Gasification processes : modeling and simulation 9783527673216, 3527673210, 9783527673209, 3527673202, 9783527673193, 3527673199, 9783527673186, 3527673180

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Edited by Petr A. Nikrityuk and Bernd Meyer Gasification Processes

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Edited by Petr A. Nikrityuk and Bernd Meyer

Gasification Processes Modeling and Simulation

The Editors

University of Alberta Department of Chemical and Materials Engineering 9107-116 Street Edmonton, Alberta, T6G 2V4 Canada

All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.

Prof. Dr. Bernd Meyer

Library of Congress Card No.: applied for

Prof. Dr. Petr A. Nikrityuk

TU Bergakademie Freiberg Department of Energy Process Engineering and Chemical Engineering Fuchsmühlenweg 9 09599 Freiberg Germany

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A catalogue record for this book is available from the British Library. Bibliographic information published by the Deutsche Nationalbibliothek

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V

Contents Preface XI List of Contributors XV Acknowledgments XVII Recommended Reading XIX Coal Gasification: Basic Terminology XXI 1

Modeling of Gasifiers: Overview of Current Developments 1 Petr A. Nikrityuk, Thomas Förster, and Bernd Meyer

1.1 1.1.1

Numerical Modeling in Engineering 1 The Role of Direct Numerical Simulation (DNS) in Particulate-Flow Modeling 3 Summary 6 CFD-based Modeling of Entrained-Flow Gasifiers 6 Mainstream Computational Submodels 8 Particle Conversion 9 Turbulence–Chemistry Interaction 12 Review of CFD-related Works 13 Noncommercial Software 13 Commercial Software 14 Summary 17 Benchmark Tests for CFD Modeling 17 British Coal Utilization Research Association Reactor (BCURA) 18 Brigham Young University Reactor (BYU) 19 Pressurized Entrained-Flow Reactor (PEFR) 22 References 24

1.2 1.2.1 1.2.1.1 1.2.1.2 1.2.2 1.2.2.1 1.2.2.2 1.3 1.3.1 1.3.2 1.3.3

2

Gasification of Solids: Past, Present, and Future Martin Gräbner

2.1 2.2 2.3 2.4

Introduction 29 Historical Background 30 Types of Gasification Reactors 33 Trends in Gasifier Development 36

29

VI

Contents

2.5

Derived Challenges for Research References 40

40

3

Modeling of Moving Particles: Review of Basic Concepts and Models 43 Sebastian Schulze, Robin Schmidt, and Petr A. Nikrityuk

3.1 3.2 3.2.1 3.2.1.1 3.2.1.2 3.2.1.3 3.2.1.4 3.2.2 3.2.2.1 3.2.2.2 3.2.2.3 3.2.3 3.2.3.1 3.2.3.2 3.2.3.3 3.3 3.3.1 3.3.2 3.3.3 3.3.4 3.3.5

Introduction 43 Soft-Sphere Model 47 Numerical Implementation 48 Contact Forces 48 Collision Parameters 49 Contact Detection 50 Time Integration 52 Validation Cases 53 Free-Falling Particle 53 Analytic Solution for the Free-falling Particle 54 Slipping Sphere on a Rough Surface 55 Illustrative Examples 56 Breaking Dam Problem 56 Rotating Drum 57 Generation of Fixed Beds 58 Hard-Sphere Model 59 Governing Equations 60 Collision Treatment in Dense Particulate Systems 62 2D Formulation of Hard-Sphere Collisions 63 Illustration of Hard-Sphere Models 65 Conclusions 68 Nomenclature 68 References 70

4

CD and Nu Closure Relations for Spherical and NonSpherical Particles 73 Kay Wittig, Andreas Richter, and Aakash Golia

4.1 4.2 4.2.1 4.3 4.4 4.4.1 4.4.2 4.4.3 4.5 4.5.1 4.5.2

Literature Review 73 Model Description 74 Numerical Scheme and Discretization 75 Code and Software Validation 78 Porous Particles 81 Geometry Assumptions 81 Heat and Fluid Flow Past Porous Particles 82 Drag and Nusselt Numbers for Porous Particles 85 Nonspherical Particles 88 Heat and Fluid Flow of Particles Oriented in the Flow Direction 88 Flow Characteristics of Particles at Different Angles of Attack 91

Contents

4.5.3 4.5.3.1 4.5.3.2 4.5.3.3 4.5.4 4.5.5

Influence of Particle Orientation on Drag Forces and Heat Transfer 95 Drag Forces 95 Heat Transfer 96 Drag Forces and Nusselt Relations for Two Rotations 97 Discussion 100 Conclusion 100 References 101

5

Single Particle Heating and Drying 105 Robin Schmidt, Kay Wittig, and Petr A. Nikrityuk

5.1 5.1.1 5.1.2 5.1.2.1 5.1.3 5.1.4 5.2 5.2.1 5.2.2 5.2.3 5.2.3.1 5.2.4 5.3 5.3.1 5.3.2 5.3.2.1 5.3.2.2 5.3.3 5.3.3.1 5.3.3.2 5.4

Nonporous Spherical Particle Heating in a Stream of Hot Air 105 State of the Art 105 Problem and Model Formulation 107 Linear Model 108 Illustration of Results and Subgrid Model 109 Semiempirical Two-Temperature Subgrid Model 114 Heating of a Porous Particle 116 Problem and Model Formulation 117 Porosity 118 Results of Simulations 119 Conclusion 123 Appendix: Analytical Model 123 Spherical Particle Drying in a Stream of Hot Air 124 CFD-based Drying Model 125 Subgrid Models 127 Standard Model 127 New Model 127 Illustration and Validation of Models 131 Results: CFD-Based Model 131 Validation of Subgrid Model 135 Conclusions 138 References 139

6

Unsteady Char Gasification/Combustion 143 Dmitry Safronov

6.1 6.2 6.2.1 6.2.2 6.2.3 6.2.4 6.3 6.3.1 6.3.1.1

Introduction 143 Modeling Approach 145 Governing Equations 146 Initial Conditions and Boundary Conditions 148 Reaction Kinetics and Transport Properties 151 Evolution of Pore Structure and Interface Tracking 152 Numerics and Code Validation 153 Results and Discussion 155 Oxidation Behavior of Porous Particles 156

VII

VIII

Contents

6.3.1.2 6.3.1.3 6.3.1.4 6.4 6.5 6.5.1 6.5.2 6.5.3

Details Inside the Particle 157 Effect of Ambient Gas Composition 157 Effect of Initial Particle Size and Ambient Gas Temperature on the Oxidation Regime 158 Advice for Beginners 160 Analytical Models 162 One-Film Model 162 Two-Film Model 165 Chemically Reacting Porous Particle 166 Nomenclature 167 References 168 171

7

Interface Tracking During Char Particle Gasification Frank Dierich and Kay Wittig

7.1 7.1.1 7.1.2 7.1.2.1 7.1.2.2 7.1.2.3 7.1.2.4 7.1.2.5 7.1.2.6 7.1.2.7 7.1.3 7.1.4 7.2

Interface and Porosity Tracking for a Moving Char Particle 171 Introduction 171 Model and Governing Equations 172 Setup 172 Governing Equations in the Gas Phase 173 Governing Equations in Porous Particles 174 Boundary Conditions at the Particle Surface 175 Reaction Kinetics 175 Change of Porous Structure and Particle Shape 176 Transport Properties 177 Numerics 178 Results and Discussion 180 3D Interface Tracking for a Porous Char Particle in the Kinetic Regime 192 Problem Description 192 Porous Particle Description 194 Internal Surface Reconstruction 196 Results 197 Conclusions 200 References 201

7.2.1 7.2.2 7.2.3 7.2.4 7.3

8

Pseudo-Steady-State Approach for Carbon Particle Combustion/Gasification 205 Matthias Kestel, Dmitry Safronov, Andreas Richter, and Petr A. Nikrityuk

8.1 8.1.1 8.1.2 8.1.3 8.1.4 8.1.5 8.1.5.1

Particle-Resolved CFD Simulations: Spherical Particles Review of the Literature 205 Setup and Model Formulation 207 Governing Equations 210 Boundary Conditions 211 Numerics and Software Validation 212 Validation against Analytical Solution 213

205

Contents

8.1.5.2 8.1.5.3 8.1.6 8.2 8.2.1 8.2.2 8.2.3 8.2.3.1 8.3 8.4 8.4.1 8.4.2 8.4.3 8.4.4

Validation against Experiments I: Laminar and Turbulent Regimes 214 Validation against Experiments II: The impact of Particle Porosity 217 Results: The Impact of Re on the Oxidation Regimes 219 Particle-Resolved CFD Simulations: Nonspherical Particles 225 Introduction 225 Shapes of Particles 227 Results 228 Integral Characteristics 231 Conclusions 235 Setup of Heterogeneous Reactions in ANSYS FLUENT 235 Step 1: Species Transport Settings 235 Step 2: Define Species and Mixtures 236 Step 3: Define Reactions 237 Step 4: Boundary Settings 238 Nomenclature 239 References 240

9

Pore-Resolved Simulation of Char Particle Combustion/Gasification 243 Andreas Richter, Matthias Kestel, and Petr A. Nikrityuk

9.1 9.2 9.2.1 9.3 9.3.1 9.4 9.4.1 9.4.2 9.5 9.6

Introduction 243 Model Assumptions and Chemistry 245 Numerical scheme, discretization, and software validation Small Porous Particle: 90 μm 249 Influence of Gas Temperature 256 Large Porous Particle: 2 mm 257 Small Reynolds Numbers 257 Large Reynolds Numbers 259 3D Simulations under Gasification Conditions 264 Conclusions 267 Nomenclature 267 References 268

10

Subgrid Models for Particle Devolatilization-Combustion-Gasification 271 Sebastian Schulze, Robin Schmidt, and Petr A. Nikrityuk

10.1

Subgrid Model for the Devolatilization/Combustion of a Moving Coal Particle 271 State of the Art 271 Model Formulation 276 Semiglobal Chemical Reactions 280 CFD-based Model 281 Model Validation 282

10.1.1 10.1.2 10.1.2.1 10.1.3 10.1.4

248

IX

X

Contents

10.2 10.2.1 10.2.1.1 10.2.1.2 10.2.2 10.2.2.1 10.2.3

Novel Intrinsic Submodel for Gasification of a Moving Char Particle 290 Model Formulation 291 Total Carbon Consumption Rate 292 Basic Equations 293 CFD-based Model 295 Numerics and Validation 297 Model Performance 297 Nomenclature 300 References 301

11

New Frontiers and Challenges in Gasification Technologies 305 Alexander Laugwitz and Bernd Meyer

11.1 11.2 11.2.1 11.2.1.1 11.2.1.2 11.2.1.3 11.2.1.4 11.2.2 11.2.2.1 11.3 11.3.1 11.3.2

Introduction 305 Trends in Gasifier Design 307 Advanced Fluidized-Bed Coal Gasifiers 308 Fluidized Bed with Slag Bath 309 Multistaged Spouted Bed with Slag Bath 310 Internal Circulating Fast Fluidized-Bed Gasifier (INCI) 311 Agglomerating Fluidized Bed with Internal Post Gasification 314 Highly Loaded Compact Gasifiers 315 Hybrid Wall Gasifier 316 Future Gasifier Simulations 319 Requirements of Proposed Future Gasifiers 319 Additional Fundamental Aspects of Future Numerical Simulations 322 References 325 Index 329

XI

Preface … in the field of coal science one can hardly distinguish between fundamental investigations, applied research and even process development. K.H. van Heek [1] This monograph aims to bridge coal gasification1) technology and computerbased modeling utilizing recent advances in computational fluid dynamics (CFD) including the methodology on numerical heat and mass transfer theories. Latest developments on coal gasification technologies around the world (e.g., China, USA, India, South Africa, Japan, Canada etc.) have demonstrated that coal-derived synthesis gas (syn-gas) utilisation for chemicals and electricity have become an indispensable part in the national energy security policies of that industrially developed countries. Because of the large reserves of coal on Earth, the importance of coal gasification will continue to increase in the future. The basic feedstock used in gasification technologies is crushed and possibly dried raw coal, which is fed into a reactor chamber, the so-called gasifier. On order to realize sustainable development of new generations of gasifiers with which it is possible to reduce their production and operation costs, it is imperative to use the so-called computer-aided design (CAD) and optimization. In this view, the bottleneck of such virtual design is a mathematical and numerical model describing physical and chemical processes inside a reactor–gasifier. Especially, simple models producing results close to reality are of great interest for the industry. However, it is impossible to develop simple models without understanding the basic fundamental processes characterizing high-temperature conversion in the gasifiers. This book is an effort to explore these fundamentals using the socalled direct or fully resolved numerical simulations of different physical processes related to interphase phenomena during the high-temperature conversion of coal and biomass particles under gasification conditions. In the design of novel gasifiers operating with solid carbonaceous fuels (particles), the important issue is the adequate prediction of the basic characteristics of such devices. Because of the complexity of the physical and chemical 1) Gasification defines a process that uses at least heat, steam, and carbon-based raw materials under high-temperature conditions to convert these materials directly into the so-called syngas composed primarily of carbon monoxide (CO) and hydrogen (H2 ).

XII

Preface

processes inside gasifiers, where high temperatures and pressures prevail, experimental studies are not always capable of characterizing the basic features of all related phenomena. One way to understand, predict, and optimize the complex processes in a gasifier is to use the CFD platform, which is based on the numerical solution of mass, momentum, energy, and chemical species conservation equations. However, the direct modeling of a gasifier resolving all the different scales, ranging from several meters for the whole reactor down to several micrometers for the coal particles, is impossible because of the lack of computing time to solve the system of equations. Therefore, the CFD modeling of a gasifier requires a multiscale approach in which the physics of the small scales is represented by submodels. In particular, typical submodels in a gasifier simulation calculate the small-scale turbulence, the chemistry–turbulence interaction, and the processes of drying, pyrolysis, and gasification/combustion of the particles. It should be noted that in spite of significant progress in the development of macroscale models for particulate flows and their numerical implementation in many commercial codes (ANSYS-Fluent, ANSYS-CFX) and open-source codes, the submodels which are used in the macroscale simulations, correspond to the models developed for coal combustion modeling in the early 1980s. Therefore, CFD-based models have become well-established tools for the understanding and optimization of fluid-particle flows in gasifiers. It is rather surprising that, in spite of the subject’s importance in the fields of chemical engineering and energy and material conversion, relatively few monographs are available on up-to-date numerical and semiempirical models describing interphase phenomena in high-temperature conversion processes such as gasification. The literature on comprehensive modeling of gasification is concentrated in conference papers or articles in scientific journals only. This book is an attempt to close the gap between the technological progress of gasification, which is well documented in the literature (e.g., see the monograph by Ch. Higman and M. van der Burg Gasification, or by J. Rezaiyan and N.P. Cheremisinoff ‘Gasification Technologies’), and the theoretical understanding and modeling of the interaction between chemically reacting solid particles and the surrounding gas, as applied to coal gasification technology. This book is designed as a specialized textbook for master’s or PhD courses in fields such as thermal sciences and chemical engineering where particulate flows with heterogeneous and homogeneous chemical reactions play a fundamental role. The purpose of this book is to present a description of a gas–particle reaction system taking into account the progress in the development of new models and numerical simulations for single-particle systems in an integrated, unified form. Special attention is paid to understanding and modeling the interaction between individual coal/char particles and a surrounding hot gas (300 K < T < 3000 K) including heterogeneous and homogeneous chemical reactions on the particle interface and near the interface between the solid and gas phases. This book, we hope, will also serve the needs of engineers from industrially oriented

Preface

R&D engaged in research, development, and design for technologies where chemically reacting particles play a significant role. This book is divided into 11 major chapters, beginning with an analysis of recent developments in computer-based simulations and mathematical multiscale models for the calculation of high-temperature conversion processes applied to gasification modeling including their validation. Next, Chapter 2 introduces a short review of the state of the art in the gasification of coal, including a brief history and analysis of existing large-scale facilities. Chapter 3 contains a review of the basic approaches used for modeling moving particles, including the treatment of particle–particle collisions and coupling with gas flows. Recommendations and illustrations for the application of these models are given at the end of the chapter. Chapter 4 is devoted to the closure relations for the drag force coefficient and the Nusselt number used for the spherical and nonspherical particles, including the influence of particle porosity on these parameters. The main new ideas, including the cornerstone of the book, are found in Chapters 5–11. Chapter 5 presents subgrid models and particle-resolved numerical simulations of coal particle heating and drying including validation against experimental data published in the literature. The new models are illustrated by comparing the results with predictions obtained using standard approaches, discussing the advantages and disadvantages of the latter with respect to the new models. Chapters 6 and 7 describe numerical models based on a fixed-grid method and the results of one-dimensional and two-dimensional numerical simulations devoted to analyzing the unsteady behavior of char particles undergoing gasification and combustion. Two models are illustrated: the so-called surface-based or shrinking-core model, and the so-called shrinking reacted-core model. The shrinking-core model is based on the assumption that the heterogeneous reactions occur at the particle surface. Thus, the carbon consumption is only related to the outer surface of the particle, whose diameter decreases over time. The shrinking reacted-core model takes into account intraparticle diffusion and intrinsic reactivity. In this case, the carbon consumption is related not only to the particle diameter but also to the particle porosity and specific surface. The particle interface tracking is treated using a sharp-interface method coupled with a fixed-grid method. Chapter 8 describes the pseudo-steady-state (PSS) approach for char particle combustion and gasification. In this chapter, a comprehensive CFD-based model is used for resolving the issues of bulk flow and boundary layer around the particle. The model comprises the Navier–Stokes equations coupled with the energy and species conservation equations. At the surface of the particle, the balance of mass, energy, and species concentration is applied to formulate the boundary conditions on the particle surface, including the effect of Stefan flow and heat loss due to radiation at the surface of the particle. The model is validated against experimental data published in the literature for the laminar and turbulent flow regimes. Finally, the influence of the Reynolds number, the ambient O2 mass fraction, and the ambient temperature on the behavior of char particle is discussed.

XIII

XIV

Preface

Chapter 9 includes descriptions of numerical simulations of the carbon conversion occurring in pores inside the particle. The PSS approach is used to explore the physics of the process. Numerous 2D and few 3D simulations are illustrated and analyzed. Chapter 10 presents advanced subgrid models for predicting the pyrolysis, gasification, and combustion of a single coal particle moving in a hot environment. Apart from the model formulation and description, this chapter includes numerical examples and validations illustrating new points and showing the robustness of the models. Finally, Chapter 11 discusses the needs and challenges in modeling the nextgeneration gasifiers which are under development or in the test phase. Finally, it should be noted that, as mathematical terms are used to introduce the models and solutions of conservation equations, the reader is expected to have a basic background in CFD (undergraduate course), including heat and mass transfer theory as applied to combustion engineering. It should also be noted that some chapters include practical recommendations for students and engineers to speed up simulations or to increase the accuracy of the models. Petr A. Nikrityuk University of Alberta Canada Bernd Meyer TU Bergakademie Freiberg Germany References 1. van Heek, K.H. (2000) Progress of coal

science in the 20th century. Fuel, 79, 1–26.

XV

List of Contributors Frank Dierich

Matthias Kestel

TU Bergakademie Freiberg Institute of Energy Process Engineering and Chemical Engineering, ZIK Virtuhcon Fuchsmühlenweg 9 09596 Freiberg Germany

TU Bergakademie Freiberg Institute of Energy Process Engineering and Chemical Engineering, ZIK Virtuhcon Fuchsmühlenweg 9 09596 Freiberg Germany

̈ Thomas Forster

Alexander Laugwitz

TU Bergakademie Freiberg Institute of Energy Process Engineering and Chemical Engineering Fuchsmühlenweg 9 09596 Freiberg Germany

TU Bergakademie Freiberg Institute of Energy Process Engineering and Chemical Engineering Fuchsmühlenweg 9 09596 Freiberg Germany

Aakash Golia

Bernd Meyer

Indian Institute of Technology Department of Mechanical Engineering Assam, 781039 Guwahati India

TU Bergakademie Freiberg Institute of Energy Process Engineering and Chemical Engineering Fuchsmühlenweg 9 09596 Freiberg Germany

̈ Martin Grabner

Air Liquide Forschung und Entwicklung GmbH FRTC – Frankfurt Research & Technology Center Gwinnerstrasse 27-33 60388 Frankfurt am Main Germany

XVI

List of Contributors

Petr A. Nikrityuk

Robin Schmidt

University of Alberta Department of Chemical and Materials Engineering 9107-116 Street, Edmonton Alberta, T6G 2V4 Canada

TU Bergakademie Freiberg Institute of Energy Process Engineering and Chemical Engineering, ZIK Virtuhcon Fuchsmühlenweg 9 09599 Freiberg Germany

Andreas Richter

TU Bergakademie Freiberg Institute of Energy Process Engineering and Chemical Engineering, ZIK Virtuhcon Fuchsmühlenweg 9 09596 Freiberg Germany

Sebastian Schulze

TU Bergakademie Freiberg Institute of Energy Process Engineering and Chemical Engineering Fuchsmühlenweg 9 09596 Freiberg Germany

Dmitry Safronov

TU Bergakademie Freiberg Institute of Energy Process Engineering and Chemical Engineering, ZIK Virtuhcon Fuchsmühlenweg 9 09596 Freiberg Germany

Kay Wittig

TU Bergakademie Freiberg Institute of Energy Process Engineering and Chemical Engineering, ZIK Virtuhcon Fuchsmühlenweg 9 09596 Freiberg Germany

XVII

Acknowledgments It is privilege of youth to look farther and to decide quicker and to give its opinion more firmly, however, they should not forget that they are standing on the shoulders of preceding generations. K.H. van Heek [1] This book is based on the basic achievements of the research carried out at the Centre for Innovation Competence (CIC) “Virtuhcon,” Group “Interphase Phenomena,” in the Department of Energy Process Engineering and Chemical Engineering, Technische Universität Bergakademie Freiberg, Germany, and financed by the Saxon Government and the Federal Ministry of Education and Research of Germany. A primary goal of the research program was devoted to the model development, simulation, and visualization of high-temperature conversion processes applied to gasification technologies. We, the editors, would like to thank Prof. M. Stelter, Prof. Ch. Prof. Brücker, Prof. D. Trimis, Prof. P. Scheller, Prof. B. Jung, Prof. M. Eiermann, and Prof. H.J. Seifert for their engagement and primary role in the initialization of CIC Virtuhcon. We also gratefully acknowledge the contribution made to the establishment of CIC Virtuhcon by Prof. W. Heschel, Dr. S. Krzack, Dr. R. Pardemann, Dr. M. Gräbner, Dr. R. Gutte, and Dr. Annett Wulkow. We also acknowledge Dr. Bernd Schumann from Project Management Jülich, Research Centre Jülich GmbH, Germany, for his valuable comments and suggestions during the project phase. We would also like to make a special mention of Prof. A. Gupta of Maryland University, USA, and Prof. R. Gupta of the University of Alberta, Canada, for their help related to productive cooperation in the field of “microscale” modeling of gasification. Finally, we would like to thank all the authors for their contributions and the audience for their discussion and comments. We hope that any colleagues whose work has not been mentioned in this acknowledgment will forgive us, since such omissions are unintentional. References 1. van Heek, K.H. (2000) Progress of coal

science in the 20th century. Fuel, 79, 1–26.

