadra, Structural theory of thermoeconomics, in: Mechanical Engineering, Energy Systems and Sustainable Development, vol. IV, EOLSS Publishers/UNESCO, 2009, p. 233. [27] Nimat Shamim, Edwin C. Thomsen, Vilayanur V. Viswanathan, David M. Reed, Vincent L. Sprenkle, Guosheng Li, Evaluating ZEBRA battery module under the peak-shaving duty cycles, Materials 14 (9) (2021) 2280. [28] Stanislaw Sieniutycz, Peter Salamon, Extended Thermodynamics Systems, vol. 7, CRC Press, 1992. [29] Páll Valdimarsson, Basic concepts of thermoeconomics, in: Short Course on Geothermal Drilling, Resource Development and Power Plants, organized by UNU-GTP and LaGeo, in Santa Tecla, El Salvador, 2011, pp. 16–22. [30] A. Valero, L. Serra, J. Uche, Fundamentals of exergy cost accounting and thermoeconomics. Part I: Theory, Journal of Energy Resource Technology (2005) 1–8, https://doi.org/10.1115/1.2134732. [31] Göran Wall, Exergy conversion in the Japanese society, Energy 15 (5) (1990) 435–444. [32] Wikipedia, Plate heat exchanger, https://commons.wikimedia.org/w/index.php?curid=68483905, 2022. (Accessed 19 January 2022). [33] Chungang Xie, Shuxin Wang, Lianhong Zhang, S. Jack Hu, Improvement of proton exchange membrane fuel cell overall efficiency by integrating heat-to-electricity conversion, Journal of Power Sources 191 (2) (2009) 433–441. [34] Takahisa Yamamoto, Tomohiko Furuhata, Norio Arai, Koichi Mori, Design and testing of the organic Rankine cycle, Energy 26 (3) (2001) 239–251. [35] E. Yantovski, What is exergy, in: Proceeding International Conference ECOS, 2004, pp. 801–817. [36] Yehia M. El-Sayed, Thermodynamics and thermoeconomics, International Journal of Thermoynamics 2 (1) (1999) 5–18. [37] Ahmet Yilanci, H. Kemal Ozturk, Oner Atalay, Ibrahim Dincer, Exergy analysis of a 1.2 kWp PEM fuel cell system, in: Proceedings of 3rd International Energy, Exergy and Environment Symposium, 2007.

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APPENDIX A

Lattice-Boltzmann codes A.1. GeometryGenerator The GeometryGenerator is designed to generate the proper geometry for a PEMFC electrode. The GDL (gas diffusion layer) of electrodes is made of carbon paper and must be correctly modeled for simulations. The whole code is presented in Program A.1. The parameters of the code are fully covered in Chapter 2.2 and are well discussed. To have a precise simulation of any PEMFC, we first need to define the geometry of the GDL. Hence the present code is a powerful tool for geometry generation, which will be used in other codes. The inputs of the code are: nx, ny, and nz define the mesh sizes in each direction; porosity describes the porosity of the GDL; β defines the anisotropic value of the electrode. Program A.1: GeometryGenerator code for generation of GDL microstructure. 1

PROGRAM GeometryGenerator

2

IMPLICIT NONE

3 4

!======================================================================

5

!

This program generates a 3D reconstructed microstructure for a PEMFC

6

!

carbon paper GDL with nx*ny*nz nodes via stochastic reconstruction

7

!

method. The two main inputs of the program are porosity and

8

!

anisotropy level (betta) of the designed carbon paper.

9

!======================================================================

10 11

integer, parameter :: nx=100, ny=120, nz=120, fiberlimit=65

12

integer, parameter :: numstep=22, numinterval=2**numstep

13

real*8, parameter :: pi=3.1415926535897932384 real*8, dimension (1:fiberlimit) :: xcfiber, ycfiber, zcfiber

14 15 16

real*8, dimension (1:fiberlimit) :: phi, p,tetta integer,dimension (0:nx,0:ny,0:nz) :: s

18

real*8:: betta,porosity,rfiber,tolsum,tolint,atotal,pin,solidity real*8:: dx,dy,dz,x0,y0,z0,u0,v0,w0

19

integer:: fibernumber,i,j,k,numsum,ii

17

20 21

dx=1.D0

22

dy=1.D0

23

dz=1.D0

24

numsum=100

25

tolsum=0.0001D0

387

388

Lattice-Boltzmann codes

26

tolint=0.000001D0

27

fibernumber=0

28

rfiber=3.5D0

29

porosity=0.78 betta=10000.

30 31

!

32

!

For Toray090 betta=10000 and porosity=0.78 For SGL10BA betta=100 and porosity=0.88 atotal=adpintegral(0.D0,pi)

33 34

open(unit=200,file=’ReconstructionData.plt’)

35 36

20

format(I7,6X,F19.6,6X,F19.6,6X,F19.6,6X,F19.6,6X,F19.6,6X,F19.6)

37 38

write(200,*)"VARIABLES =FiberNo,X0,Y0,Z0,U0,V0,Wo"

39

write(200,*)"ZONE ","I=",fiberlimit,",","F=POINT"

40 41

call RANDOM_NUMBER(xcfiber)

42

call RANDOM_NUMBER(ycfiber)

43

call RANDOM_NUMBER(zcfiber)

44

call RANDOM_NUMBER(phi)

45

call RANDOM_NUMBER(p)

46 47

do i=1,fiberlimit

48

xcfiber(i)=xcfiber(i)*nx

49

ycfiber(i)=ycfiber(i)*ny

50

zcfiber(i)=zcfiber(i)*nz

51 52

phi(i)=phi(i)*2.D0*pi p(i)=p(i)*atotal

53

pin=p(i)

54

tetta(i)=solvetetta(pin)

55

print *,"P & Tetta are",p(i), tetta(i)

56

end do

57

do ii=1,fiberlimit

58

x0=xcfiber(ii)

59

y0=ycfiber(ii)

60

z0=zcfiber(ii)

61 62

u0=sin(tetta(ii))*cos(phi(ii)) v0=sin(tetta(ii))*sin(phi(ii))

63

w0=cos(tetta(ii))

64

write(200,20) ii,x0,y0,z0,u0,v0,w0

65

call fibermaker(x0,y0,z0,u0,v0,w0)

66 67

print *, "fiber number",ii,"was created." solidity=0

68

do i=0,nx

69 70 71 72 73 74

do j=0,ny do k=0,nz if (s(i,j,k)==1) solidity=solidity+1 end do end do end do

Lattice-Boltzmann codes

75 76

if ((solidity*1.D0/nx/ny/nz)>=(1.D0-porosity)) then print*, "Porosity condition is satisfied."

77

exit

78

end if

79

end do

80 81

print*, "Porosity is",1.0-solidity*1.D0/nx/ny/nz call showgeometry

82

pause

83

close(200)

84 85

contains

86 87 88 89

real*8 function solvetetta(pppin) real*8 :: pppin,a,b,c,sum integer :: countsum

90 91

a=0.D0

92

b=pi

93

c=(a+b)/2.D0

94

countsum=0

95 96

do

97

sum=adpintegral(0.D0,c)

98

if ((abs(sum-pppin)=numsum)) then

99 100 101 102 103 104 105 106 107 108 109

exit else if (sum>pppin) then b=c c=(a+c)/2.D0 elseif (sum