XIX

Recommended Reading The main goal of this book is to introduce closure submodels for the description of a single coal particle’s behavior in an entrained-flow gasifier. These submodels were developed on the basis of the so-called particle-resolved simulations carried out using computational fluid dynamics (CFD) coupled with heat and chemical species conservation equations. It should be emphasized that no attempt has been made in this book to explore all details of the numerical algorithms used in CFDbased simulations. The reader who requires a more comprehensive explanation of numerical algorithms used in CFD and a theory of chemically reacting flow is referred to other textbooks for a complete understanding of numerical algorithms applied in this monograph. The following list of books provides a source of references for a more detail study of the various areas related to computational heat and mass transfer applied to high-temperature conversion of coal into syngas. The books are grouped by three subject areas: theory of gasification, computational fluid dynamics, and chemical engineering.

• Theory of gasification –Overview of gasification technologies and phenomenological description of gasification [1, 2]. –Gasification theory for engineers [3]. –Fundamentals of coal combustion and gasification [4]. –Modeling approaches to gasification [5]. –Physics and models of gas–solid reactions [6, 7]. –Fundamentals of chemically reacting flows [8]. • Computational Fluid Dynamics and transport phenomena –Numerical approaches to heat and fluid flow for beginners [9]. –Computational methods for fluid dynamics [10, 11]. –Phenomenological and semianalytical description of transport phenomena [12, 13]. –Computational gas–solid flows and reacting systems [14]. –The fluid dynamics, heat and mass transfer of single bubbles, drops and particles [15].

XX

Recommended Reading

• Chemical engineering –Chemical reactor modeling [16]. –Fluid dynamics in chemical engineering [17, 18]. –Particle technologies in chemical engineering [19]. Finally, it should be noted that none of these books fully discusses the particleresolved simulations of coal gasification or computational submodels for hightemperature coal conversion. As a result, this book can be considered as additional material for the theory of gasification on a particulate level including computational heat and mass transfer theory applied to chemically reacting particles moving in a hot ambient gas. References 1. Higman, C. and van der Burgt, M.

2.

3.

4.

5.

6.

7.

8. 9.

10.

(2008) Gasification, 2nd edn, Elsevier GPP, Gulf Professional Publishing, Amsterdam U.A. Krzack, S. (2008) Die Veredlung und Umwandlung von Kohle, Technologien und Projekte 1970 bis 2000 in Deutschland, chapter Grundlagen der Vergasung, DGMK Deutsche Wissenschaftliche Gesellschaft für Erdöl, Erdgas und Kohle e.V., pp. 299–306. Rezaiyan, J. and Cheremisinoff, N.P. (2005) Gasification Technologies: A Primer for Engineers and Scientists, CRC Press. Smoot, L.D. and Smith, P.J. (1985) Coal Combustion and Gasification, The Plenum Chemical Engineering Series, Springer. de Souza-Santos, M.L. (2010) Solid Fuels Combustion and Gasification: Modeling, Simulation, and Equipment Operation (Mechanical Engineering), 2nd edn, CRC Press, Boca Raton, FL, pp. 33487–2742. Szekely, J., Evans, J.W., and Sohn, H.Y. (1976) Gas-Solid Reactions, Academic Press Inc. Kee, R.J., Coltrin, M.E., and Glarborg, P. (2003) Chemically Reacting Flow, Theory & Practice, John Wiley & Sons, Inc., Hoboken, NJ. Turns, S.R. (2006) An Introduction to Combustion, 2nd edn, McGraw-Hill. Patankar, S.V. (1980) Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corporation. Ferziger, J.H. and Peric, M. (2002) Computational Methods for Fluid Dynamics,

11.

12.

13.

14.

15.

16.

17.

18.

19.

3rd edn, Springer-Verlag, Berlin, Heidelberg, New York. Versteeg, H.K. and Malalasekera W. (2007) An Introduction to Computational Fluid Dynamics: The Finite Volume Method, 2nd edn, Pearson/Prentice Hall. Bird, R.B., Stewart, W.E., and Lightfoot, E.N. (2007) Transport Phenomena, 2nd edn, John Wiley & Sons, Inc. Plaswsky, J. (2010) Transport Phenomena Fundamentals, 2nd edn, CRC Press, Boca Raton, FL. Sreekanth, P., Syamlal, M., and O’Brien, T.J. (2011) Computational Gas-Solid Flows and Reacting Systems: Theory, Methods and Practice, IGI Global. Clift, R., Grace, J., and Weber, M.E. (1978) Bubbles, Drops, and Particles, Dover Publications, Inc., Minaola, New York. Rawlings, J.B. and Ekerdt, J.G. (2012) Chemical Reactor Analysis and Design Fundamentals, 2nd edn, Nob Hill Publishing, Madison, WI. Carberry, J.J. (2001) Chemical and Catalytic Reaction Engineering, 1st edn, Dover Publications, Inc., Mineola, New York. Rosner, D.E. (2000) Transport Processes in Chemically Reacting Flow Systems, 1st edn, Dover Publications, Inc., Mineola, New York. Richardson, J.F., Harker, J.H., and Backhurst, J.R. (2002) Chemical Engineering, Vol. 2, 5th edn, Butterworth Heinemann, Linarce House, Oxford.

XXI

Coal Gasification: Basic Terminology ...oil and gas last for decades, but coal for centuries K.H. van Heek [7] Before a reader jumps into reading this book, it is advantageous to introduce a short overview of the basic terminology and phenomena characterizing the gasification of solid carbonaceous materials. Finally, the interested readers may look up details in [1–3]. Strictly speaking in the literature there are several definitions of the word “gasification.” De Souza-Santos in his book [3] gives a general definition of gasification as transformation of solid fuel components into gases under high-temperature conditions. According to Rezaiyan and Cheremisinoff ’s book Rezaiyan, the gasification characterizes a process for converting carbonaceous materials into a combustible or synthetic gas (e.g., H2 , CO, CO2 , CH4 ). Finally, a short and very precise definition of gasification is given by Higman and van der Burgt Higman2008:In its widest sense, the term gasification covers the conversion of any carbonaceous fuel to a gaseous product with a usable heating value. It can be seen that this short and very valuable definition of gasification excludes combustion because the product gas of combustion does not have “residual heating value.” However, generally the term gasification applied to industrial usage of this technology is a very complex process, which includes many subprocesses which are defined by the transformation of a raw coal particle during its heating up to 2500 K. In particular, when a coal particle (moving or nonmoving) is heated, it passes through different thermal and chemical processes: drying, devolatilization (pyrolysis), combustion/gasification/, ash melting, fragmentation. These processes may occur sequentially or simultaneously, depending on the size of particle, heating rate, and the coal composition. Figure1 illustrates schematically these basic processes. Drying As a raw coal particle is heated and its temperature is increased, the evap-

oration of the water, which is present in a coal particle, occurs. The drying temperature depends strongly on the pressure in the reactor. In particular, the typical operation conditions are 1–60 bar [1], which corresponds to the boiling temperatures of water between 373 and 550 K [4]. In line with [2], drying can be expressed

XXII

Coal Gasification: Basic Terminology

Tboil < T < Tgasific Pyrolysis

Ash melting T> = Tmelt

T < Tboil Heating and drying

Coal particle

Gasification and combustion T > Tgasific Figure 1

Different states of a coal particle heated continuously up to 2000 K.

as follows: Coalwet + Heat −−−−→ Coaldry + Vapor Generally, drying of a solid particle is a complex, multiphase process, which combines three coexisting phases: liquid water, vapor(gas), and the porous solid through which the liquid and vapor migrate to the surface Devolatilization/Pyrolysis As a particle temperature increases, the so-called

devolatilization or pyrolysis,1) may start. The temperature when devolatilization occurs varies between 600 and 900 K [1, 2]. The process of devolatilization consists of the release of mixtures of organic and inorganic gases, which are called volatile gases, and liquids from the particle into the ambient atmosphere. In particular, during the heating the network of molecules that forms the coal particle becomes unstable and the volatile gases are split up [5]. The coal is converted to char. Typical gases are methane, carbon monoxide, carbon dioxide, water, hydrogen, nitrogen, hydrogen sulfide, ammonia, some unsaturated hydrocarbons, and tar [2]. This process can be summarized by the following equation [2]: Coaldry + Heat −−−−→ Char + Volatiles After devolatilization, the remaining char consists mostly of carbon and ash including small amounts of minerals, sulfur, and other elements, and accounts for 30–70% by weight of the original coal [5]. Gasification and Combustion Further increase of the char particle tempera-

ture leads to the the remaining conversion step, the so-called gasification or combustion depending on the ambient gas composition and the temperature. Particle gasification takes place by endothermic heterogeneous reactions of the char carbon with steam and carbon dioxide, or by exothermic reaction with hydrogen [5]. In a simplified notation, these reactions have the form 1) In the literature, the term pyrolysis is often used as a synonym for devolatilization, but strictly speaking this term means devolatilization in an inert gas atmosphere [1].

Coal Gasification: Basic Terminology

• the water gas reaction Cchar + H2 O + Heat −−−−→ CO + H2

• the Boudouard reaction Cchar + CO2 + Heat −−−−→ 2CO

• the methanation reaction Cchar + 2H2 −−−−→ CH4 + Heat As a rough rule, gasification reactions are endothermic. That is why basically in many gasification processes, the energy, which is necessary to start gasification, comes from the partial combustion of the solid fuel [3]. The main heterogeneous combustion reactions are the oxidation of carbon leading to carbon monoxide and carbon dioxide: Cchar + O2 −−−−→ CO2 + Heat Cchar + 1∕2O2 −−−−→ CO + Heat The following semi-global homogeneous reactions plays an important role during coal gasification:

• oxidation of carbon monoxide −−−−−−− ⇀ CO + O2 ↽ − CO2 + Heat

• the water-gas shift reaction −−−−−−− ⇀ CO + H2 O ↽ − CO2 + H2 + Heat

• the steam methane reforming reaction −−−−−−− ⇀ CH4 + H2 O + Heat ↽ − CO + 3H2 Further details on all these reactions can be found in Chapters 2 and 6–10. It should be noted here that, theoretically, in an ideal case, gasification processes can be organized in a such way that the heat released from oxidation (exothermic) reactions balances the heat needed for endothermic gasification reactions. However, in a real practice, the processes in a gasifier are controlled by many physical phenomena including interaction between particles and gas flow, where the flow is often turbulent. Moreover, all chemical reactions may take place simultaneously in a gasifier. This fact makes the design of a gasifier a nontrivial task. The maximum efficiency of a gasifier may be predicted using thermodynamic equilibrium models [6]. That is why the so-called computational fluid dynamics is becoming more and more popular in the industry in order to determine the optimum operating conditions of gasifiers, see Chapter 1 for details.

XXIII

XXIV

Coal Gasification: Basic Terminology

Ash Melting If a coal particle reaches a temperature higher than the so-called slag-

ging temperature, then ash or mineral matter2) on the particle surface or even inside the particle can be melted during high-temperature carbon conversion. The melting temperature of ash is strongly dependent on its multicomponent composition. Thus, generally, it is not a trivial task to predict accurately the melting temperature under different ambient gas conditions and compositions. The melting temperature of ash is a very important characteristic of any coal used for gasification, because it defines the maximum gas temperature needed or allowed to be reached in a gasifier, see Chapters 2 and 11 for details. Types of Gasifiers In the last century, the design of gasifiers has converged to three basic technical implementations. In particular, gasifiers can be divided into three main categories: moving(fixed)-bed, fluidized-bed, and entrained-flow gasifiers [1].

• In moving-bed or fixed-bed reactors, the coal bed moves under the influence of gravity. The gasification agent is normally injected at the bottom, and the flow of the gas counter to the moving bed leads to the formation of spatially localized zones of drying/devolatilization, gasification, and oxidation. Characteristic size of the particles used in fixed-bed gasifiers is between 1 and 10 cm [5]. Movingbed processes are normally operated at low maximum temperatures and low gas exit temperatures to avoid slagging and agglomeration of the particles. One main drawback of the method is that the pyrolysis gases are transported with the syngas and are not converted [1]. • In fluidized-bed processes, the mixing in the reactor and the heat and mass transfer to the particles lead to its maximum efficiency. The reactor is normally operated below the slagging (ash melting) temperature to avoid agglomeration of the particles. Characteristic size of particles used in fluidized-bed gasifiers is between 1 mm and 1 cm [5]. In fluidized beds, the carbon conversion is limited because fine carbon particles leave the reactor with the ash. To increase the carbon conversion, some attempts have been made to operate in an ash-softening zone to avoid the removal of fine carbon particles, see Chapters 2 and 11. • Entrained-flow gasifiers are characterized by a co-flow of the particles and the gas. The particles are fine, with the initial size 0.01 [3]. 5) Dilute particulate flows are defined by � < 0.01.

3

4

1 Modeling of Gasifiers: Overview of Current Developments

models: a hard-sphere and a soft-sphere model (see Chapter 3). However, recently, the soft-sphere model originally proposed by Cundall and Strack [32] has become more popular for UDPM simulations using a large number of particles (Np > 106 ) [33]. In spite of the significant success of coupled DPM/DEM CFD models in the prediction of fluidized-bed systems, one of the limitations of this class of models is the use of the so-called subgrid zero equation (0-D) models for the modeling of hydrodynamic forces acting on the particles, and heat and mass transfer between the particles and the fluid. Applied to heat transfer calculation using UDPM CFD models, the temperature evolution of the particles is basically calculated using a simplified semiempirical model where the effective heat transfer coefficient is calculated using a Nusselt number relation (e.g., Ranz–Marshall equation for a spherical particle [34]). This simplification is justified by the fact that the cell size of an Eulerian grid is larger than the size of the particles. Thus, to model fluid–particle interaction or the particle temperature, one requires closure correlations to describe the momentum exchange between the particle and the fluid or the heat transfer between the particle and the surrounding fluid (see [29]). In contrast to the UDPM, the resolved discrete particle model (RDPM) uses an Eulerian grid, with cells about one order of magnitude smaller than the size of the particles (see [26] for details). Both the particle–particle and particle–fluid interactions are modeled directly using hard-sphere/soft-sphere models and surface integrals, respectively. From this point of view, in the literature RDPM is often known as the direct numerical simulation (DNS) model or particle-resolved simulation (PRS). It should be noted that originally the term “DNS” came from turbulence modeling (e.g., see [35]), where it was implied that the size of the smallest turbulent vortices (Kolmogorov scale) is larger than the smallest cell in a computational grid. Applied to simulations of moving particles, the main idea of DNS models is to embed an irregular solid particle/particles into a larger, simple domain and to specify no-slip boundary conditions on the particle boundaries. Thus the fluid flow is computed only between the solid particles. The forces acting on each particle are calculated directly by taking the surface integrals over each particle. Generally, the so-called immersed boundary (IB) method is used for the DNS of particulate flows. For a review of IB methods, we refer the reader to the work by Mittal and Iccarino [36]. Examples of DNS-based models for particulate flows can be found in representative works by Pan et al. [37] (isothermal particulate flows) and by Deen et al. [38] (nonisothermal particulate flows), where corresponding reviews of the fundamental work in this area are given in detail. An alternative to the classical DNS Euler–Lagrange models of particulate flows is the combination of the lattice Boltzmann method [39, 40], which is used to solve the fluid flow between the solid particles, and an Euler method, which is applied to solve a convection–diffusion equation for a passive scalar such as the temperature or species concentration (e.g., see [41]).

1.1

Numerical Modeling in Engineering

Applied to the modeling of gas–solid chemically reacting flows in gasifiers or combustors, the UDPM-based Euler–Lagrange models have become wellestablished tools for macroscale simulations of transport processes, whereas DNS-Euler–Lagrange approaches are used for understanding the micro- and mesoscale processes by resolving single or several chemically reacting particles including intraparticle diffusion of chemical species and heat transfer. In this view, DNS of fluid–particle flows allows the prediction of parameters and the “observation” of processes, which are almost impossible or very expensive to measure in experimental studies. Hence, DNS plays the role of a numerical experiment. For example, in the case of nonisothermal gas–solid flows, DNS can deliver the heat transfer coefficient between the fluid and the particles, which can be utilized in the development of closure correlations (submodels) to describe the heat transfer exchange between the particle and the fluid (e.g., see [38]). Hence, new submodels play the role of scale “bridges” between microscale (e.g., interfacial phenomena) and macroscale simulations (e.g., reactor-scale simulations). Finally, utilization of the so-called submodels allows one to take into account the multiscale character of gas–solid flows. However, in the development of submodels, the following requirements should be kept in mind:

• Simple submodels are of great importance because anybody can understand them and they are basically fast and robust in simulations. However, too simple a model may provide only superficial information. • At the same time, too sophisticated a submodel may take years to develop and it can cause difficulties in computations (e.g., the convergence problem). Here, it should be noted that, generally, submodels have to be run many times until the macroscale simulation converges. An example of a such multiscale modeling strategy for particulate flows in chemical reactors is shown in Figure 1.1. The different scales to be modeled in a gasifier are shown in Figure 1.2. The sequential use of all steps shown in the figure may significantly reduce the errors or uncertainties in model development and, hence, enhance the reliability of the final results and models. As an example, the work by Agrawal et al. [42] provides a thorough review of a similar multiscale approach. Particle resolved simulations (PRS) Direct numerical simulation (DNS)

Experiments Reality

Understanding of reality

Sho

rt w

CFD-based simulations of a large-scale facility

ay Development of submodels

Modeling of reality New "submodels" are the key

Figure 1.1 Principal scheme of a multiscale modeling strategy for particulate flows in a chemical reactor.

5

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1 Modeling of Gasifiers: Overview of Current Developments

Summary

DNS using new numerical and postprocessing algorithms has the potential to transform significantly the current model development. In particular, it is advantageous that, compared to advanced/expensive experimental techniques, DNS can enable researchers to have access to the microscale and mesoscale characteristics of chemically reacting gas–solid flows. This strategy allows engineers to make new model designs in a timely and cost-efficient manner. Moreover, carrying out “numerical experiments” for different input parameters can provide a better understanding of the problem to be solved. And by knowing a “physically accurate” numerical solution, it is possible to find a semiempirical approach to solving the same problem using less computational time and resources. In this book, this relatively new approach is illustrated by numerous examples.

1.2 CFD-based Modeling of Entrained-Flow Gasifiers

Applied to industrial companies engaged in the development and production of industrial-scale gasifiers, CFD – and especially commercial CFD software – has only recently been explored as a powerful tool in designing and optimizing gasifiers and their working parameters. It is evident now that the coupling of CFD with a chemical reaction engineering theory has the potential to reduce the need for expensive and time-consuming large-scale tests. Especially, in the last 20 years significant improvements in the CFD modeling and computational hardware and combustion/gasification model development have made it possible to gain insights into the influence of design variables, coal properties, and processing conditions on the gasifier performance. In the first line, it concerns entrainedflow gasifiers because of their several advantages over fluidized-bed or fixed-bed systems. In particular, entrained-flow gasifiers are becoming popular in the coal conversion into synthetic fuels because they produce higher coal gasification rates and are easier in operation than other reactors due to their simple design and reliability (see Chapters 2 and 11 for details). Moreover, entrained-flow gasifiers are easier to model compared to fluidized-bed and fixed-bed gasifiers because of the dilute particulate flows, where particle collisions can be neglected. The principal scheme of an entrained-flow gasifier including different scales of modeling concepts is shown in Figure 1.2. In an entrained-flow reactor, small (O(10−4 )m) coal particles (solid or as a slurry) are injected into a moving gaseous medium which enhances the dispersion of particles over the reactor. This effect provides the largest solid–gas reactive surface area, which promotes the chemical reaction between the solid and gas phases. As gaseous medium, oxygen (air) and steam are introduced simultaneously to the coal particles. Near the inlet of fuel in the zone of coal–oxygen mixing, extremely high temperature is to be expected as a result of the relatively high oxygen concentration and the combustion of volatiles produced during the devolatilization of coal. Strictly

1.2

Coal

CFD-based Modeling of Entrained-Flow Gasifiers

Steam, oxygen/air

Eulerian grid Lagrange particles

ΔX ~ 10−3 – 10−2 m

DNS ΔX ~ 10−5 – 10−6 m

Slag

Slag

Syngas Figure 1.2 Principal scheme of an entrained-flow gasifier and different scales of modeling concepts.

speaking, this phenomenon is very similar to the processes occurring near the inlet of a coal combustor. The heat produced by oxidation of coal supports the endothermic gasification reactions. Theoretically, in an ideal case, gasification processes can be organized in a such way that the heat release from oxidation (exothermic) reactions balances the heat needed for the endothermic gasification reactions. However, in real practice, all chemical reactions may take place simultaneously in a gasifier because of the impact of gas flow and turbulence. In this view, a CFD-based modeling coupled with heterogeneous and homogeneous chemistry is necessary to understand and then to optimize the dynamics of coal conversion under entrained-flow conditions. In general, CFD simulation serves as a preliminary part for complex design studies or to investigate phenomena in a known gasifier setup. A wide range of boundary as well as model conditions are needed to define the proper CFD simulation framework, which requires the understanding of the processes to be modeled. In particular, many physical effects have to be taken into account such as turbulent flow, coal particle conversion reactions, homogeneous chemistry, particle–flow interactions, radiation, and so on. For simplicity, each effect can be subdivided into complex subprocesses in order to be able to develop a final overall model for a numerical investigation of a

7

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1 Modeling of Gasifiers: Overview of Current Developments

reactor. Such module-based principle is often used in computational engineering. Applied to modeling of a gasifiers, the following multiscale phenomena have to be taken into account [43, 44]:

• • • •

HHI – heterogeneous and homogeneous chemistry interaction. TCI – turbulence–chemistry interaction. PTU – particle–turbulence interaction. PGI – particle–gas interaction including the following processes: –heating and moisture evaporation of coal particles, –coal devolatilization and char formation, –char oxidation and gasification. • PWI – particle–wall interaction. • PPI – particle–particle interaction6) . It should be noted that the direct modeling of a gasifier resolving the particles and all turbulence scales (e.g., see [18]), ranging from several meters for the whole reactor to several micrometers for the coal particles (Figure 1.2) is impossible nowadays because of the lack of computing power. For example, to carry out CFD-based particle-resolved simulation of an entrained-flow gasifier with a height of 10 m and a radius of 1 m, we need a grid with more than 1017 control volumes assuming an average grid spacing of Δx = 10−5 m. Therefore, recent CFD studies of an entrained-flow gasifier include models for turbulence, radiation heat transfer, coal drying and devolatilization, and char combustion/gasification. Analysis of recent publications [14, 15, 18–22, 45, 46] including a recent review paper [44] shows that, to describe multiscale phenomena in chemically reacting entrained pulverized coal flows, the following mainstream models are used. For a detailed review of the models used for stochastic tracking of particles in an entrained-flow gasifier including its coupling with different turbulence models (SST k − �, standard, and realizable k − �, LES), the reader is referred to the work of Kumar and Ghoniem [47]. 1.2.1 Mainstream Computational Submodels

Pulverized coal combustion/gasification is basically modeled as a dilute solid–gas reacting flow utilizing an Eulerian–Lagrangian approach, (e.g., see [43, 44]). The so-called particle-source-in-cell method [48] is used to calculate the interaction between a moving particle and gas, governed by mass, momentum, energy, and species conservation through various particle source terms. To illustrate the main idea of this method, we write the mass conservation equation for the gas phase as follows [48]: ) np ( 1 ∑ mp,i,out − mp,i,in ∂ ( ) ∂ � ui = − (1.1) (�) + ∂t ∂xi Vcv i=1 Δt 6) Because of the low values of volume fraction of the solid, PPI can be neglected in the modeling of entrained-flow gasifiers.

1.2

CFD-based Modeling of Entrained-Flow Gasifiers

Control volume

Out Particle In

Figure 1.3 Illustration of mass, heat, and momentum exchange between continuous (gas) and discrete (particle) phases.

where mp,i,out is the mass of the ith particle at the cell exit [kg], mp,i,in is mass of the ith particle at the cell entry [kg], and np is the number of particles inside the control volume Vcv (see Figure 1.3). The momentum transfer from the continuous phase to the discrete phase is computed by summing the change in momentum of each particle passing through a control volume: ] np [ ∑ ) 18 � CD Re ( F g−s = up,i − u + F other . (1.2) 24 �p dp2 i=1 The change in thermal energy of each particle passing through a control volume is given by n

Q̇ cv =

p ∑ )( )] 1 [( mp,i,in − mp,i,out −Δhfg + Δhdevot + Δhhet − Δt i=1 np [ ] Tp,in Tp,out ∑ 1 mp,i,out c dT cp dT − mp,i,in − ∫Tref p ∫Tref Δt i=1

(1.3)

where CD is the drag force coefficient, � is the viscosity of the gas, �p is the density of the particle, dp is the diameter of the particle, Re is the relative Reynolds number, up is the velocity of the particle, u is the velocity of the gas phase, Tp,out is the temperature of the ith particle at the cell exit, Tp,in is the temperature of the ith particle at the cell entry, and Δhfg , Δhdevot , Δhhet are the enthalpies of moisture evaporation, devolatilization, and heterogeneous reactions, respectively. 1.2.1.1 Particle Conversion

Generally, the rates of particle conversion processes such as drying, devolatilization, and gasification are heterogeneous reactions and are slow compared to the turbulence timescale [15]. In this case, the conversion fluxes are calculated using the mean gas properties.

9

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1 Modeling of Gasifiers: Overview of Current Developments

• For the prediction of a particle drying, basically, one uses the so-called surfacebased model, which assumes that the moisture content is located on the particle surface [1, 44]. Thus, the drying can be described by utilizing a theory used for droplet evaporation. Energy conservation equation for the particle during heating and drying has the form ( ) dTp ) ( 4 = Ap � T∞ − Tp + Ap �S � T∞ − Tp4 −ṁ ⋅ Δhfg (1.4) mp cp dt ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ conv.−diffus.

radiation

where Ap is the particle surface area. If Tp < Tboil , the moisture flux can be calculated using the semiempirical relation for interfacial species mass balance, given by ( ) ′′ ′′ (1.5) ṁ = YH∗ O ṁ + �g � YH∗ O − YH2 O,∞ 2 2 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ conv.−diffus.

where

YH∗ O 2

is the interfacial mass fraction of steam, YH2 O,∞ is the steam mass P

MW

fraction in a control volume of the CFD grid, YH∗ O = Psat MWH2 O , and Psat is the 2 mix vapor pressure. If Tp > Tboil , the evaporation rate is governed by heat transfer, and therefore the quasi-steady-state model for droplet evaporation can be used [49]: ( ) ′′ � ṁ = ln 1 + Bq (1.6) ṁ = Ap cp, g where Bq =

( ) cp,g T∞ + Tboil Δhfg −

4 +T 4 � �S (T∞ boil )

(1.7)

ṁ ′′

with T∞ being the temperature of gas in a control volume of the CFD grid. The heat transfer coefficient � is defined as follows: Nu � , Nu = 2 + 0.6Rep1∕3 Pr1∕3 . (1.8) �= dp Here, the Nusselt number is calculated using the Ranz–Marshall relation [34]. For a detailed review of the basic models, see Chapter 5. • At high ambient temperatures, after the drying is completed7) the particle temperature increases. As a result of the thermal decomposition of organic compounds inside a coal particle, the so-called volatile matter “leaves” the particle. This process is called devolatilization. Devolatilization kinetics and yields are strongly dependent on the heating rate, the ambient gas, and the ambient pressure (e.g., see the recent two-dimensional CFD-based simulation of a coal particle ignition [51]). Some of the most usable devolatilization models are as follows [1]: 7) For large particles, dp > 10−3 m, drying and devolatilization can occur simultaneously. But in the case surface-based drying model, this is not valid anymore [50].

1.2

CFD-based Modeling of Entrained-Flow Gasifiers

– The single-global reaction rate model [52]. The thermal decomposition rate of dry coal particles is described as dmp )( )] [ ( = −ṁ p = k mp − mp,0 ⋅ 1 − fv,0 1 − fw,0 (1.9) − dt where mp and mp,0 are the current and initial particle mass, fv,0 is the mass fraction of volatiles on a dry basis, and fw,0 is the mass fraction of moisture initially present in the coal particle as received. The rate constant k has the form ] [ E k = Ak ⋅ exp − d (1.10) Ru Tp where Ru is the ideal gas constant, and Tp is the particle temperature. – The multiple-reaction model [53]. – The Kobayashi model (for details, see [1]). – The CPD (chemical percolation devolatilization model [54]. This model characterizes the devolatilization behavior of rapidly heated coal based on the physical and chemical transformations of the coal structure. Basically the volatile matter contained in the coal are assumed to be composed of CO, CO2 , H2 , CH4 , H2 O, and Cx Hy as a heavy fraction. For a detailed review of the basic models, see Chapter 10. • Generally, to predict the char consumption by gasification/combustion, the socalled Baum and Street model [55] is used. Smith [56] generalized this approach for simplified multispecies surface reactions represented by three simple global heterogeneous reactions: 2Cchar + O2 → 2CO + Heat

(1.11)

Cchar + H2 O + Heat → CO + H2

(1.12)

Cchar + CO2 + Heat → 2CO.

(1.13)

This model belongs to the class of one-film models and considers that heterogeneous reactions take place on the surface of the particle (in most cases a sphere). In the literature, this approach is referred to as the diffusion kinetic single film (DKSF) or the kinetic/diffusion model. The species O2 , CO2 , and H2 O are considered to react heterogeneously with char after diffusion to the particle surface through the boundary layer. The kinetic/diffusion-limited rate model uses harmonic average weighting between diffusion and kinetic defined rates: kdiff,i ⋅ kkin,i kiS = (1.14) kdiff,i + kkin,i where kdiff,i and kkin,i are the diffusion and kinetic rate constants for the ith reaction, respectively: [( ) ]0.75 Tp + T∞ ∕2 kdiff,i = Ci , (1.15) dp ( ) −EAi n (1.16) kkin,i = AE T exp ( ) Ru Tp

11

12

1 Modeling of Gasifiers: Overview of Current Developments

where Ci is the overall mass diffusion-limited constant [44]: Ci =

𝜈i MWC MW MWi R T01.75

⋅ Sh ⋅ Di,0 ⋅

P0 P

(1.17)

where 𝜈i is the stoichiometric coefficient in the ith reaction, and MW is the average molecular weight of the gas mixture in a control volume of the CFD grid. The constant Ci depends on a heterogeneous reaction [44]. In the CFD software ANSYS-Fluent® [57], this constant has a default value of about 10−12 s K −0.75 . Taking into account basic heterogeneous surface reactions, the carbon consumption rate has the form ṁ C = Ap

3 ∑

Pi,g kiS .

(1.18)

i=1

For a detailed review of the basic models and description of the new intrinsicbased model, see Chapter 10. 1.2.1.2 Turbulence–Chemistry Interaction

In industrial-scale gasifiers/combustors, the gas flow inside a reactor is always turbulent, which makes the numerical modeling a nontrivial task. The so-called DNS cannot be used for the whole gasifier (see discussion in previous section). In this case, turbulence models (e.g., RANS) have to be applied to account for the effect of turbulence on the transport processes including chemistry–turbulence interaction. Applied to CFD-based modeling of gasifiers, the impact of turbulence on the homogenous chemistry has been modeled using the so-called eddy dissipation model (EDM) and the eddy dissipation concept (EDC) models coupled with RANS or LES. The EDM model [58] assumes that the chemical reaction is faster than the timescale of the turbulence mixing of the species, which is governed by the large eddy-mixing time, k∕�, as originally proposed by Spalding [59]. Thus, a homogeneous chemical reaction is supposed to occur instantaneously when the reactants are brought into contact. This assumption makes it unnecessary to use finite-rate kinetics. An enhanced version of the EDM takes the finite-rate chemistry into account. Finally, the smaller reaction rate given by the Arrhenius rate and turbulent mixing rate is chosen for homogeneous reactions (e.g., see [21]). The important tuning parameters in this model are the so-called Magnussen’s empirical constants A and B (default: A = 4.0, B = 0.5), for the reactant and the product, respectively. Their variation can significantly change the final results (e.g., see [44]). A significant limitation of this model is that only two reactions can be considered whereas, in fact, there are different Arrhenius rates for a multistep mechanism [57]. As the next extension of the EDM model for the case of multistep chemical kinetics, the so-called EDC model was used in many works on combustion/gasification [24, 60, 61]. The EDC model is based on the original work of Magnussen [62].

1.2

CFD-based Modeling of Entrained-Flow Gasifiers

It should be noted that in the EDC model a scalar equation is solved for each chemical species. Thus, in comparison to the EDM model, the EDC model needs a relatively high calculation time for integrating the chemistry. An alternative and very promising model for coal combustion/gasification is the flamelet model [63], where the fuel and the oxidizer are supplied separately to the reaction zone. The distinguishing feature of this model as applied to CFD-based combustion modeling is that there is no need to calculate scalar equations for the species. Instead of chemical species transport equations, one needs to solve only two equations for the mean mixture fraction and the mixture variance. In particular, recently Prieler et al. [61] carried out a CFD analysis of an 11.5 kW lab-scale furnace with oxygen–natural gas combustion for a high-temperature process using three different TCI models: two-step EDM, EDC with 17 species and 46 reversible reactions, and the steady laminar flamelet model (SFM). It was shown that EDM was unable to predict the oxygen–fuel combustion correctly. In contrast to the EDM results, temperatures calculated using EDC and SFM showed close agreement with the measured data in the furnace. However, using SFM the computational time was decreased from 3 weeks needed for EDC model to 4 days on an 8 CPU-core computer. Finally, it should be noted that the so-called advanced TCI models, such as flamelet or PDF models, for coal combustion/gasification are in the development phase. However, some promising results obtained using flamelet-based models have been published recently [51, 64]. 1.2.2 Review of CFD-related Works

Next, we present a brief review of the recent literature devoted to CFD-based modeling of entrained-flow gasifiers and related processes. Here, the focus is on entrained-flow coal gasification since 1990. A comprehensive review of the basic works devoted to the modeling and simulations of entrained-flow gasifiers published before 1990 can be found in [65, 66]. This short review is divided into the analysis of CFD-based works that used noncommercial software and the commercial ANSYS-Fluent®software. 1.2.2.1 Noncommercial Software

At the beginning of 1990, Sijerˇci´c and Hanjal´c developed a two-dimensional code for the modeling of an entrained-flow gasifier [13, 67, 68]. The gas phase was described in the Eulerian frame and the discrete phase in the Lagrangian frame, taking into account heat and mass transfer exchange between the phases using the particle-source-in-cell method [48]. Four heterogeneous chemical reactions of coal were considered in a kinetic–diffusion regime accounting for the impact of particle velocity on the heat and mass transfer between a particle and a gas using the Ranz–Marshall relation. However, concentrations of chemical species on the particle surface were neglected in the surface-based burnout model. The distinguishing feature of the model of Sijerˇci´c and Hanjal´c is the use of a transport

13

14

1 Modeling of Gasifiers: Overview of Current Developments

equation for the particle number density Np to calculate the particle concentration field necessary for the prediction of radiative heat transfer coefficients. The code was validated against published experimental data for the BCURA reactor [69]. More information concerning the BCURA reactor is given in the next section. One of the first published results on the three-dimensional simulation of an industrial-scale 200 tpd (tons per day) two-stage air-blown entrained-flow coal gasifier was by Chen and coworkers [14–16]. An extended coal–gas mixture fraction model with the “multi solids progress variables method” was utilized to simulate the gasification reaction and the reactant mixing process. The model tracked 11 500 particle trajectories, and a 21 × 21 × 62 grid mesh was used. It was shown that the three different zones, namely the devolatilization, the combustion, and the gasification zones, have complex contours in the gasifier. Moreover, it was demonstrated that turbulent fluctuations in the volatile and the char–oxygen reaction have a significant impact on the temperature and gas composition. In 2013, Abani and Ghoniem [22] published one of the first LES calculations of a lab-scale entrained-flow gasifier (BYU gasifier) operating at atmospheric pressure. They used the open-source CFD software OpenFOAM with a standard kinetic/diffusion approach. The total computational mesh size consisted of 0.33 × 106 cells. The Rosin–Rammler distribution [70] (with a distribution index of 3.5) was used to represent the variation in particle size with a minimum diameter of 10 μm and a maximum diameter of 80 μm. The LES/RANS results showed that in the combustion zone RANS calculation overpredicted the mixing rate, which led to higher combustion temperatures and it did not capture accurately the unsteady characteristics of the two-phase mixing in the gasification zone, which, on the other hand, was important for modeling char consumption. LES calculation resulted in a longer combustion zone and a more uniform species distribution in the gasification zone. The overall results of the LES simulations showed a more accurate prediction of the scalar fields compared to similar RANS calculations. 1.2.2.2 Commercial Software

With significant development and progress in the commercial CFD software ANSYS-Fluent®, several papers on entrained-flow gasifiers have been published recently. Silaen and Wang [19] effectively employed the DPM-CFD gasification model available in ANSYS-Fluent®to investigate the influence of different submodels on gasification performance including five turbulence models, four devolatilization models, and three solid coal sizes. Three-dimensional simulations were carried out using the following RANS turbulence models: Standard k − �, RNG k − �, Standard k − � Model, SST k − � Model, and Reynolds Stress Model (RSM). The results showed that the standard k-� and the RSM turbulence models gave consistent results. Concerning devolatilization rates, chemical percolation devolatilization (CPD) and the single-rate models reproduced more moderate results and the devolatilization rates were not as slow as those of the Kobayashi model. Recently, Lu and Wang [45] carried out investigations on three-dimensional simulations of a two-stage slagging-type entrained-flow gasifier (operating

1.2

CFD-based Modeling of Entrained-Flow Gasifiers

pressure 24 atm, 1700 tpd, 190 MW energy output) using five different radiation models available in the ANSYS-Fluent®software [57]: discrete transfer radiation model (DTRM), P-1 radiation model, Rosseland radiation model, surface-tosurface (S2S) radiation model, and discrete ordinates (DO) radiation model. The commercial software ANSYS-Fluent®-Version 12. was utilized. The computational grid consisted of 1.1 × 106 unstructured tetrahedral cells. For TCI modeling, both EDM and finite-rate models were used to calculate the reaction rates. It was shown that the P-1 model was more robust and stable in predicting the syngas temperature and composition compared to the other four models used. However, the P-1 model resulted in the lowest temperature of the inner wall of the gasifier. The DO and DTRM models took about twice the CPU time as the other radiation models. The assumptions that surface-based heterogeneous kinetics does not adequately represent the gasification process advocate that investigations in the area of heterogeneous reactions submodels should be the focus for gasification reactor modeling [p.94] [65]. In this context, Australian researchers of the Cooperative Research Centre for Coal in Sustainable Development (CCSD) incorporated new submodels in a two-dimensional RANS simulation in the CFD software ANSYSFluent®for better predictions of the drying, pyrolysis, and heterogeneous coal gas–char reactions. The results of numerical investigations were compared with experimental data [71–74]. In particular, in a series of conference papers, Hla et al. coupled successfully the intrinsic heterogeneous reaction rates at elevated pressure with a model proposed by Laurendeau [75]. The intrinsic character of coal was accounted for by the random pore model developed by Bhatia and Perlmutter [76]. The simulation results indicated that the used models could predict a more realistic image of the gasifier performance. Not only the trends could be reproduced but also good agreements of the experimental data for different types of coal were reached for CO, CO2 , and H2 species concentrations along the axis. Good agreement with experimental data was reached especially with anthracite coals. However, it was found out that the boundary conditions (e.g., wall temperature) had a great impact on the final results concerning their agreement with experiments [73]. Kumar and Ghoniem [77] modified the DKSF submodel by Baum and Street using an additional term characterizing a moving flame front (MFF) introduced by Zhang et al. [78]. The overall results showed that the use of the MFF model gave more accurate results reflecting better physics of particle burn-up history. The main idea of MFF model is to vary the flame front radius up to several (up to 50) particle radii to fit the burnout curve to the experimental data. However, it is well known that the ratio between the flame radius and the radius of a carbon particle oxidizing in an O2 -based atmosphere cannot exceed the value of 2. This value can be derived using the classical two-film model (see Chapters 6 and 8). It can be seen that significant progress was achieved in the commercial CFD software ANSYS-Fluent®concerning the prediction of heat and fluid flow in pulverized coal jets (e.g., see the comparison of different CFD software [79]) and in entrained-flow gasifiers. On the other hand, the development of improved

15

16

1 Modeling of Gasifiers: Overview of Current Developments

submodels describing particles conversion was slower and thus the submodels developed in the 1980s are still basically to describe the particle–gas interaction in gasifiers. Only recently, new, advanced submodels developed in this century received more attention in CFD-related predictions of chemically reacting flows in gasifiers and pulverized coal combustors. For instance, Vascellari et al. published a series of papers devoted to CFD-based simulation of pulverized coal MILD combustion [80] and the BYU entrained-flow gasifier [81] using advanced coal/charconversion submodels [23, 24]. In particular, numerical simulations carried out in [80] revealed that the use of new virtual homogeneous-zone single-film submodel (H-zone model), originally developed by Schulze et al. [82], produces results that are closer to the experimental data in comparison to the standard Baum and Street burnout submodel. The distinguishing feature of the H-zone model [82] is the coupling of homogeneous CO oxidation reaction with heterogeneous gasification reactions for the calculation of particle temperature and carbon conversion rate. Additionally, this new “surface-based” subgrid model considers a detailed description of the transport phenomena in the proximity of the particles under convective environmental conditions. Further developments of this char-conversion submodel is presented in Chapter 10. Recently, Vascellari et al. [24] implemented a single Nth-order reaction (SNOR) model originally developed by Liu and Niksa [83] into the ANSYSFluent®software using the user-defined function (UDF). This model is an intrinsic-based model which takes into account random pore evolution and char density changes. The CBK/E [84] and CBK/G [83] models for char oxidation and gasification, respectively, were used for calibrating the SNOR kinetic model. Turbulence was modeled using the realizable k − � approach coupled with the EDC model accounting for the TCI in combination with a detailed kinetic mechanism (for details, see [24]). Radiation was modeled via the P1 model available in ANSYS-Fluent®. Comparison with the experimental data for the BYU entrained-flow gasifier [81] showed good agreement for gas composition and carbon conversion. However, the main disadvantage of the SNOR submodel is the need for calibration with the CBK/G model. Moreover, for the calculation of mass conversion rates for each heterogeneous reaction, the model uses an empirical factor which accounts for the physical evolution effects such as char density changes and pore evolution. This factor is a function of the char-conversion rate X, which has the form of a fifth-order polynomial correlation for oxidation and gasification reactions separately. The char density was calculated as a function of X according to [83] m (1.19) �c = �c,0 (1 − X)�n , X = 1 − c mc,0 where �n is an empirical model parameter. It can be seen that this model does not account for the simultaneous change of particle density and particle diameter. It is a well-known fact that, during the oxidation of char, the so-called diffusion-controlled regimes govern the char conversion, where the particle diameter changes instead of the density [75, 85, 86].

1.3

Benchmark Tests for CFD Modeling

From this point of view, further developments of intrinsic-based submodels are needed to avoid the use of many unphysical input empirical model parameters and to account for intraparticle diffusion and heat transfer coefficients into such models. Summary

The analysis of the literature shows that in most recent simulations the commercial CFD code ANSYS-Fluent®was utilized using more advanced submodels implemented via UDF. As discussed at the beginning of this chapter, those software packages greatly reduce the effort in developing better strategies for modeling the complex physics in gasifiers. Despite the rapid growth in the availability and speed of computer technologies, there is only a slow transition from 2D to 3D RANS or even LES calculations. The recent research focus has been on improving the turbulent nature of two-phase flows and incorporating and validating new submodels to account for the intrinsic nature of gasification. Abani et al. [22] demonstrated that a good estimation of the unsteady characteristics of the turbulent flow field can yield a better description of the combustion and gasification processes. The works by Hla et al., Kumar et al., and Vascellari et al. [24] use heterogeneous reaction models for CFD in their gasifier simulations and showed the intrinsic behavior of coal. There is a great need for new developments in this area because most of the presented CFD predictions are based on nonintrinsic combustion assumptions that do not capture accurately the gasification behavior of coal. In the future, slag behavior and CFD gas–particle interaction (dense particle flows) need to be the new focus points for further research. In addition to new CFD tools, there is a great need for validation cases from char to high-ash and high-volatile yield coals under varying operation conditions such as high pressures to be able to accomplish future developments. Finally, it should be emphasized that any successful application of a CFD software requires good understanding of the models and assumptions that will be used in the simulations. However, in many cases commercial CFD codes are black boxes, where it is impossible to “read” the model and equations in the code. From this point of view, it is extremely important to validate the software before actual studies can be carried out. At the same time, the parametric runs can help understand the basic assumptions in the model used.

1.3 Benchmark Tests for CFD Modeling

A review of recent works devoted to CFD-based modeling of entrained-flow gasifires revealed the importance of models and software validation against experimental data published in the literature. In the following section, we analyze experimental data for lab-scale gasifiers published in the open literature. It should be stated that the proximate and ultimate analyses are based on either as

17

1 Modeling of Gasifiers: Overview of Current Developments

received (ar) or as dry and ash-free (daf ) state. Those input data are important to characterize, among others, devolatilization processes that have a great impact on the overall carbon consumption. 1.3.1 British Coal Utilization Research Association Reactor (BCURA)

The BCURA pilot-scale combustion reactor is an air-blown furnace which was operated at ambient pressure. The system could process more than 9 tpd of coal. A detailed description of the experimental setup and results are documented in the article by Gibson and Morgan [69] and Baker et al. [87]. A first approach of the mathematical model of the reactor was proposed by Field in his book (see Appendix U [88]). Several reports have been published by BCURA that are not part of this review and may provide further details and experimental results. For further information, see the references in [69]. The BCURA pilot-scale combustor consists of a horizontal cylindrical chamber with two inlets for coal, and primary and secondary air. The reactor has a height of 6.1 m and a diameter of 1.1 m. All basic geometric parameters are given in Figure 1.4. The presented setup and results are taken from the experiment “Flame 49” [69]. In Tables 1.1 and 1.2, the boundary conditions are given. The inner wall temperature of the chamber is a complex function of the estimated heat loss and depends on the used models (e.g., radiation model) and boundary conditions. The outer wall temperature can be assumed with 400 K. The injection speed of the particles may be estimated with 21.9 m s−1 if you consider that the fluid and particle flow field are equal at the entry point. The proximate and ultimate analysis are given in Table 1.3. The used coal has a high-ash and a fixed-carbon content and can be classified as a low-rank bituminous coal. No data is available for heterogeneous kinetics and only limited experimental data for the BCURA rig are documented in literature. Experimental results of “Flame 49” are illustrated in Figures 1.5 and 1.6. The measured overall heat loss adds up to 1350 kW. A simplified contour plot of temperature isolines is shown in Figure 1.6. A flame zone appears at approximately 1 m [69].

R550

A

A

Secondary air 6100

Figure 1.4 Geometry of the BCURA rig (in mm).

R150 R40

Primary air + coal R190

18

1.3

Table 1.1

Boundary conditions [69].

BC

Value

ṁ 1 T1 ṁ 2 T2 Twall poperation �wall Q̇ loss

0.104 kg/s 373 K 0.822 kg/s 626 K Twall (Q̇ loss )b 1 bar 0.7 1350 kW

a b

… XN2 ∕XO2 = 0.79∕0.21 … Twall min = 400 K

Table 1.2

Simulation setup parameters [69].

BC

Value

ṁ Fuel X-Velocity R-Velocity Tfuel Min. diameter Max. diameter Mean diameter Spread par.

0.086 kg/s 21.9 m/s 0 m/s 373 K 2e-6 m 200e-6 m 43e-6 m 1.0b

a

Benchmark Tests for CFD Modeling

… Δu ≈ uair − uCoal = 0 assumed

b…

Table 1.3

Coal properties [69].

Proximate analysis (wt%) Moist 4.10

Ultimate analysis (daf, wt%)

HHV (ar)

FC

VM

Ash

C

H

O

S

N

(Mj/kg)

53.59

32.01

10.30

80.60

5.14

11.59

1.86

0.81

27.9

1.3.2 Brigham Young University Reactor (BYU)

The BYU reactor is an oxygen-fed lab-scale gasifier (0.6 tpd) operating at ambient pressure (Tables 1.4–1.6). This experimental rig is very well documented. Therefore, it is well suited as a validation setup. But you need to be aware that only highly volatile coals were considered in the past surveys. Therefore, a good

19

1 Modeling of Gasifiers: Overview of Current Developments 100

Burnout (daf) (%)

Heat flux to the wall (kW m2)

75 50 25 0

0

1

(a)

2

3

4

5

Distance from burner (m)

6

100 75 50 25 0

0

(b)

1

2

3

4

5

6

Distance from burner (m)

Figure 1.5 Measured date of “Flame 49” with (a) heat flux through the walls and (b) burnout of the coal along the axis. Graphs based on the data taken from [69].

1

2

3

00

14

00

00 12

0.5 m 0.4 0.3 0.2 0.1 0.0

10

20

4

5

6m

14

00

1600

Figure 1.6 Contour plots of the measured temperature field of Flame 49 (in K). Graphs based on the data in [69]. Table 1.4

Boundary conditions for the BYU reactor [81].

BC

Value

ṁ 1 T1 XO2 ∕XAr ∕XH2 O ṁ 2 T2 XH2 O Twall poperation

7.290 g/s 367 K 0.850/0.126/0.024 1.840 g/s 450 K 1 unkown 1 bar

devolatilization prediction for this reactor model is necessary. Four types of coals were investigated, and experimental results in the axial and radial directions for different species concentration have been reported by Brown et al. [81]. Additional information on conducted BYU experiments can be found, among others, in the journal papers of Soelberg, Smoot, and Smith et al. [89, 90]. The experimental rig consists of six horizontally oriented sections with a length of 305 mm and one section of 153 mm which is partially illustrated in Figure 1.7. The effective length of the reactor chamber is specified as 1890 mm. A tube-intube configuration separates the primary from the secondary inlet streams with a diameter of 4.6 and 28.6 mm, respectively; for details see [81].

1.3

Table 1.5

21

Injection conditions for the BYU reactor [81].

BC

Value

ṁ fuel X-Velocity R-Velocity Tfuel Min. diameter Max. diameter Mean diameter Spread par.

6.634 g/s 50.6 m/s 0 m/s 367 K 3e-6 m 35e-6 m 80e-6 m 1.0b

a

Benchmark Tests for CFD Modeling

… Δu ≈ ugas − ufuel = 0 assumed

b…

Table 1.6

Coal properties Utah bituminous coal [81].

Proximate analysis (wt%) Moist 2.4

Ultimate analysis (daf, wt%)

HHV (db)

FC

VM

Ash

C

H

O

S

N

Mj/kg

43.7

45.6

8.3

77.60

6.56

13.88

1.42

0.55

29.8

153

ϕ200

A

ϕ28.6 ϕ 4.6

A

305

Secondary stream Primary stream + coal

1890

Figure 1.7 Geometry of the BYU rig (in mm) [81].

Several experiments have been conducted by the BYU. The presented boundary conditions and obtained results focus only on experiments with Utah bituminous coal. Other coals (Wyoming subbituminous, North Dakota lignite, and Illinois No. 6 bituminous coals) and the corresponding kinetic parameters for the heterogeneous reaction are documented by Brown et al. [81]. Wall temperatures were not mentioned directly in this work, but it can be assumed that rig walls are nonadiabatic (e.g., see [23, 81]. Several parametric studies for the BYU lab-scale reactor were performed [89]. Molar concentrations of CO, CO2 , H2 , and H2 O in the radial and axial direction are documented for the Utah bituminous coal, and an example is illustrated in

1 Modeling of Gasifiers: Overview of Current Developments

40 Species concentration (mol%)

30

20

H2 CO H2O CO2

10

0

0

1.5 0.5 1 Distance from burner in axial direction (m)

Figure 1.8 Molar concentrations of CO, CO2 , H2 , and H2 O along the axis for Utah bituminous coal. Graph based on the data in [81].

A

ϕ70

22

A Secondary stream Primary stream + coal

2100

Figure 1.9 Geometry of the PEFR rig (in mm).

Figure 1.8 for the axial distribution of different species. The exit temperatures are also measured to have an additional parameter to fit the used boundary conditions. For the Utah coal, the exit gas temperature is estimated between 1350 and 1400 K. Further information on the experimental results of Utah coal is presented by Soelberg et al. [90]. He has included contour plots for the obtained species concentrations in the BYU reactor. 1.3.3 Pressurized Entrained-Flow Reactor (PEFR)

The PEFR is a small lab-scale reactor (0.1 tpd) and is part of a project developed by the Cooperative Research Centre for Coal in Sustainable Development (CCSD) in cooperation with the Commonwealth Scientific and Industrial Research Organization (CSIRO) during the late 1990s. The special feature of this reactor is the directly measured high-pressure heterogeneous intrinsic kinetics of the used Australian coals at 20 bar. Hla et al. have described all parameters for setting up a CFD calculation [72, 73]. More information concerning the gasification behavior of Australian coals can be found in the article by Harris et al. [71] and supplemented in the research reports of the CCSD [74, 91, 92].

1.3

Table 1.7

Simulations boundary conditions for the PEFR reactor [74].

BC

Value

ṁ 1 T1 XN 2 ṁ 2 T2 XN2 ∕XO2 Twall inlet Twall reactor poperation

1.680 g/s 298.15 K 1.0 15.385 g/s 1275 K 0.973/0.027 Adiabat 1673 K 20 bar

Table 1.8

Benchmark Tests for CFD Modeling

Setup parameters for the PEFR reactor [74].

BC

Value

ṁ fuel X-Velocitya R-Velocity Temperature Min. diameter Max. diameter Mean diameter Spread par.b

0.511 g/s 1.8 m/s 0 m/s 298.15 K 20e-6 m 250e-6 m 177e-6 m 1.12

a … Δu b

≈ u1 − ufuel = 0 … assumed

Table 1.9

Coal properties of CRC252 coal.

Proximate analysis (wt%) Moist 10.7 a

Ultimate analysis (daf, wt%)

HHV (db)

FC

VM

Ash

C

H

O

S

N

(MJ/kg)

39.11

38.85

10.34

78.1

5.9

14.4

0.5

1.1

25.7a

… value taken from Harris et al. [71].

The gasification reactor consists of a horizontal cylindrical chamber with two inlets for coal and primary and secondary gas streams. The vertically oriented reaction chamber is 2100 mm long and has a diameter of 70 mm. No information on the nozzle geometry is available in the literature. The basic geometric parameters are shown in Figure 1.9. The presented boundary conditions are taken from experiments for the coal type CRC252 and are listed in the articles of Hla and

23

1 Modeling of Gasifiers: Overview of Current Developments

4 Species concentration (vol%)

24

H2 CO CO2

3

2

1

0

0

0.5

1.5 1 Distance from burner in axial direction (m)

2

Figure 1.10 Distribution of the different species along the axis of CRC252 in the PEFR. Graph based on the data in [72].

Harris et al. (for details, see Tables 1.7–1.9) [72, 73]. The particle size distributions are documented in the research report of Harris et al. (see Table 1.8) [92]. Carbon conversion, particle diameter, and species molar concentration along the reactor axis have been described by Hla et al. [72] for six types of coal. The described Australian coals are characterized by an identification number, for example, CRC 281. The given proximate and ultimate analyses indicate a broad selection of different types of coal ranging from high-ash, high-volatile, and anthracite coals. In Figure 1.10, the axial distributions of the different species are illustrated for the gasification experiments of the coal CRC 252. The lines in Figure 1.10 should illustrate the general trends of the measured data. Further information is given in [72–74]. References 1. de Souza-Santos, M.L. (2010) Solid Fuels

5. Scharff, M.F., Chan, R.K.C., Chiou, M.J.,

Combustion and Gasification: Modeling, Simulation, and Equipment Operation, Mechanical Engineering, 2nd edn, CRC Press, Taylor & Francis Group, Boca Raton, FL, pp. 33487–32742. 2. Antia, H.M. (2002) Numerical Methods for Scientists and Engineers, 2nd edn, Birkhäuser Verlag, Boston, MA, Basel, Berlin. 3. Sreekanth, P., Syamlal, M., and O’Brien, T.J. (2011) Computational Gas-Solid Flows and Reacting Systems: Theory, Methods and Practice, IGI Global. 4. Garg, S.K. and Pritchett, J.W. (1975) Dynamics of gas-fluidized beds. Journal of Applied Physics, 46 (10), 4493–4500.

Dietrich, D.E., Dion, D.D., Klein, H.H., Laird, D.H., Levine, H.B., Meister, C.A., Srinivas, B. (1982) Computer modeling of mixing and agglomeration in coal conversion reactors. Technical Report No. DOE/ET/10329-1211, DOE. 6. Syamlal, M., Rogers, W., and O’Brien, T.J. (1993) MFIX documentation: theory guide. Technical Report No. DOE/METC-94/1004(DE94000087), DOE–Morgantown Energy Technology Center. 7. Syamlal, M. (1998) MFIX documentation: numerical techniques. Technical Report DOE/MC-313465824.NTIS/DE98002029, U.S. Department of Energy.

References 8. The National Energy Technol-

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ogy Laboratory, Multiphase flow with interphase exchanges (MFIX), https://mfix.netl.doe.gov/. Garg, R., Galvin, J., Li, T., and Pannala, S. (2012) Open-source MFIX-DEM software for gas-solids flows: Part IVerification studies. Powder Technology, 220, 122–137. Selected Papers from the 2010 NETL Multiphase Flow Workshop. Syamlal, M., Guenther, C., Gel, A., and Pannala, S. (2009) Advanced coal gasifier designs using large-scale simulations. Journal of Physics: Conference Series, 180, 012034. Guenther, C., Shahnam, M., Syamlal, M., Longanbach, J., Cicero, D., and Smith, P. (2002) CFD modeling of a transport gasifier. Proceedings of the 19th Annual Pittsburgh Coal Conference, Pittsburgh, PA, September 2002. Guenther, C., Syamlal, M., Longanbach, J., and Smith, P. (2003) CFD modeling of a transport gasifier Part II. Proceedings of the 20th Annual Pittsburgh Coal Conference, Pittsburgh, PA, September 2003. Sijerˇci´c, M. and Hanjali´c, K. (1994) Application of computer simulation in a design study of a new concept of pulverized coal gasification, Part II: model of coal reactions and discussion of results. Combustion Science and Technology, 97, 351–375. Chen, C., Horio, M., and Kojima, T. (2001) Use of numerical modeling in the design and scale-up of entrained flow coal gasifiers. Fuel, 80 (10), 1513–1523. Chen, C., Horio, M., and Kojima, T. (2000) Numerical simulation of entrained flow coal gasifiers. Part I: modeling of coal gasification in an entrained flow gasifier. Chemical Engineering Science, 55 (18), 3861–3874. Chen, C., Horio, M., and Kojima, T. (2000) Numerical simulation of entrained flow coal gasifiers. Part II: effects of operating conditions on gasifier performance. Chemical Engineering Science, 55 (18), 3875–3883. Chen, C., Miyoshi, T., Kamiya, H., Horio, M., and Kojima, T. (1999) On the scaling-up of a two-stage air blown entrained flow coal gasifier. Canadian

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29

2 Gasification of Solids: Past, Present, and Future Martin Gräbner

In view of the huge subject matter, the science of coal has gathered from its different disciplines during a century and in different institutions around the globe ..... K.H. van Heek [39] 2.1 Introduction

The history of the industrial gasification of coal goes back to more than 100 years. The great interest in this process can be explained easily by looking at its basic thermodynamics. During the incomplete oxidation of carbon to carbon monoxide (Eq. 2.1), only 28% of the coal’s heating value is released as thermal energy. Hence, the major part remains as chemically bonded energy in the gas phase. C + 0.5O2 −−−−→ CO

ΔR H ◦ = −110.5 kJ mol−1

(2.1)

C + CO2 ⇐==⇒ 2CO

ΔR H = +172.4 kJ mol

−1

(2.2)

C + H2 O ⇐==⇒ CO + H2

ΔR H = +131.3 kJ mol

−1

(2.3)

◦

◦

If the released heat from the partial oxidation could be used to supply other carbon-consuming endothermic reactions such as the Boudouard (Eq. 2.2) or heterogeneous water-gas reaction (Eq. 2.3), then 73–89% of coal’s heating value could be conserved in the gas phase. The produced gas consists mainly of H2 and CO and can be used in various ways as fuel gas or as synthesis gas (syngas). Consequently, an investment of only 11–27% of the feedstock energy is necessary to convert heterogeneous solids such as coal or biomass to a flexible, ash-free, pipeline-transportable energy carrier. Looking at the overall process of gasification, various other beneficial aspects ensue, which can be summarized as follows:

• The gas produced by gasification is a primary feedstock for the synthesis of base and fine chemicals or liquid fuels such as gasoline or diesel. This is in contrast to conventional coal utilization that aims only at the production of electricity through combustion. However, coal gasification can be integrated into a power Gasification Processes: Modeling and Simulation, First Edition. Edited by Petr A. Nikrityuk and Bernd Meyer. © 2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA.

30

2 Gasification of Solids: Past, Present, and Future

plant using a combined cycle for gas combustion. These so-called integrated gasification combined cycle (IGCC) power plants have a much higher efficiency potential than conventional coal combustion steam power plants. • In contrast to combustion, gasification permits much more efficient removal of pollutants from the raw gas due to the operation at high pressure. • Gasification also allows waste substances to be converted into syngas, thereby deteriorating critical hazardous substances (e.g., dioxin) because of the nonoxidizing conditions in the gasifier. These unique features have been promoting gasification technologies for decades which still remain the engine for today’s developments [1, 2].

2.2 Historical Background

The first developments made in gasification technologies relate closely to the combustion of gaseous fuels. In particular, one of the first practical uses of the so-called coal gas was the illumination of homes and foundries. In 1792, a Scottish engineer, Murdoch, pyrolyzed coal in an iron retort to produce coal gas to light his home [3]. The Westminster Bridge was lighted using the gas produced from coal by the first gas company established in 1812 in London. In continental Europe, Wilhelm August Lampadius introduced the first coal gas street light in 1811 in Freiberg, Germany. Soon, the streets of the city of Baltimore in the United States were lighted utilizing gas manufactured from coal. By 1875, in the United States the coal or town gas was widely applied for home lighting [3]. This town gas production was mainly carried out in atmospheric fixed-bed water-gas producers which were alternately fed with air and steam. The first industrial production of gas from coal traces also back to atmospheric systems. Some examples for this first generation of developments are the classical pressureless rotating-grate fixed-bed gasifiers and the fluidized-bed Winkler gasifier that was first put into commercial use in 1926 at the Leuna site near Leipzig, Germany. This is also the context in which the term synthesis gas or syngas emerged, since the gas was used for chemical syntheses. But also a technology for coal dust gasification – the Koppers–Totzek atmospheric entrained-flow process – was developed in Germany in the 1940s. It was Prof. Rudolf Drawe (1877–1967) who first saw the great potential of replacing air as the commonly used gasifying agent with oxygen and steam mixtures. This became possible after the invention of the Linde–Fränkel air separation process. In 1927, the German engineering company Lurgi patented the first pressurized oxygen-blown fixed-bed gasifier, which was put into commercial use from 1936 in Hirschfelde near Dresden, Germany. Besides town gas production and the upcoming NH3 market, the fast development in Germany was mainly

2.2

Historical Background

driven by the need, induced by the two World Wars, to produce liquid fuels from domestic sources such as brown coal. Of these early developments, the pressurized Lurgi fixed-bed dry-bottom (FBDB) technology was the most successful in the past. It was used for town gas production in Eastern Germany and Czechoslovakia, for synthetic natural gas production in the United States, and for ammonia production in Shanxi, China. Moreover, it also demonstrated its suitability for large-scale Fischer–Tropsch fuel production in South Africa, and even to fuel an IGCC power plant for electricity production in Vˇresová, the Czech Republic, or at the Schwarze Pumpe site in Germany until 2007. On the Chinese market, the Second Design Institute (Sedin) successfully introduced a domestic version of the FBDB technology. Lurgi recently extended the scope of the technology to provide gas for the direct reduction of iron ore in India. However, the main disadvantages of the pressurized fixed-bed process remain its limited particle size range (>5 mm), low gas quality (tar production), high steam requirements, and high wastewater production [4, 5]. In spite of these minor individual factors, the oil crises led to the development of a second generation of the coal gasification process. The global targets of the development, which took place from the 1970s to the 1990s, can be summarized as follows [6–8]:

• Increase in gasification pressure to 20–40 bar to reach higher single-unit capac• • • • •

ities (up to 500 MW) and lower capital investment; Increase in carbon conversion rates and cold gas efficiencies; Integration of heat recovery for steam generation; Low emission levels; High on-stream time; Operational flexibility regarding feedstock and load variation.

This second generation of coal gasifiers comprises numerous technologies such as the BGL (British Gas/Lurgi) [9], HTW (high-temperature Winkler) [10], U-Gas [11], KRW (Kellog, Rust, and Westinghouse) [12], Texaco [13], GSP (Gaskombinat Schwarze Pumpe) [14], E-Gas (Dow Chemical) [15], Shell [16], and Prenflo (Uhde) [17] processes. Most of them have become mature technologies and have been successfully implemented in IGCC plants (Wabash River, Polk, Puertollano, Buggenum) as well as for methanol, ammonia, and acetic acid anhydride syntheses. Most of these technologies have featured in the substantial growth over the last 10 years in China. Within the last decade, increasing interest in the utilization of low-grade coals (especially those with high ash contents) in emerging nations, sustained efforts to reduce CO2 emissions, the substitution of crude oil by other energy sources (natural gas, biomass, coal) to guarantee future energy sources, and high power generation fluctuations due to the increasing share of renewable energy sources again gave rise to investigations in gasification processes. Numerous design variations

31

32

2 Gasification of Solids: Past, Present, and Future

of second-generation gasification processes have been suggested for the state-ofthe-art processes used by Shell [18], Uhde [19], Siemens (formerly GSP) [20], GE (formerly Texaco) [21], Lurgi FBDB [22], and CB&I (E-Gas) [23]. In addition to this, a third generation of newly developed gasification processes has emerged, such as KBR (Kellog, Brown, and Root) [24], PWR (Pratt and Whitney Rocketdyne) [25], MHI (Mitsubishi Heavy Industries) [26], and the INCI concept (Internal circulation gasifier, TU Freiberg) [27]. In parallel, five new Chinese processes have been brought to commercial reality in China: the HT-L (Hangtian Lu) technology, the two-stage oxygen gasifier from the Tsinghua University, the OMB (opposed multiple burner) gasifier from the East China University of Science and Technology (ECUST), the TPRI two-stage coal gasifier from the Thermal Power Research Institute, and the MCSG (Multi Component Slurry Gasification) technology from the Northwest Research Institute of Chemical Industry. The Chinese developments mainly are a result of politically motivated support for the production of chemicals (e.g., ammonia, Shell, NL

Prenflo (Uhde), DE

GE Energy (Texaco), US

CB&I (E-Gas), US

HT-L,

Siemens (GSP), DE

CN

Mitsubishi (MHI), JP

MCSG (NRICI), CN

Tsinghua two stage oxygen, CN

OMB (ECUST), CN

TPRI two-stagecoal, CN

Entrained bed HTW (Uhde), DE

U-Gas (GTI), US

TRIG (KBR), US

Sasol, Sedin, or Lurgi dry ash, ZA, CN, DE

BGL (Envirotherm), DE, CN

Moving bed

Fluidized bed

Figure 2.1 Commercially available gasification technologies (NL – The Netherlands, DE – Germany, CN – China, US – United States of America, JP – Japan, ZA – South Africa). Abbreviations: GSP – Gaskombinat Schwarze Pumpe; HT-L – Hangtian Lu; GE – General Electric; MHI –Mitsubishi Heavy Industries; MCSG – Multi Component Slurry Gasifier; NRICI – Northwest Research Institute

of Chemical Industry; TPRI – Thermal Power Research Institute; OMB – Opposite Multiple Burner; ECUST – East China University of Science and Technology; BGL – British Gas/Lurgi; HTW – High-Temperature Winkler; GTI – Gas Technology Institute; TRIG – Transport Reactor Integrated Gasifier; KBR – Kellog-Brown-Root.

2.3

Types of Gasification Reactors

33

methanol, polyethylene) from domestically available coal resources in order to decrease dependency on petrol imports. Figure 2.1 shows the currently available commercial gasification technologies as a result of the described development. 2.3 Types of Gasification Reactors

In order to understand the potential of the various gasification technologies, it is instructive to classify the processes according to their bed type, as indicated in Figure 2.1. The type of bed chosen places restrictions on many process parameters. A summary is given in Table 2.1. Moving-bed gasifiers – often also called fixed-bed gasifiers – exhibit advantages in terms of simple fuel preparation, feeding method, process control, and oxygen consumption. The low exit gas temperature causes considerable amounts of higher hydrocarbons (tars) in the gas, which not only increases the amount of Table 2.1

Gasifier classification by bed type [3, 4, 28].

Flow types Typical feed size Feed preparation Feeding system Solid residue Oxygen consumptiona Steam consumptionb Exit temperature Carbon conversion Tar deterioration Residence time Specific capacity

Moving-bed

Fluidized-bed

Entrained-flow

Co/countercurrent Coarse-grained (3–60 mm) Screening, agglomeration Sluices Dry ashc , slagd 0.19c –0.53d 2.0c –0.4d 350–800 ◦ C 80–90% Barely Hours Low/moderate

Stationary, circulating Small-grained (0–6 mm) Crushing Gravity pipes,screw feeders Dry ash, agglomerates 0.4–0.7 0.2–0.6 800–1000 ◦ C 80–95% Predominantly Minutes Moderate/high

Up-/down-flow Pulverized (95% Completely Seconds Very high

Typical gas composition (25 bar, H2 O/O2 -blown lignite gasification, dry vol%) Example Lurgi FBDB HTW CO 10 38 38 32 H2 13 5 CH4 39 25 CO2 Typical processes

a m3 (STP)/kg(waf ). b kg/kg(waf ). c Lurgi d BGL

FBDB technology. technology

Lurgi FBDBc , BGLd

HTW, U-Gas, TRIG, etc.

Siemens 59 34 0 s Overlap > 0

m2

u′1

m1

m1

u′ is function of

m1

change in momentum coefficient of normal and tangetial restitution friction coefficient

m2

u′1 u′2

Post-contact

m

m2

u′2 Post-contact u′ is function of

m1

m2

contact forces calculated using “overlap” + inertia

Figure 3.2 Illustration of the basic assumptions used in “hard-sphere” and “soft-sphere” approaches used in the DEM model. (Reproduced with changes from [6].)

this “soft” particle boundary, the particles can overlap slightly. In this approach, friction and elastic restitution come into effect only when the spheres penetrate each other. In comparison to the hard-sphere model, where only one binary collision is considered at each time step, the soft-sphere model is able to handle multiple simultaneous contacts. The work of Cundall and Strack [16] was one of the first to use the method for the simulation of packed particles. The main difference between the different soft-sphere models reported in the literature is in the contact force scheme. Various popular schemes are given in [17, 18]. Before each model is described in more detail, we will outline some specific advantages and disadvantages of the soft-sphere and hard-sphere models. In the view of computational efficiency, the hard-sphere model needs fewer computations because the post-collision velocities are directly calculated based on the pre-collision velocities. This guarantees the conservation of energy. However, one of the significant limitations of this model is that only binary collisions can be calculated and, thus, naturally, event-driven schemes are used where the simulation jumps from one collision to the next collision (e.g., [10, 11, 19, 20]). On one hand, this scheme is suitable for systems with a low particle density and thus a low collision frequency because the time between the collisions is large. On the other hand, for dense systems with a high collision frequency, the time step approaches zero. However, there are some modifications where the so-called time-driven schemes has been used also (e.g., [12, 21–23]).

3.2

Soft-Sphere Model

Using the soft-sphere model, the collision of multiple particles is not a problem because the forces are simply added, which allows furthermore the straightforward implementation of additional forces such as the van der Waals force. A drawback of the soft-sphere model is that small time steps must be used in order to resolve the time history of the particle overlap and to conserve the energy during the collision.

3.2 Soft-Sphere Model

The soft-sphere model is a time-driven model in which, after a fixed time step Δt, the current state of the particles are updated. The time step has to be much smaller than the time of contact between two entities. Furthermore, in the soft-sphere model, the entities are allowed to overlap. This approach enables one to model multiple contacts at one time, and the sum of contact forces results in a change of velocity. Therefore, the soft-sphere model is, apart from its computational costs, highly suitable to model particle–particle interactions in dense systems. Elgobashi proposed the classification of interphase coupling as shown in Figure 3.3. It can be seen that with a high solid volume fraction (�S > 10−3 ) the momentum transport due to particle–particle interaction has to be considered. These dense gas–solid processes are widely applied in the gasification industry having in mind fluidized-bed and moving/fixed-bed reactors. In this section, we present the soft-sphere model. First, the basic equations for the calculation of the contact forces are given. The estimations of the key parameters and optimization to reduce computational cost are described as well. Analytical and experimental cases of particle interactions are presented to validate the introduced form of the model and to give the readers the opportunity to check their own implementation. Finally, we show some examples of simulations to give some insights into the capabilities of soft-sphere approach.

One-way coupling

Two-way coupling

Four-way coupling

Fluid

Fluid

Fluid

Particles Particle collison modeling Particles 10−9

10−7

Particles 10−5

Particles 10−3

10−1

εS

Figure 3.3 Interphase coupling of multiphase flows. (Partially adapted from [24, 25].)

47

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3 Modeling of Moving Particles: Review of Basic Concepts and Models

3.2.1 Numerical Implementation 3.2.1.1 Contact Forces

The contact mechanics of interparticle forces are quite complex. Cundall and Strack [16] introduced the first simplified model. They used a linear spring and dashpot to model the contact forces. This model is represented schematically in Figure 3.4. More sophisticated models [26, 27] using Hertz’s theory [28] for the normal direction and forms of Mindlin and Deresiewicz [29] model for the tangential forces have been developed. Hertz proposed that the normal force is a nonlinear function of the normal displacement. Mindlin and Deresiewicz developed a general model for the tangential forces. The capabilities of the different contact models have been investigated by Di Renzo and Di Maio, F.P [30]. In this chapter, we limit ourselves to the linear model since it is computationally less expensive. The contact forces are divided into normal and tangential forces, that is ) ∑( F⃗ij,n + F⃗ij,t (3.7) F⃗contact,i = j

The normal component of the contact force between two particles can be calculated as F⃗ij,n = kn �n n⃗ ij − �n u⃗ ij,n

(3.8)

where kn is the normal spring stiffness, �n is the normal overlap, and n⃗ ij is the normal unit vector. These are used to describe the force due to elastic deformation (Hook’s law) represented by the spring. The viscous dissipation represented by the dashpot is dependent on the damping coefficient �n and the normal relative velocity u⃗ ij,n . The relative velocity at the contact points of the particles i and j is formulated as ) ) ( ( (3.9) u⃗ ij = u⃗ i − u⃗ j + ⃗rP,i Ω⃗i + ⃗rP ,j Ω⃗ j × n⃗ ij

Dashpot Center of mass

(a)

Spring

Center of mass Spring

Dashpot

Friction slider

(b)

Figure 3.4 Schematic representation of spring–dashpot contact model for (a) the normal component of the force and (b) the tangential component of the force. (Adapted with changes from Tsuji [26].)

3.2

Soft-Sphere Model

⃗ is the angular where u⃗ is the particle velocity, rP is the particle radius, and Ω particle velocity. The tangential component of the force is limited by Coulombic type of friction and is given by { F⃗ij,t =

−kt �⃗t − �t u⃗ ij,t , −�|F⃗ij,n |⃗tij ,

if |F⃗ij,t | ≤ �|F⃗ij,n | if |F⃗ij,t | > �|F⃗ij,n |

(3.10)

where kt , �⃗t , �t , and u⃗ ij,t are the tangential stiffness, displacement, damping coefficient, and relative tangential velocity, respectively. The friction coefficient is described by �, and ⃗tij is the tangential unit vector. The calculation of the tangential displacement is more extensive because the instantaneous displacement is dependent on the previous value of the tangential displacement. Furthermore, the contact plane between the two particles in which the tangential unit vector is situated may change during the contact and a rotation of the reference frame is applied. Thus, the formulation of the tangential displacement takes the following form, which is adapted from [1]: { �⃗t =

t �⃗t,0 H + ∫t0 v⃗ij,t dt, � ⃗ − k |Fij,n |⃗tij , t

if |F⃗ij,t | ≤ �|F⃗ij,n | if |F⃗ij,t | > �|F⃗ij,n |

(3.11)

The calculation of the tangential displacement in case of sliding was proposed by Brendel and Dippel [31]. The rotation matrix H [1] is 2 ⎡ qhx + c ⎢ H = qhx hy + shz ⎢ ⎣ qhx hz − shy

qhx hy − shz qh2y + c qhy hz + shx

⃗ c, s, and q are defined as where h, n⃗ ij ×⃗nij,0 h⃗ = |⃗n ×⃗n | , c = cos �, s = sin �, ij

ij,0

qhx hz + shy qhy hz − shx qh2z + c

⎤ ⎥ ⎥ ⎦

(3.12)

( ) q = 1 − c with � = arcsin |⃗nij × n⃗ ij,0 |

3.2.1.2 Collision Parameters

After modeling the contact forces, it is necessary to determine the spring stiffnesses kn and kt , the damping coefficients �n and �t , and the friction coefficient �. The fully elastic and the fully inelastic impacts are the two limiting cases of particle impact. Since most impacts are between these two cases, the loss of kinetic energy has to be taken into account using viscous damping expressed with the coefficient of restitution. The coefficient of restitution is defined as the ratio of the relative velocities at the start and at the end of the contact of two entities. Therefore, this parameter is easily accessible from experiments and is used to formulate

49

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3 Modeling of Moving Particles: Review of Basic Concepts and Models

the damping coefficient. The normal damping coefficient is obtained from ⎧ −2 ln en √mijk n ⎪ √ , π2 +ln2 en ⎪ �n = ⎨ √ ⎪ ⎪ 2 mij kn , ⎩

if en ≠ 0 (3.13) if en = 0

and the tangential damping coefficient from ⎧ −2 ln e √ 2 m ⎪ √ t 7 ijkt , ⎪ π2 +ln2 et �t = ⎨ ⎪ √2 ⎪ 2 7 mij kn , ⎩

if et ≠ 0 (3.14) if et = 0

Here, en and et are the normal and tangential coefficients of restitution, respectively. The effective mass mij is given by )−1 ( 1 1 + (3.15) mij = mi mj In the linear spring–damper system, the use of a lower spring stiffness is preferred to employ larger time steps and optimize the computation time. However, the spring stiffness determines the overlap of the particles, which should be < 1% of the particle diameter [25, 32]. On the other hand, the contact time should be considerably larger than the computational time step. For the determination of the calculation time step, the following relation is widely used: √ m (3.16) Δt ≤ C k where C is a constant that can be varied. Common values for C are 0.2π or 0.2 [33].

Note: If a nonlinear Hertzian spring–damper system is used to model the contact forces instead of a linear one, the spring stiffness is commonly calculated from the particles’ material properties, namely the Young’s modulus, the Poisson’s ratio, and the particle radius.

3.2.1.3 Contact Detection

After choosing the right the time step, an effective detection of collisions is essential to speed up computations. In particular, this is true for the simulation of a large number of particles since the workload has the order of  (NP2 ) for the naive collision check. We will present the combined linked cell–neighbor list because it is easy to implement and comprehensible. This approach was introduced to the MD simulation [8]. The first part of the method was suggested by Verlet [34] and

3.2

Soft-Sphere Model

dNB

Figure 3.5 Scheme of the combined neighbor and linked cell list. The observed particle (black) and its neighbors (gray) are situated in a frame representing the linked cells without boundaries.

therefore also known as the Verlet list. In this method, a list of neighbors is maintained which is updated only after a certain number of time steps. A particle is considered as a neighbor when its distance from the particle of interest is less than dNB (see Figure 3.5). The optimum distance dNB considered for constructing the neighbor list is dependent on the particle velocity and the density of the system. However, a suitable value seems to be 2rP . If the sum of the maximum moving path of the particles, drmax , for each time step is larger than dNB ∕2, the neighbor list has to be updated, that is ∑ 2 drmax ≥ dNB (3.17) ti

Although only collisions between the particles stored in the list are checked, the number of operations is still of the order  (NP2 ). The linked cell list [8] follows a different approach. Here, the system is divided into a lattice of subcells. Figure 3.5 exemplarily shows the lattice for the 2D case. In three dimensions, the considered neighbors are searched only in the 26 neighboring cells and in the cell where the particle of interest is itself placed. Therefore, each particle has to be sorted into the corresponding cell. The sorting is a rather fast process and the information is stored in the two arrays HEAD and NEXT (see Figure 3.6). The array HEAD has the size of the total cell number, and in each element one particle number is stored. Further particles in the cell are stored in the array NEXT. The particle number from the array HEAD is used as the index of the array NEXT to address the next particle number in the cell. Subsequently, the stored NEXT array element is the next index in this array next particle in the cell, and so on, till eventually the element in array NEXT is zero. This means that no further particles are found in that cell and the process is continued for the next cell in the array HEAD. Figure 3.6 demonstrates an example of this approach. The application of the linked cell list leads to 27 NP ⋅ N P,C operations to detect interparticle collisions, where N P,C is the average number of particles within one cell.

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3 Modeling of Moving Particles: Review of Basic Concepts and Models

Cell 1

5

Cell 3

HEAD(NCell)

Cell 6

NEXT(NP)

9

1 10 Cell 4

Cell 2

6

4 Cell 5 8

2

1 2 3 4 5 6 10 9 0 3 8 7 1 2 3 4 5 6 7 8 9 10 0 0 0 0 1 5 0 2 4 6

7

3

(a)

HEAD(1)

NEXT(10)

NEXT(6)

NEXT(5)

NEXT(2)

Empty

(b)

Figure 3.6 Example of the generation of the linked cell list stored in the arrays NEXT(NCell ) and NEXT(NP ). (a) Distribution of particles in the cells and (b) the corresponding arrays with the particle sequence for cell 1.

This means that the number of operations is now  (NP2 ) for the collision check. We mention two further advantages of the method. First, compared to the Verlet list, the computationally expensive distance check is avoided, which requires a square root operation. Second, ghost cells can be easily created at the boundaries. This enables one to apply periodic boundary conditions. The combination of the two approaches is beneficial. The neighbor list does not have to be constructed at each time step, and not every particle pair has to be checked. One has to pay attention to the size of the cell. The edge length of one cell should be equal to or larger than dNB + rP,max .

Note: The computations can be further accelerated when neighbor lists for the walls are implemented.

3.2.1.4 Time Integration

The new positions and velocities of the particles are obtained from the calculated forces by integrating the equations of motion (see Figure 3.7). An appropriate time integration scheme is essential to gain stable and efficient computations and to ensure energy conservation. Therefore, the time integration is a widely investigated area going back to the field of MD [8, 34–36]. The time integration scheme for DEM was reviewed and compared by Rougier et al. [37], Fraige and Langston [38], and Kruggel-Emden et al. [39]. The numerical schemes were divided into three categories: one-step, multistep, and predictor–corrector methods [39]. For the examples presented in this chapter, a Beeman–Verlet scheme was applied [25]. The particle coordinates and velocities are calculated as ( ) 1 4a(t) − a(t − Δt) Δt 2 + O(Δt 4 ) (3.18) x(t + Δt) = x(t) + u(t)Δt + 6 ( ) 1 2a(t + Δt) + 5a(t) − a(t − Δt) Δt + O(Δt 3 ) (3.19) u(t + Δt) = u(t) + 6

3.2

Soft-Sphere Model

Initializing Definition of geometry and particle properties Definition of particle coordinates and velocities

i = 1, number of time steps

Path of particle > cut radius

no

Yes Update of neighbour list j = 1, number of particles in neighbourlist

Collison detection and calculation of overlap

Calculation of forces and moments acting on particles

j = 1, number of particles

Calculation of acceleration

Integration Update of particle velocities and coordinates

End

Figure 3.7 Program flowchart of the soft-sphere model.

3.2.2 Validation Cases

Next, we present some single-particle examples to validate and verify the softsphere model described. 3.2.2.1 Free-Falling Particle

A spherical particle falls from the initial height h0 because of gravity g on a plane wall. Figure 3.8a gives its schematic representation. The particle motion can be divided into free fall, contact, and rebound. For each stage, an analytical expression

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3 Modeling of Moving Particles: Review of Basic Concepts and Models

3 0.5 2

mg y

0.4

1

0.3

0 −1

0.2

−2

0.1

−3

0

0.1

x (a)

(b)

0.2

0.3

0.4

0.5

y (m)

h0

u (m s−1)

54

0

t (s)

Figure 3.8 (a) Scheme of a spherical particle falling onto a wall (adapted from [41]) and (b) comparison of DEM results and analytic solution for the particle velocity and position. The DEM and analytic results are represented by symbols and lines, respectively.

can be found in [40, 41] and the equations are rewritten in the next paragraph. In Figure 3.8b, the results of the DEM simulation and the analytic solution are presented for a spherical particle (rP = 0.1 m, �P = 2600 kg m−3 ) falling from an initial height (h0 = 0.5 m, g = 9.81 m s−2 ). The normal spring stiffness and the normal restitution coefficient take the values 4 × 104 N m−1 and 0.9, respectively. It is obvious that the data from DEM simulation agree well with the results of the analytic solution. 3.2.2.2 Analytic Solution for the Free-falling Particle

The equations for the velocity and the position in the free-fall stage with the initial ̇ = 0) = 0 and y(t = 0) = h0 are as follows: conditions y(t ẏ =

∫

ÿ dt = −gt

(3.20)

y=

∫

1 ̇ = h0 − gt ydt 2

(3.21)

The free-fall stage ends when the sphere hits the wall. Then the center position √ is equal to the particle radius and this gives the start time of contact tC = 2(h0 − rP )∕g. According to the linear spring–dashpot model, the particle acceleration is expressed as ( ) ÿ = −g − �0 y − rP − 2��0 ẏ

(3.22)

√ √ where the expressions � = �n ∕2 kn mP and �0 = kn ∕mP are introduced. For � < 1, the particle position and velocity can be calculated as

3.2

Soft-Sphere Model

√ �g ⎤ ⎡ −2 2g(h0 − rp ) + � ) )⎥ (√ ( √ ⎢ g 0 2 2 1 − � �0 t + 1 − � �0 t ⎥ sin y = ⎢ 2 cos √ �0 1 − � 2 ⎥ ⎢ �0 ⎦ ⎣ ( ) (√ ) g × exp 1 − � 2 �0 t + rp − 2 (3.23) �0 √ ( ) ⎡ √ ( ) 2g h0 − rP − g �� √ 0 ( ) ⎢ 1 − � 2 �0 t + ẏ = ⎢− 2g h0 − rP cos √ �0 1 − � 2 ⎢ ⎣ )] (√ ) (√ 1 − � 2 �0 t exp 1 − � 2 �0 t (3.24) × sin The rebound stage starts when the spherical particle makes contact with the wall. Here, the center position of the particle is again equal to the particle radius. ̇ = tr ) = ur and Thus, the equations of motion differ in the initial conditions (y(t y(t = tr ) = rP ) compared to the ones for the free fall. ẏ =

∫

ÿ dt = −gt + ur

(3.25)

y=

∫

1 ̇ = rP + ur t − gt 2 ydt 2

(3.26)

3.2.2.3 Slipping Sphere on a Rough Surface

A spherical particle is set on a horizontal surface with an initial translational velocity, whereas the rotational velocity is zero [40, 41] (see Figure 3.9a). A tangential force at the contact point starts to act in the opposite direction of the translational motion due to the sliding friction. That is why the translational velocity is reduced, whereas an angular velocity is induced until the relative velocity is zero at the contact point (u = �rP ). The equations of translational and angular acceleration take the forms u mP x = Ft = −�SF mP g (3.27) dt d� = �SF mP g rP (3.28) I dt where � is the angular velocity of the spherical particle, shown in Figure 3.9a. Both equations can be integrated, and the time when the particle ceases to slide can be calculated as tS = 2u0 ∕7�SF g. The nondimensional velocities u′x and �′ are u �r �′ = P (3.29) u′x = x , u0 u0 The nondimensional velocities and time of the DEM simulation and the analytical solution are now presented. Figure 3.9b displays the nondimensional velocities at the time tS and the nondimensional time t ′ = �SF g∕u0 tS obtained from DEM simulation and from the analytical solution. The sliding friction coefficients are varied from 0.1 to 0.9. It can be seen that the results of nondimensional time

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3 Modeling of Moving Particles: Review of Basic Concepts and Models

0.7145 t′ Analytic t′ DEM

ux′, ωy′ Analytic

0.7144

ωy′ DEM

0.2865

ux′ DEM

ω

0.286 0.7142 0.2855

0.7141

v

y

0.7143 t′

ux′, ωy′

56

Ft

0.714

x

0.2

0.4

0.6

0.8

0.285

μSF (a)

(b)

Figure 3.9 (a) Scheme of a spherical particle with finite translational velocity and zero angular velocity slipping on a surface. (Adapted from [41].) (b) Comparison of the results of DEM simulation for nondimensional time t′ and velocities u′ and �′ with the analytic solution for different coefficients of sliding friction.

from DEM are in good agreement with the analytic solution. Small deviations are seen for the nondimensional velocities. In the DEM simulations, the translational velocity is slightly overestimated whereas the angular velocity is lower than the analytic solution. 3.2.3 Illustrative Examples 3.2.3.1 Breaking Dam Problem

Numerical experiments were performed to study the particulate flow in the wellknown dam-break in a rectangular channel. The results of the numerical simulations are compared with those of an experiment [42]. The dynamics of the particles depend particularly on the initial ratio a, which is defined as the ratio of column height H and length L. The principal setup is shown in Figure 3.10. The particles are placed uniformly behind a gate in the rectangular channel. After the gate is opened, the particles form an avalanche because of gravity. The aspect Granular material

Plexyglass gate

Rough surface H W = 45 mm L Figure 3.10 Scheme of the experimental setup. (Adapted from [42].)

3.2

Table 3.1

Soft-Sphere Model

Parameters and particle properties [43].

Parameters

a = 0.6

a = 16.7

L NP dP �P e �SF �RF

0.102 m 13 300 3 × 10−3 m 2.5 × 103 kg m−3 0.75 0.5 0.133

0.01 m 4810 3 × 10−3 m 2.5 × 103 kg m−3 0.75 0.5 0.133

ratio a is varied from a = 0.6 to a = 16.7. The parameters used in the simulations are presented in Table 3.1. The rolling friction, described with the rolling friction coefficient, is applied only between the wall and the particles [43]. Figure 3.11 illustrates the deposit morphologies of the particles at certain times. 3.2.3.2 Rotating Drum

Now we present the example of a rotating drum. The particulate flow is determined by the properties of the particles and the rotating drum. The system comprises 5000 spherical particles (dP = 7 mm, �P = 2000 kg m−3 , en = et = 0.9) in a rotating drum with periodic boundaries in the axial direction. Sliding friction is applied and the friction coefficient � is chosen to take the value 0.3 for the particles and the wall. The rotation velocity of the drum with diameter DW = 0.238

(a)

a = 0.6

(b)

a = 16.7

Figure 3.11 Sequence of images of the granular collapse of stacks of spherical particles with aspect ratios (a) a = 0.6 and (b) a = 16.7 in the experiment (black background [42]) and in √numerical simulations. The time interval between each image corresponds to �C = H∕g (Images of experiments shown in left figures of (a) and (b) are reprinted with permission from [42]. Copyright © 2005, AIP Publishing LLC.)

57

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3 Modeling of Moving Particles: Review of Basic Concepts and Models

→

|u|∗ (a)

0 0.2 0.4 0.6 0.8 1 1.2 1.4

40 rpm, Rolling

→

|u|∗ 0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 (b)

150 rpm, Cascading

Figure 3.12 Particulate flow in a rotating drum at different rotation velocities where the particles are shaded according their nondimensional velocity magnitude |⃗u|∗ .

is varied. The computational time step was about 4 × 10−6 s. Figure 3.12 shows snapshots of the particles shaded by the nondimensional translation velocity |⃗u|∗ according to Eq. (3.30): |⃗u|∗i =

|⃗u|i 0.5 DW ΩW

(3.30)

where ΩW is the angular velocity of the drum. The regimes of rolling and cascading can be seen, which are established according to the angular velocities of 40 and 150 rpm, respectively. 3.2.3.3 Generation of Fixed Beds

In the next example, we present the influence of the dispersion of granular media on the porosity distribution in fixed beds. Therefore, fixed beds with a diameter of 0.6 m have been generated numerically. The dispersion of the packing is varied by the ratio of largest to smallest particle, RdP . The ratio is unity in the monodisperse case and ranges from 2 to 10 in the polydisperse case. The particle size distribution (PSD) is determined by the well-known Rosin–Rammler distribution [44]. In Eq. (3.31), the cumulative sum function is described. [( )nPSD ] dP Q3 (dP ) = 1 − exp (3.31) dP63.3 where dP63.3 and nPSD are the particle sizes, for which the cumulative sum is 63.3%, and the spread parameter of size fractions, respectively. Figure 3.13 depicts the columns of spherical particles with different particle diameter ratios. Figure 3.14a shows the porosity distribution of monodisperse packing. It can be seen that the

3.3

Particle diameter (m)

Hard-Sphere Model

Particle diameter (m) 0

0.02 0.03 0.04 0.05

0

0.02 0.03 0.04 0.05

z

z

x

(a)

RdP = 2, NP = 9 × 104

y

x

(b)

y

RdP = 10, NP = 2 × 105

Figure 3.13 Columns of polydisperse particles with different diameter ratios RdP .

DEM simulation shows good agreement with results from Eq. (3.32) derived in [45] from experimental data. 2 ⎧ 2.14z − 2.53z + 1, ⎪ �(r) = ⎨ � + 0.29 exp (−0.6z) [cos (2.3�(z − 0.16))] B ⎪ ⎩ + 0.15 exp (−0.9z) ,

z ≤ 0.637 (3.32) z > 0.637

In the above, z describes the dimensionless radial distance, z = (R − r)∕dP , and �B is the bulk porosity. In Figure 3.14b, the radial porosities are given for different RdP values. It can be seen that the overall porosity decreases as RdP increases. Furthermore, it can be observed that the sinusoidal behavior is less pronounced for polydisperse packings. Substantial deviations from the overall porosities can be observed within a distance of one particle diameter from the wall.

3.3 Hard-Sphere Model

In this section, we describe the basic equations and numeral calculations of the hard-sphere model. Afterward, we illustrate the performance of the collision model using 2D particle-resolved computational fluid dynamics (CFD) simulation in air or under vacuum.

59

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3 Modeling of Moving Particles: Review of Basic Concepts and Models

1

1

0.8

ε

ε

0.8

DEM deKlerk

0.8

1

0.6

0.6

0.6

Rdp = 1

0.4

Rdp = 2

0.2 13

Rdp = 10

15

0.4

0.4

0.2

14

6

8

10

12

r/dP

(a)

0.2

14

6

8

10

12

14

r/dP

(b)

Figure 3.14 (a) Radial porosity distribution for monodisperse packing compared with de Klerk’s equation and (b) polydisperse packings with different particle diameter ratios RdP .

3.3.1 Governing Equations

A collision between the particles can alter their direction of flight, velocity, and rotation. In hard-sphere collision models, the impulse transfer during the collision of two particles is instantaneous and the velocity change of a binary collision can be directly calculated [1, 7, 14]. The situations before and after the collision are sketched in Figure 3.15. Before ⃗ the collision, the system is characterized by the translational (⃗u) and rotational (Ω) velocities of the particles i and j. During the collision, the system is defined by the relative velocity h⃗ and the normalized collision direction n⃗ at the point of contact: n⃗ =

x⃗j − x⃗i

(3.33)

|⃗xj − x⃗i |

⃗ i + rj Ω ⃗ j ) × n⃗ h⃗ = (⃗ui − u⃗ j ) − (ri Ω

(3.34)

with the position of the particles x⃗ and their radius r. z

→ Ωi

→ uj

Ri

→ n

Rj

→ xi

→ ui → Ωj

→ h

→ xj x y Figure 3.15 Scheme of the collision between two particles. (Adapted from [1, 7].)

3.3

Hard-Sphere Model

⃗ and after (h⃗ ′ ) The relation between the normal and tangential velocity before (h) the collision is defined by the restitution coefficient in the normal direction � and in the tangential direction � [7, 13, 46]: ⃗ n⃗ ⋅ h⃗ ′ = −� (⃗n ⋅ h)

(3.35)

⃗ × n⃗ (⃗n × h⃗ ′ ) × n⃗ = � (⃗n × h)

(3.36)

The coefficient � takes values between zero and unity, where for � = 1 the collision is elastic and for � < 1 the collision is inelastic. The coefficient � can vary in the range −1 ≤ � ≤ 1, and it describes the surface roughness of the particles. In detail, the case � = −1 corresponds to the collision of smooth particles, where the particles slip over one another and the tangential velocity at the contact point does not change during the collision. As � increases, the surface roughness increases and the tangential velocity after the collision decreases. At � = 0, the tangential velocity becomes zero during the collision. However, when � increases further, the direction of the tangential velocity is reversed and the magnitude of the velocity increases as � increases. From the physics point, in this case the particle surfaces are so rough that they grip on each other. As a result, the particles deform and the kinetic energy is transferred into strain, which is converted back as the particle shape recovers. However, the direction of the tangential velocity is reversed. The kinetic energy of the system is conserved in the case of perfectly elastic (� = 1), perfectly smooth (� = −1), or perfectly rough (� = 1) particles [7, 13, 15]. For particles with mass m and the mass moment of inertia I, the post-collision ⃗ ′ ) are calculated as follows [7, 14, 15]: velocities (⃗u′ , Ω �⃗ (3.37) J u⃗ ′i = u⃗ i − mi ij u⃗ ′j = u⃗ j +

�⃗ J mj ij

(3.38)

⃗′ = Ω ⃗ i − � ri n⃗ × ⃗Jij Ω i Ii ⃗′ = Ω ⃗j − Ω j

� rj Ij

n⃗ × ⃗Jij

(3.39)

(3.40)

where � = 2 mi mj ∕mi + mj is the reduced mass, and ⃗Jij is the collision impulse given by ( ( ) ) ⃗Jij = �1 n⃗ ⋅ h⃗ n⃗ + �2 n⃗ × h⃗ × n⃗ (3.41) with the dissipation parameters �1 and �2 , which are calculated as follows [13, 14]: �1 =

�+1 2

(3.42)

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3 Modeling of Moving Particles: Review of Basic Concepts and Models

�2 =

(� + 1) k 2 (1 + k)

(3.43)

with the coefficient of the moment of inertia k = I∕(m r2 ), which is 1∕2 in the case of a 2D cylinder or 2∕5 in the case of a sphere. Collision of a particle with a wall can be calculated assuming the wall as the particle j with infinite mass. In the next section, we use the collision model by assuming the restitution coefficients � and � to be constant. However, it must be noted that � depends on the impact velocity [7, 14, 47, 48] and � depends on the Coulomb friction. A collision model with a dynamic calculation of � was developed by Walton [46]. In his model, � is not a constant but rather a function of the ratio of the incident normal to tangential velocities [46]: or in other words, the angle between n⃗ and h⃗ [49]. After that, � is utilized to decide whether a sliding or sticking type of collision takes place (for further details, see [7, 20, 46, 49]). 3.3.2 Collision Treatment in Dense Particulate Systems

In comparison to the soft-sphere model, the hard-sphere model suffers from the basic disadvantage that only binary collisions can be resolved. Thus, naturally, the collision modeling is done in an “event-driven” scheme where the simulation proceeds from one collision event to the next one using a variable time step for the collision model (e.g., see [10, 11, 19, 20]). However, the method becomes ineffective if the particle density is high and a large number of collisions take place in fast sequence. As a result, the time between the collision events approaches zero. For dense particulate systems, the so-called “time-driven” schemes have an advantage because the time step is constant. However, a particle can collide with many particles in one time step. This is no drawback when using a soft-sphere model but it is when using a hard-sphere model. Nevertheless, time-driven schemes have been used in combination with hard-sphere models (e.g., [12, 21, 22]). A time-driven collision scheme was also used by Schmidt and Nikrityuk [23, 50] and Schmidt [51]. In their model, multiple collisions could take place in one time step, where a multiparticle collision is calculated as a sequence of binary collision. The scheme of the so-called “time-splitting” algorithm is shown in Figure 3.16. To illustrate the method, Figure 3.17 shows the sequence of collisions in a case of five smooth (� = −1) particles colliding simultaneously [23]. The collision sequence at t = tcoll (Figure 3.17b–h) is highlighted in gray. It can be seen that, after the sequence of six collisions, the particles move away from each other (Figure 3.17i). At this point, it must be noted that the example shown in Figure 3.17 is symmetric and works only with smooth particles. For rough particles (� > −1), the algorithm predicts wrong results because, during the sequence of binary collisions, a tangential impulse starts particle rotation (e.g., see the situation in Figure 3.17c). However, for real particulate systems, the symmetry in multiparticle collision is smaller and the overall movement of a particulate bed can be predicted by the algorithm.

3.3

Hard-Sphere Model

Search for neighbouring particles

i = 0, i < Npart Calculate collisions

yes

More than one collision per particle

no Time integration

Figure 3.16 Overview of the multiparticle collision algorithm. (Taken from [23, 51].)

3.3.3 2D Formulation of Hard-Sphere Collisions

The post-collision velocities of the particles i and j are calculated according to their mass m, translation velocity u⃗ , rotation velocity Ω, radius R, and the center of mass coordinates x⃗. The first step is the determination of the relative velocity of ⃗ the contact point h: ) ( ( ) j hx = uix − ux − ny Ri Ωi + rj Ωj ) ( ( ) j hy = uiy − uy − nx Ri Ωi + rj Ωj

(3.44) (3.45)

with the normalized collision vector n⃗ . j

nx = √

xx − xix j

(3.46) j

(xx − xix )2 + (xy − xiy )2 j

ny = √

xy − xiy j

(3.47) j

(xx − xix )2 + (xy − xiy )2

63

64

3 Modeling of Moving Particles: Review of Basic Concepts and Models

2

5

1

2

2

3

3

3

5

1

4

4 (a)

4

(b)

2

(c)

2

2

3

5

1

5

1

3

3

5

1

4

5

1

4

(d)

4

(e)

(f) 2

2

2

5

1

5

1

5

1

4

4

(g)

3

3

3

(h)

4 (i)

Figure 3.17 Detailed view of the simultaneous collision of five smooth particles: (a) t < tcoll , (b–h) t = tcoll (the colliding particles are highlighted in gray), and (i) t > tcoll .

The dissipation factors �1 and �2 are calculated from the restitution coefficient � and the surface roughness �. �+1 2 (� + 1) k �2 = 2 (1 + k) �1 =

(3.48) (3.49)

In the case of cylindrical particles, the coefficient of the moment of inertia k equals 1∕2, and for spheres the value of k equals 2/5. The components of the collision impulse take the form ( ) ( ) Jx = �1 kx hx + ky hy kx + �2 (ky hx − kx hy )ky ( ) ( ) Jy = �1 kx hx + ky hy ky + �2 (kx hy − ky hx )kx

(3.50) (3.51)

3.3

Hard-Sphere Model

The post-collision velocities u⃗ ′ and Ω′ can be calculated as follows: 𝜉 J mi x 𝜉 i u′y = uiy − i Jy m 𝜉 j j u′x = ux + j Jx m 𝜉 j j u′y = uy + j Jy m 𝜉 ′i i Ω =Ω − (n J − ny Jx ) kI mi Ri x y 𝜉 j (n J − ny Jx ) Ω′ = Ωj − kI mj Rj x y i

u′x = uix −

(3.52) (3.53) (3.54) (3.55) (3.56) (3.57)

where 𝜉=

2 mi mj mi + mj

(3.58)

3.3.4 Illustration of Hard-Sphere Models

In this section, we demonstrate the applicability of the “time-splitting” method for dense particulate flows using a 2D dam-break and a rotary kiln as examples. The dam-break of alumina cylinders in air was used to validate the collision algorithm (see [23, 50, 51]). In detail, the system under consideration consist of 33 aluminum (�Al = 2700 kg m−3 , Dp = 0.01 m) cylinders arranged in six layers with air (�Air = 1.1885 kg m−3 , � = 18.205 Pa s) as ambient fluid. The collapse is modeled using a particle-resolved finite-volume CFD code with stair-step representation of the particles and a particle resolution of 38 control volumes per particle diameter. Using the fact that the particles are resolved on the computational grid, a particle collision is calculated when two particles occupy neighboring control volumes and when the dot product of their velocities is less than zero. The overall size of the calculation domain was L1 = 0.26 m and L2 = 0.08 m in the x- and y-direction, respectively. Figure 3.18a shows the initial setup of the system. The collisions are modeled using the hard-sphere model with time splitting, where the coefficient � is set at 0.9 and � = −0.5. The time step of the simulation is 1 × 10−5 s. In Figure 3.18, the numerical results are compared with the experimental results of Zhang et al. [52]. Good agreement can be seen in the snapshots of the particles (Figure 3.18a) and for the time history of the mean particle position in the x- and y-direction (Figure 3.18 b,c). This illustration case shows that the time splitting works well for particle columns. Next, the performance of the time-splitting algorithm is illustrated in the case of a moving particulate bed in a 2D rotating drum, which has been analyzed by Schmidt [23, 51]. The particulate flow in rotating drums is a good test case

65

3 Modeling of Moving Particles: Review of Basic Concepts and Models

Experiment

Numeric

0.0 s

0.1 s

0.3 s

0.5 s

(b)

1

0.32 0.30 0.28 0.26 0.24 0.22 0.20 0.18 0.16 0.14 0.12 0.10 0.0

y/L

1

(a)

x/L

66

Experiment Numeric 0.1

0.2

0.3 t (s)

0.4

0.5

(c)

0.12 0.11 0.10 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.0

Experiment Numeric 0.1

0.2

0.3

0.4

0.5

t (s)

Figure 3.18 Comparison of experimental [52] and numerical results for the collapse of stacked cylinders. (a) Snapshots of the system. (b, c) Time history of the mean center of mass positions in x- and y-direction, respectively. (The images of experiments shown in the left figures of (a) are reproduced with permission. Copyright © 2009 Elsevier [52].)

for the collision algorithm because, depending on the rotational velocity, the properties of the granular material, and the drum filling level, six different flow regimes can be observed [53–55]. The investigated system contains 400 particles (Dp = 7 mm, �p = 2000 kg m−3 , � = 0.9, � = 0.0) in a rotating drum with diameter DW = 0.238 m. To obtain different flow regimes, the rotation velocity of

3.3

1 rpm, Slumping (a)

40 rpm, Cascading (d)

Hard-Sphere Model

1 rpm, Slumping

5 rpm, Rolling

(b)

70 rpm, Cataracting

67

(c)

150 rpm, Centrifuging

(e)

(f)

0 0.15 0.3 0.45 0.6 0.75 0.9 1.05 1.2 1.35 1.5 →

|u|∗ Figure 3.19 Particulate flow in a rotating drum at different rotation velocities where the particles are shaded according their nondimensional velocity magnitude |⃗u|∗ . The time difference between (a) and (b) is 0.75 s.

the drum is varied from 1 to 150 rpm. The particulate flow is modeled in vacuum using a particle-resolved CFD, where the particles are represented using a stair-step approximation (for details, see [23, 51]). The particle diameter and grid resolution correspond to 22 control volumes per particle diameter. The regimes slumping, rolling, cascading, cataracting, and centrifuging can be successfully reproduced using � = 0.9 and � = 0 (see Figure 3.19) [23, 51]. The figure shows snapshots of the particulate bed where the particles are shaded according their nondimensional translation velocity u∗ calculated by Eq. (3.30). The presented snapshots show the characteristic features of the different regimes. In detail, at a low rotation velocity of 1 rpm, the slumping regime is

68

3 Modeling of Moving Particles: Review of Basic Concepts and Models

visible where the particulate bed is lifted as the drum rotates (see Figure 3.19a). At this moment, the angle of response of the particulate bed is around 30◦ and an avalanche of fast (white colored) particles moves downhill. The situation after the avalanche is shown in Figure 3.19b. It can be seen that the angle of response is lower at around 21◦ . As the rotation velocity of the kiln increases to 5 rpm, the rolling regime is obtained (Figure 3.19c), where the particles in the so-called cascading layer move permanently downhill and a static angle of response is observed. In this regime, the core of the particulate bed is nearly at rest. Further increases of the rotation velocity results in higher particle trajectories, and at 40 rpm (Figure 3.19d) the particulate bed is kidney shaped, which is characteristic for the cascading regime. The regimes of rolling and cascading have high mixing rates and they are mainly used in rotary kilns [55]. Figure 3.19e depicts the cataracting regime, where the particles detach from the particulate bed and fly downhill. As result of the high particle velocity and thus high collision forces, the cataracting regime is used in ball mills to comminute the particulate material [55]. Finally, if the centripetal force of the particles exceeds the gravitational force, the centrifuging regime is observed where the particles stick to the wall. The above short discussion of the modeled flow regimes shows that the time-splitting method works well. Here, it must be noted that the slipping regime was not observed using rough particles defined by � = 0. However, using ideally smooth particles (� = −1), slipping can be also reproduced.

3.3.5 Conclusions

In this section, a simple hard-sphere model and the so-called time-splitting method were introduced. The performance of the model was demonstrated using two examples: the collapse of a cylinder column and the flow structure in a rotary kiln. The flow structure was well predicted by the model. However, it must be pointed out that the hard-sphere collision algorithm is not optimal to model the dynamics of particles that are in permanent contact. In this case, the particles can penetrate each other, which is nonphysical. Thus, for the modeling of, for example, particle columns or particles in hoppers, the soft-sphere model is recommended, but for systems where the particles are in permanent movement like in fluidized beds, the hard-sphere model can be an effective method to predict the particulate flow, especially when the so-called particle-resolved CFD-based models are used.

3.3

Hard-Sphere Model

Nomenclature Symbols Symbol

Unit

Description

a d dP63.3 DW e F⃗ g h⃗ I ⃗J k m n⃗ nPSD NP Q3 r ⃗t t T⃗ u⃗ , u

m s−2 m m m – N m s−2 m s−1 kg m2 kg m s−1 N m−1 kg – – – – m N s Nm m s−1

Acceleration Diameter Representative diameter for particle size distribution Diameter of drum Coefficient of restitution Force Gravitational acceleration Relative velocity at contact point Moment of inertia Impulse Spring stiffness Mass Normal unit vector Spread factor of particle size distribution Number of particles Cumulative sum of relative mass Radius Tangential unit vector Time Torque Translational velocity

Greek symbols Greek symbol

� � � � � �S � � ⃗ Ω ΩW

Unit

Description

m N s m−1 – – – kg m−3 s−1 s−1

Restitution coefficient in normal direction Tangential restitution coefficient Overlap Damping coefficient Porosity Volume fraction of solid Friction coefficient Density Angular velocity Angular velocity of drum

69

70

3 Modeling of Moving Particles: Review of Basic Concepts and Models

Subscripts Subscript

Description

n t i, j P

Normal Tangential Particle indices Particle

Superscripts Superscript

Description

References 1. Deen, N.G., van Sint Annaland, M.,

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3.

4.

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6.

7.

van der Hoef, M.A., and Kuipers, J.A.M. (2007) Review of discrete particle modeling of fluidized beds. Chemical Engineering Science, 62(1–2), 28–44. van der Hoef, M., van Sint Annaland, M., Deen, N., and Kuipers, J. (2008) Numerical simulation of dense gas-solid fluidized beds: A multiscale modeling strategy. Annual Review of Fluid Mechanics, 40, 47–70. Zhu, H.P., Zhou, Z.Y., Yang, R.Y., and Yu, A.B. (2008) Discrete particle simulation of particulate systems: A review of major applications and findings. Chemical Engineering Science, 63, 5728–5770. Clift, R., Grace, J., and Weber, M.E. (1978) Bubbles, Drops, and Particles, Dover Publications, Inc., Minaola, New York. Sreekanth, P., Syamlal, M., and O’Brien, T.J. (2011) Computational Gas-Solid Flows and Reacting Systems: Theory, Methods and Practice, IGI Global. O’Sullivan, K. (2011) Particulate Discrete Element Modelling: A Geomechanics Perspective, Spon Press, London and New York. Rao, K. and Nott, P. (2008) An Introduction to Granular Flow, Cambridge University Press, Cambridge.

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4 CD and Nu Closure Relations for Spherical and Nonspherical Particles Kay Wittig, Andreas Richter, and Aakash Golia

Understanding the behavior of single particles is, however, a solid foundation upon which to build knowledge of multiple-partcile systems Clift, Grace & Weber [49]

4.1 Literature Review

The rates of heat and mass transfer for spherical and nonspherical particles suspended in agitated fluids are important for the adequate modeling of fluid– particle flows in many industrial applications, especially fluidized-bed reactors. Computational models such as discrete particle models (DPMs) [1–3] or Euler–Euler-based models have become well-established tools for analyzing and studying the parameters of fluid–particle flows in fluidized-bed or entrained-flow reactors [3]. However, the behavior of fluid–particle flows in this class of reactors is still not completely understood (e.g., see [1]). Further, closure relations for the particle–fluid interaction and the heat transfer between the particle and the fluid, which are very often represented in the form of the drag coefficient cD and the Nusselt number Nu, respectively, are important to obtain realistic results using DPM. With the increase in computational resources, numerical methods have made it possible to develop closure relations for cD and Nu directly using computational fluid dynamics (CFD). In addition, this is possible at a far lower cost compared to experimental measurements. Basically, most drag models and the heat transfer coefficient closures are obtained empirically for ideal spherical particles. Early computational work on heat and fluid flow around a sphere for low to moderate Reynolds numbers was carried out by Dennis et al. [4]. Later, Johnson and Patel [5] carried out both numerical and experimental investigations in fluid flow past a sphere for Re ≤ 300. Different flow regimes were described depending on Re. Recently, Dixon et al. [6] provided a systematic numerical study of heat and fluid flow past a single sphere up to Re > 20 000. On the other hand, concerning heat transfer Gasification Processes: Modeling and Simulation, First Edition. Edited by Petr A. Nikrityuk and Bernd Meyer. © 2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA.

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4 CD and Nu Closure Relations for Spherical and Nonspherical Particles

past a sphere, it has been shown that the well-known Ranz and Marshall [7] and Whitaker [8] relations are in acceptable agreement with numerical predictions. In contrast to the work on spherical particles, there are fewer investigations on flow past nonspherical particles [9, 10]. For a recent review on drag correlations for nonspherical particles, see the work by Hölzer and Sommerfeld [11]. Based on over 2000 values for particles of different shapes, they developed a formulation for the drag coefficient as a function of the Reynolds number as well as sphericity, lengthwise sphericity, and crosswise sphericity. Concerning the characteristics of flow past a cube, Raul et al. [12] investigated the flow around a cube for 10 < Re < 100. Saha [13, 14] numerically investigated fluid flow and heat transfer past a cube. Moreover, Richter and Nikrityuk [9, 15] and Wittig et al. [10] calculated fluid flow and heat transfer past spheres, ellipsoids, and cubes for Reynolds numbers in the range 10 < Re < 250, developing a new regression formula for the Nusselt number and drag coefficient in this range. Additionally, Richter and Nikrityuk [15] and Wittig et al. [10] investigated the influence of one or two rotation angles on the fluid flow and heat transfer of nonspherical particles. In reality, however, particles are often porous initially, or they become porous, for example, as a result of drying, pyrolysis, or conversion processes [16]. For porous particles, several authors have studied the influence of a sphere’s porosity on the drag force [17, 18]. However, there are few works devoted to the study of heat flow past a porous sphere for moderate Reynolds numbers [16, 19, 20] in spite of the subject’s relevance for many industrial applications, for example, the gasification of coal or biomass. From this point of view, heat and fluid flow past nonspherical and porous particles are discussed below. The main aim is to develop closure relations for the drag force coefficient cD and the Nusselt number Nu as a function of the Reynolds and Prandtl numbers taking into account the particle shape.

4.2 Model Description

All investigations in this chapter are aimed at analyzing the effect of various geometric properties of the particle on the flow resistance and heat transfer of the particle. Therefore, the investigated particle is placed in a homogeneous crossflow of air with a constant inflow temperature T∞ , whereby the particle’s surface is assumed to be isothermal at a temperature Ts . A schematic view of the resulting computational domain is given in Figure 4.1. The length of each particle Dref is correlated to its volume, such that the particle volume for nonspherical particles equals that of a sphere with a radius D = 1 mm. The inflow and outflow boundaries are shown in Figure 4.1, where the value of the inflow velocity is u∞ . All remaining boundaries have a symmetric boundary condition. As suggested by Richter and Nikrityuk [9], the chosen domain size is at least 40 D × 20 D × 20 D in order to avoid blockage effects. The particle is located 10D short of the inflow and centered in the y- and z-direction, respectively. The

4.2

Symmetry

Model Description

Outflow

z y

x u

20 Inflow 20

40 Projected particle

Figure 4.1 Computational domain for flow past an arbitrarily shaped particle.

following assumptions are made: The gas flow is incompressible, and both viscous heating and buoyancy effects are negligible. All hydrothermal properties of the fluid are assumed to be constant and correspond to air at T∞ = 300 K, resulting in the Prandtl number Pr = 0.744. In addition, the temperature field does not influence the velocity field and behaves like a passive scalar. The Reynolds number characterizing the flow field is defined by u∞ D . (4.1) � In the following, two approaches are described for the discretization of the problem. The first uses Cartesian grids and the second is the conventional CFD approach with a body-fitted mesh. When using a Cartesian grid to set the no-slip and the thermal Dirichlet boundary conditions on the particle surface, a so-called immersed boundary (IB) method was utilized in the continuous forcing mode. With the assumptions made above, the governing equations for mass, momentum, and energy conservation written in Eulerian form read as Re =

∇ ⋅ u⃗ = 0, 1 1 ∂ u⃗ + (⃗u ⋅ ∇)⃗u = − ∇p + �∇2 u⃗ + F⃗IB , ∂t � �

(4.2) (4.3)

∂ (4.4) T + �cp u⃗ ⋅ ∇T = �∇2 T + QIB . ∂t The continuous force is represented by a Darcy-like term which vanishes for bodyfitted approaches where the solid phase is not resolved [21]. More details on the definition of the IB terms FIB and QIB in Eqs (4.3) and (4.4) can be found in Wittig et al. [10]. �cp

4.2.1 Numerical Scheme and Discretization

Two finite-volume solvers are employed to solve the three-dimensional Navier– Stokes equations with heat transfer, Eqs (4.2–4.4). The first solver is a 3D

75

76

4 CD and Nu Closure Relations for Spherical and Nonspherical Particles

finite-volume open-source code for Cartesian grids based on Ferziger and Peric [22], where the energy conservation equation and the IB method are implemented. In particular, for the immersed body reconstruction on a fixed Cartesian grid, a so-called supersampling method is used (see [16]). The pressure correction on the staggered grid arrangement was calculated with the SIMPLE algorithm (Semi-Implicit Method for Pressure-Linked Equations) [23], and pressure–velocity decoupling is prevented using the Rhie and Chow stabilization [24]. The evolving coefficient matrix is then solved with the SIP algorithm (see [25] for more details). All viscous fluxes are discretized using a central differencing scheme. The convective fluxes are approximated using a deferred correction scheme, which converges to a central differencing scheme. The source terms FIB and QIB in Eqs (4.3) and (4.4) are linearized as recommended by Patankar [23] (see [16] for a more detailed description). As mentioned before, the IB geometry is reconstructed using a so-called supersampling method, which was introduced in Wittig et al. [16]. In a traditional stair-step approximation of the geometry, a control volume (CV) can only take the values 1 for a fluid CV or 0 for a solid CV. However, both solid and liquid phases are present in the interface cells. For these cells, the supersampling method calculates a good approximation of the fractions of the two phases [16]. With a volume fraction of fluid between 0 and 1, the interface cells become porous cells and are modeled in the Navier–Stokes equations by a Darcy-like term. In order to estimate the optimal grid resolution for the IB approach, several grid resolutions were examined in [10]. The resulting grid is shown in Figure 4.2. The second solver is the Ansys Fluent® [26], which is a commercial CFD solver using meshes that are fitted to the geometry under investigation. The pressurebased solver was applied; this denotes the sequential solution of the governing equations. Since the equations are nonlinear, they are linearized beforehand using an implicit method. A point-implicit linear equation solver (Gauß–Seidel) is then applied in conjunction with an algebraic multigrid method (AMG) to solve the resulting system of linear equations. Since all components of the velocity vector are solved sequentially, the resultant velocity field may not fulfill the continuity equation. To overcome this, the SIMPLE algorithm is used, which combines

Figure 4.2 Middle plane of the rectilinear grid employed in the IB code.

4.2

Model Description

pressure and velocity corrections to enforce mass conservation [23]. For the spatial discretization of the convective terms, the quadratic upstream interpolation for convective kinematics (QUICK) scheme [27] was applied. In the steady-state regime, only the steady-state equations are solved, so the time derivatives in the Navier–Stokes equations are neglected. For higher Reynolds numbers, the flow field becomes unsteady. In that case, the unsteady equations are solved using a second-order implicit time integration scheme. Though a Cartesian grid is used, the structured mesh generation for a domain with complex geometries, for example, a sphere, is not a trivial task. A domain has to be decomposed into several “regular” block subdomains, where structured grids can be generated. At the same time, it is a well-known fact that when heat and fluid flow problems are being calculated, the solution depends significantly on the quality of the underlying numerical mesh. For example, the use of unstructured grids may lead to increased numerical diffusion [22] and very often does not guarantee proper accuracy and convergence of a flow solution. Thus, although unstructured meshes are easy to use, they should be applied very carefully. At the same time, it is a well-known fact that velocity and temperature gradients within boundary layers have to be resolved adequately, since they directly correlate with drag coefficients and Nusselt numbers. Equally, an insufficient domain size causes blocking effects, which also influence the flow characteristics (e.g., see [28]). Thus, resolving both the boundary layer and the domain size adequately may lead to CVs that offer very high aspect ratios and avoid the convergence of the whole system [9]. In order to ensure that the mesh is of high quality with sufficiently fine resolution near the wall, body-fitted, hexahedral meshes are applied (see Figure 4.3 for a example). Since the preparation of such numerical grids is difficult if a series of different angles of attack are considered, sliding surfaces are deployed. To achieve this, the whole domain is divided into two parts, which are separated by a spherical surface with a radius of 6d. The inner domain and the outer domain feature

®

Figure 4.3 Middle plane of the body-fitted structured grid used with Ansys Fluent .

77

78

4 CD and Nu Closure Relations for Spherical and Nonspherical Particles

a structured grid, but the meshes on the inner and outer side of the interface surface are not identical. This means that one or more surfaces from finite volumes outside the interface are attached to one surface of a finite volume that is located at the inner side of the interface. At the interface, the numerical flux is calculated from the weighted average of all fluxes coming from the neighboring volumes. This technique allows the inner domain to be rotated, so the angle of attack can be changed without the need to fully create a new mesh. In Richter and Nikrityuk [9], the authors discretized an irregularly shaped particle with two structured, hexahedral meshes, one mesh with and the other without a sliding interface. A comparison of the calculated drag coefficients and Nusselt numbers demonstrated the validity of this approach.

4.3 Code and Software Validation

For every CFD simulation, it is necessary to perform an extensive validation for the flow and heat transfer [29]. For the case of heat and fluid flow past a sphere, the following presents, first, a comparison of the two solvers and, second, a comparison of the solvers with cD and Nu predictions from the literature. First, the flow around a sphere is examined for Reynolds numbers between 10 and 250. Before proceeding with the validation of integral values such as the drag coefficient and Nusselt number, a comparison of the local values for the flow and the temperature field is shown in Figure 4.4. Next, the drag coefficient and surface-averaged Nusselt number of the flow around a sphere were validated against numerical results and semiempirical relations given in the literature. Based on different references, Clift et al. [30] and Richter and Nikrityuk [9] summarized the drag force correlations for the desired Reynolds number range. The drag correlation reported by Haider and Levenspiel [31] was used. In this case, we adopt a correlation found by Turton 1

IB Fluent

0.8

IB Fluent

310

T (K)

u(x) (m s−1)

308 0.6 0.4

306 304

0.2 302 0 300 −0.2 −1

(a)

−0.5

0

0.5 x/D

1

1.5

−1

2

(b)

−0.5

0

0.5

1

1.5

2

x/D

Figure 4.4 (a, b) Comparison of the velocity in the x-direction and the temperature along the x-axis (z = 0, y = 0) in front of and behind the sphere calculated for Re = 100 using two solvers.

4.3

Code and Software Validation

79

and Levenspiel [32] which, however, is suitable for spheres only. This correlation takes the following form: cD =

) 0.4251 24 ( 1 + 0.1806 ⋅ Re0.6459 + . Re 1 + 6880.95 Re

(4.5)

Moreover, the values of cD were compared with those from an empirical relationship developed by Almedeij [33] for the drag coefficient asymptotically approaching the wide trend for all Reynolds numbers. Other recent numerical results can be found in the literature [34, 37]. In Table 4.1 and Figure 4.5, we compare the drag coefficients determined by the two solvers with those taken from the literature. Good agreement can be seen between the two numerical approaches, namely the body-fitted code and the IB code. The maximum deviation between the two Table 4.1 Flow around a sphere at different Reynolds numbers: comparison of drag coefficients from different references with results from the present study. The results referred to as [36–38] correspond to data obtained numerically. Re

[30]

[38]

[33]

[35]

[31]

[36]

[37]

Fluent Solver

®

IB Solver

10 25 50 75 100 150 200 250

4.151 2.275 1.539 1.254 1.094 0.914 0.81 0.741

— — 1.57 1.26 1.09 — — 0.68

4.491 2.431 1.533 1.18 0.993 0.804 0.711 0.654

4.35 2.42 1.61 — 1.11 0.918 0.807 —

4.319 2.348 1.568 1.264 1.095 0.904 0.796 0.724

4.318 2.37 1.579 — 1.09 0.889 0.772 —

4.255 — 1.532 — 1.104 0.901 0.784 —

4.312 2.367 1.577 1.266 1.09 0.891 0.776 0.712

4.465 2.424 1.604 1.268 1.098 0.892 0.772 0.701

5

14

4

CD

3

10

1

(a)

×

8 ×

6

× ×

4 0

50

100

150 Re

200

× ×

×

2

0

Ranz and Marshall [7] VDI Bagchi [40] Whitaker [8] × Fluent IB

12

Nu

Clift [30] Mittal [38] Haider [31] Shirayama [37] Tabata [36] Schlichting [35] Almedeij [33] Fluent IB

2

250 (b)

0

50

100

150 Re

Figure 4.5 Comparison of the drag coefficients (cD ) (a) and Nusselt numbers (Nu) (b) from the literature and the present study for flow around the sphere at different Reynolds numbers. (Reproduced with permission from [10]. Copyright © 2012 Begell House Publishing.)

200

250

80

4 CD and Nu Closure Relations for Spherical and Nonspherical Particles

codes is 3.5% at Re = 10, and the mean deviation is 0.9%. The maximum error compared with data in the literature is 7% at Re = 10. Numerical results and semiempirical relations are also given in the literature for the surface-averaged Nusselt number. Here again, Richter and Nikrityuk [9] provide a summary based on different references. Contrary to the results for the drag coefficient, the literature data for the surface-averaged Nusselt number are subject to some variation. The data from the literature and those using Fluent and IB solvers are given in Table 4.2 and Figure 4.5. There, the Nusselt numbers based on the two solvers are in a good agreement. The corresponding maximum deviation between the two codes is 1.1%, with the mean deviation amounting to 0.5%. The highest deviations from the literature data were obtained in the case of the predictions by the VDI [41], which, as can be seen in Figure 4.5, seem to overpredict the main trend. On checking these results against other tabulated data, only the main characteristics are comparable. More precisely, the absolute values for the Nusselt numbers vary for all Reynolds numbers analyzed. For instance, the values found by Whitaker [8] are 7% below the average across all sources, and the values from the VDI [41] are 12.8% above. The results found by Kramers [39] are well above the mean of the other data and were neglected. It can be concluded that, although the near-wall resolution of the body-fitted mesh was much higher than that of the Cartesian grid belonging to the IB code, there was good agreement between the Fluent and the IB solver for the drag coefficient and Nusselt numbers, which justifies the use of the IB approach for the given class of problems. Both approaches are suitable to solve the given problem. However, this study has shown that the use of structured grids is connected with lower numerical costs regardless of the grid size. For instance, in the case of the flow past a sphere at Re = 100, the computational time when using the Fluent solver is 11 times higher than taken by the IB solver. For that reason, in the following, the results are based on the IB solver.

Table 4.2 Flow around a sphere at different Reynolds numbers: comparison of Nusselt numbers from different references with results from the present study. Here, only the work by Bagchi et al. [40] contains numerical results. Re

[8]

10 25 50 75 100 150 200 250

3.371 4.233 5.237 6.026 6.703 7.858 8.85 9.735

[39]

[40]

5.148 6.255 7.502 8.459 9.266 10.62 11.761 12.766

— — 5.4 — 6.91 9.12 9.95 —

[7] 3.719 4.719 5.845 6.709 7.437 8.659 9.689 10.597

[41] 3.921 5.055 6.348 7.351 8.205 9.653 10.889 11.99

Fluent Solver

®

IB Solver

3.362 4.37 5.49 6.331 7.035 8.251 9.324 10.014

3.384 4.393 5.514 6.357 7.059 8.26 9.303 10.124

4.4

Porous Particles

4.4 Porous Particles 4.4.1 Geometry Assumptions

In this section, we investigate the heat and fluid flow past and through a porous spherical particle. The porous particles are modeled as a three-dimensional array of differently sized and shaped grains. The complete array is of spherical shape. Thus the porous particle would become a solid sphere if its porosity � is decreased to zero, where the porosity � = Vfluid ∕V�=0 is the void fraction of the porous particle. To be able to vary the porosity over a significant range, constituent grains were chosen with a considerably smaller diameter than that of the porous particle. Additionally, the grain diameter d is not constant, in order to take into account porous particles having the same porosity � but different inner surfaces. Hence, the particle surface ratio �p describes the ratio between a porous particle’s surface and the surface area of a full sphere with the same diameter: �p = Sporous ∕Ssolid . Thus, it could also be called porous sphericity. Two samples of the porous particles under investigation are shown in Figure 4.6. To construct such porous particles, first, the grains are arranged in concentric circles, their diameters being equally distributed. Defining nc as the number of circles used in this first step, the diameters of the circles take the values D − d∕nc , 2(D − d)∕nc , · · · , n(D − d)∕nc . Here, the diameter of the largest circle D − d ensures that the external dimension of the porous particle along any axis is D and therefore independent of the size and the shape of the grains. Additionally, the separation between each pair of adjacent grains on the circular rings is the same. The values of d for the different porosity shapes can be found in Table 4.3. Second, the circular rings are rotated nc times, whereby the grains are distributed in a geometry similar to the lines of longitude. The corresponding total number of grains as well as the accompanying porosity is also given in Table 4.3.

(a)

(b)

Figure 4.6 Zoomed view of a porous particle consisting of (a) small spheres and (b) small cubes.

81

82

4 CD and Nu Closure Relations for Spherical and Nonspherical Particles

Table 4.3 Description of the particle agglomerates investigated. Cases (1–16) correspond to spheres chosen as a basic geometry, and cases (17–32) correspond to cubes as a basic geometry. n is the total number of grains used to generate the porosity, d∕D is the ratio between the grain and particle diameter, � is the void fraction, and the last two values stand for the number of concentric shells the grains are placed on and the surface ratio �p . Case 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

n 219 175 153 132 93 67 43 77 54 35 489 415 347 278 227 159

d∕D 0.127 0.127 0.127 0.127 0.127 0.127 0.127 0.178 0.178 0.178 0.0952 0.0952 0.0952 0.0952 0.0952 0.0952

� 0.603 0.663 0.699 0.747 0.812 0.868 0.912 0.596 0.703 0.804 0.636 0.700 0.731 0.780 0.813 0.870

Layers 4 4 4 3 3 3 3 3 3 3 5 5 5 5 4 4

�p 2.779 2.509 2.246 1.862 1.438 0.9909 0.6925 2.127 1.572 1.105 3.341 2.171 2.562 2.137 1.853 1.307

Case 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

n 219 175 153 132 93 67 43 77 54 35 489 415 347 278 227 159

d∕D 0.102 0.102 0.102 0.102 0.102 0.102 0.102 0.147 0.147 0.147 0.0787 0.0787 0.0787 0.0787 0.0787 0.0787

� 0.605 0.665 0.700 0.748 0.812 0.867 0.912 0.599 0.701 0.804 0.638 0.702 0.731 0.781 0.814 0.870

Layers 4 4 4 3 3 3 3 3 3 3 5 5 5 5 4 4

�p 3.322 3.081 2.752 2.297 1.783 1.221 0.8592 2.598 1.927 1.371 4.123 3.245 3.197 2.625 2.263 1.599

Finally, as constituent components of the array, the grains chosen are spheres and cubes because there are numerous works on the flow past these bodies (see [5, 9, 10]). In order to change the surface ratio �p while leaving the porosity � constant when using grains of different shapes, the diameter of a sphere has to be 1.2407 times the edge length of the corresponding cube. 4.4.2 Heat and Fluid Flow Past Porous Particles

The following presents the results of three-dimensional simulations of the heat and fluid flow past porous particles. The results of the approach presented here were previously compared with a macroscale model approach proposed in Wittig et al. [16]. Figure 4.7 presents the vector and contour plots of the nondimensional temperature Tn =

(T − T∞ ) (Tpart − T∞ )

(4.6)

for different particle porosities and Reynolds numbers, respectively. Here, T∞ is the free-stream temperature, and Tpart is the surface temperature of the particle. It can be seen that, if the particle porosity has a value of about 0.6 corresponding to the Darcy number Da ≈ 10−4 , the gas does not penetrate the particle for almost all the Re numbers under consideration because of the high flow resistance.

4.4

(a)

ε = 0.6, Re = 200

(b)

ε = 0.75, Re = 300

(c)

Porous Particles

83

ε = 0.87, Re = 450

Figure 4.7 (a–c) Distribution of the nondimensional temperature Tn predicted for different particle porosities and Re numbers (x − y plane, z = 0) using a 3D IB solver.

In particular, it is evident that for � = 0.6 and Re = 200 the flow patterns are similar to the heat and fluid flow past a solid sphere predicted by Bagchi et al. [40]. The only difference is that both solvers predict insignificant flow penetration into the porous sphere near the forward stagnation point. The increase in porosity up to � = 0.75 leads to a “flow-through” effect when the cold gas flows through the hot porous sphere (see Figure 4.7b). As a result, the gas inside the sphere is cooled down and, at the same time, more gas is heated behind the sphere. This effect is illustrated in Figure 4.8a, which shows the nondimensional temperature distribution and a zoomed view of flow and temperature inside the particle. These results were predicted using the commercial CFD software Fluent® [42]. The next increase in the porosity, up to � = 0.87, allows more gas to pass through the porous particle. It should be noted that, as the porosity increases, the socalled flow relaminarization occurs. This effect increases the value of the critical Re when the flow becomes unsteady. This finding is in good agreement with the results in [43, 44], in which a similar effect was discovered for a porous cylinder using a macroscale model. However, as shown in [16], at high Reynolds numbers (Re > 300) and � ≥ 0.75, the flow field predicted using a macroscale model differs completely from that obtained with the microscale model. The effect of flow penetration into the particle and its impact on the gas heating could be clearly seen in Figure 4.8b, which depicts the nondimensional temperature distribution and a zoomed view of flow and temperature inside a particle with porosity � = 0.92. The simulations were carried out using the commercial software Fluent® [42]. It can be seen that the flow past this particle has changed, making the hot track behind the particle thinner in comparison with denser particles (e.g., see Figure 4.8a). Returning to the gas moving through the particle, this effect is illustrated more qualitatively in Figure 4.9. There, for two different particle porosities, the gas flow rate moving through the porous particles is shown normalized with respect to the inflow rate V̇ in . As shown in Figure 4.9a, for � = 0.6 and Re ≥ 25, the flow penetrates the particle from the upstream region, while at the same time the recirculating wake enters the particle from behind and causes negative flow rates. A

4 CD and Nu Closure Relations for Spherical and Nonspherical Particles

15

5

Y/D

10 0.95 0.85 0.74 0.64 0.54 0.43 0.33 0.23 0.12 0.02

0

(a)

−5 15

1.00 0.94 0.87 0.81 0.74 0.68 0.61 0.55 0.49 0.42 0.36 0.29 0.23 0.16 0.10

10

5

Y/D

84

0

(b)

−5

Figure 4.8 Distribution of the nondimensional temperature

T−T∞ TS −T∞

predicted for

(a) � = 0.67, Re = 266 and (b) � = 0.92, Re = 427, (x − y plane, z = 0) using the Fluent ware.

® soft-

4.4

Porous Particles

85

0.7 0.15

0.05

Re 10 Re 100 Re 250 Re 300 Re 400 Re 450

0.5 Vparticle/Vin

0.1

Vparticle/Vin

0.6

Re 10 Re 25 Re 50 Re 75 Re 100 Re 150 Re 200 Re 250

0

0.4 0.3 0.2 0.1 0

−0.05

−0.4

−0.2

(a)

0

0.2

−0.4

0.4

Position on x-axis/D

(b)

ε = 0.6, n = 219

−0.2

0

0.2

Position on x-axis/D

ε = 0.87, n = 159

Figure 4.9 (a,b) Dimensionless flow rates through the particle on slices perpendicular to the flow direction. The upstream region and the wake region also penetrate the particle.

totally different behavior was observed for the low Re = 10, where no recirculating wake appears either for high porosities � = 0.87 and Re ≥ 250. In the latter case, the previously described flow-through effect can be seen as the gas moves through the particle and causes the recirculating wake to detach. This is also in good agreement with the findings of [44]. 4.4.3 Drag and Nusselt Numbers for Porous Particles

The drag coefficient of a porous particle can be calculated by cD =

2FD , �u2∞ Aref

(4.7)

where Aref is chosen to be the projected frontal area of the particle and the drag force FD is calculated by integrating the incompressible stress tensor over the particle surface as follows: FD = �

∫S

u⃗ ⋅ n⃗ dS −

∫S

p⃗i ⋅ n⃗ dS.

(4.8)

Here, ⃗i is the unit vector in the x-direction. In particular, because of the constant material properties, the inflow velocity for equal Reynolds numbers is the same as in the previous section. In Figures 4.10 and 4.11, the drag coefficients for porous particles are compared for different Re as a function of the particle porosities. More precisely, while Figure 4.10a shows simply the drag coefficients of a porous particle, on Figure 4.10b this drag coefficient is normalized with respect to that of a solid sphere. While the cD values are almost constant for small porosities � < 0.7, at higher values (� > 0.7) the drag coefficient decreases for low Re and the cD value increases for Re > 100. In Figure 4.11, the specific surface area of the porous particles is varied.

0.4

86

4 CD and Nu Closure Relations for Spherical and Nonspherical Particles 1.4

5 Re 10 Re 25 Re 50 Re 100 Re 200

3

1.3 cD / cD , Sphere

cD

4

Re 10 Re 25 Re 50 Re 100 Re 200

2

1.2 1.1 1

1 0.6

0.65

0.7

0.75

(a)

0.8

0.85

0.9

0.9

0.6

0.65

0.7

(b)

Porosity ε

0.75

0.8

0.85

0.9

Porosity ε

Figure 4.10 (a) Drag coefficients for the flow past a porous particle as a function of the porosity. (b) The drag coefficients are divided by that of a solid sphere. 1.2

1.7 1.6

1.15

cD

1.4 1.3 1.2

cD/cD, Sphere

1.5 Re 50, s Re 50, c Re 150, s Re 150, c

1.1

Re 50, s Re 50, c Re 150, s Re 150, c

1.1 1.05

1

1

0.9 0.8 0.6

(a)

0.65

0.7

0.75 0.8 Porosity ε

0.95 0.6

0.9

0.85

(b)

Figure 4.11 (a) Drag coefficients for the flow past a porous particle as a function of the porosity with different basic geometries. The small letters in the legend indicate

0.65

0.7

0.75 0.8 Porosity ε

0.85

0.9

whether the grains are (c) cubes or (s) spheres. (b) The cD values are again divided by that of a solid sphere.

When the small grains are exchanged such that the particle consists of cubes instead of spheres, the specific surface area �p increases (see Table 4.3). As shown in Figure 4.11, a higher specific surface area generally leads to a higher value of cD . At the same time, the behavior of cD when � and Re are changed to high values does not seem to depend on the grain shape of the porous particle. The final aim was to compare the surface-averaged Nusselt numbers for different Re and �. Additionally, the well-known semi-empirical equations in [7] and [45] are used in this comparison, which take the following form for a nonpermeable sphere: 1

1

Nu = 2 + 0.6 Re 2 Pr 3

(4.9)

and for a porous particle ) ( ) ( ) ( 1 1 1 Nu = 7 − 10 � + 5 �2 ⋅ 1 + 0.7 Re 5 Pr 3 + 1.33 − 2.4 � + 1.2 �2 Re0.7 Pr 3 , (4.10)

4.4

Porous Particles

87

respectively. Here, Gunn’s correlation, Eq. (4.10), is based on a stochastic model of the packed-bed configuration (see [45]). In the frame of the porous particle discussed here, the surface-averaged Nusselt number is derived as follows: Dref ∇T ⋅ n⃗ dS, (4.11) Nu = Sref (TS − T∞ ) ∫S where Sref is the surface of the reference sphere. Further, in [16] the value of Nu has been approximated in accordance with ) ( ) ( 1 (4.12) Nu = 4.31 − 12.71 � + 9.81 �2 ⋅ 1 + 0.8 Re0.6 Pr 3 . The variations of the surface-averaged Nu as a function of the Reynolds number evaluated are depicted in Figures 4.12 and 4.13. In these figures, the correlation obtained by Eq. (4.12) is referred to as the Golia relation. In particular, from Figure 4.12a, it can be seen that the Nusselt number calculated from the Golia relation is in good agreement with the Nu predicted by Eq. (4.11). However, the values of Nu lie below those of Ranz and Marshall. Interestingly, it was found that by multiplying the surface-averaged Nusselt number with the surface enlargement factor �p = Sporous ∕Ssolid , that is Nuequiv =

Sporous Ssolid

⋅ Nu = �p ⋅ Nu

(4.13)

the Nusselt number predictions are in good agreement with the Ranz and Marshall relation (see Figure 4.12b). However, the increase in the porosity value leads the recent prediction to deviate from the Ranz and Marshall relation (see Figure 4.13). Although the agreement between the Ranz and Marshall relation and corrected Nu values using Eq. (4.13) is good for low Re (≤150), there is deviation in the Nu predictions for Re > 150. 25

25 Wittig [16]

Nu

15 10

n = 219, øp = 2.8 n = 219, øp = 3.3 n = 77, øp = 2.1 n = 77, øp = 2.6 Gunn [45] Ranz [7] Golia

20 15 Nu

20

10 5

5 0

50

100

150

200

0

250

ε = 0.6

50

100

150 Re

Re

(a)

Wittig [16] n = 219, øp = 2.8 n = 219, øp = 3.3 n = 77, øp = 2.1 n = 77, øp = 2.6 Gunn [45] Ranz [7] Golia

(b)

ε = 0.6, Nuequiv

Figure 4.12 Surface-averaged Nusselt numbers as a function of the Reynolds number compared with the semiempirical Ranz and Marshall and Gunn relations. (a) With surface correction; (b) without. Nuequiv = Sporous ∕Ssphere ⋅ Nu.

200

250

88

4 CD and Nu Closure Relations for Spherical and Nonspherical Particles

16

Nu

12 10

16 Wittig [16]

14

n = 132, øp = 1.9 n = 132, øp = 2.3 Gunn [45] Ranz [7]

12 Nu

14

10

8

8

6

6

4

4

2

50

100

150

200

2

250

Wittig [16] n = 67, øp = 1.0 n = 67, øp = 1.2 n = 159, øp = 1.3 n = 159, øp = 1.6 Gunn [45] Ranz [7]

50

100

Re

(a)

ε = 0.75

150

200

250

Re

(b)

ε = 0.87

Figure 4.13 (a,b) Surface-averaged Nusselt numbers as a function of the Reynolds number compared with the semiempirical Ranz and Marshall and Gunn relations.

At this Re, the porous particle no longer behaves as a nonpermeable sphere and is penetrated by the surrounding gas flow. It should be noted that the comparison of Nu behavior predicted for � = 0.87 using Gunn’s relation and recent prediction models showed good agreement. To sum up, discussion is needed on the possible influence of the arrangement of particles inside the global sphere and the particle shape on the convective heat transfer between the gas and porous media. It is well known that a real porous medium, for example, a porous coal particle with its open pores inside the solid phase, cannot be described using a regular cluster of equal-sized spheres as used in this work. At the same time, it is a well-known fact that a fully resolved numerical modeling of a real porous medium is not possible because of the complex micropore structure of a real char particle. From these points of view, the use of a simple representation of porosity using equal-sized spherical grains in a regular packing is a compromise between physical accuracy and the ability to achieve computational results within a reasonable time. The analysis of three-dimensional simulations showed that a porous spherical particle with the value of porosity below 0.7 can be represented as an almost solid sphere. Thus, the shape of the pores inside the sphere will not influence the drag coefficient. However, because of the change in the pore shape, the surface enlargement ratio must be taken into account when calculating the Nusselt number (see Eq. (4.13)). 4.5 Nonspherical Particles 4.5.1 Heat and Fluid Flow of Particles Oriented in the Flow Direction

This section is dedicated to nonspherical particles, namely spheroids and cubes. As will be shown later on, the orientation of these particles may influence

4.5

Nonspherical Particles

the fluid flow and heat transfer. For this reason, we intend to, first, examine nonrotating particles that are positioned in the flow direction. Richter and Nikrityuk [9] have studied in detail the heat and fluid flow past ellipsoidal and cubic particles that are oriented in the flow direction and developed closure correlations for cD and Nu as a function of different shape factors. In this section, individual correlations are developed for the drag coefficient and Nusselt number of ellipsoidal and cubic particles oriented in the flow direction. The subsequent sections then focus on different angles of attack. The ellipsoidal particle has an aspect ratio of 2. For details about the physics of flow of such particles without an angle of attack, see Richter and Nikrityuk [9], and for the cube see Wittig et al. [10]. In accordance with the previous sections, all particles have as feature an identical volume (1 mm3 ). The coefficients for the heat and fluid flow are based on the diameter D and on the projected area Aref of the volume-equivalent sphere. The Reynolds number is also based on the diameter Dref of the volume-equivalent sphere using Eq. (4.1). The drag coefficient is defined by Eq. (4.7). For the drag coefficient of a sphere placed in a laminar, homogeneous flow, several correlations between cD and Re are given in the literature. Based on a generic form for the drag coefficient [46] cD =

a b + √ + const, Re Re

(4.14)

individual correlations for spherical, ellipsoidal, and cubic particles are developed in order to quantify the correlation between cD and Re. To do so, a statistical software was applied (for more details, see [47]). The resultant correlations are summarized in Table 4.4. The maximum deviation between the correlations and the numerical data is less than 0.6%, and the average deviation is 0.2%. Figure 4.14 illustrates the validity of the given correlations by comparing them with numerical data. From Table 4.4, it can be seen that the coefficients a and b are similar for all particle shapes. The constant value c, on the other hand, is much higher for the cube compared to the other particle shapes. Depending on the Reynolds

Table 4.4 Regression formulae for the drag coefficient as a function of the particle

shape. Particle shape

Spherical Ellipsoidal Cubic

Correlation 17.9 7.47 + 0.162 +√ Re Re 18.9 6.25 cD,ellipsoid = Re + √ + 0.0316 Re 8.19 √ cD,cube = 20.4 + + 0.216 Re Re

cD,sphere =

89

4 CD and Nu Closure Relations for Spherical and Nonspherical Particles

5

Cube Sphere Ellipsoid

4.5 4

cD

3.5 3 2.5 2 1.5 1 0.5

0

20 40 60 80 100 120 140 160 180 200 Re

Figure 4.14 Drag coefficient for different particle shapes based on new regression formulae (see Table 4.4). The particles are oriented in the direction of flow. Symbols: numerical results; solid lines: regression curves.

number, the drag coefficient of an ellipsoidal particle is reduced by 9.5–27% (Re = 10 − 200), and for a cubic particle it is increased by about 12–14%, compared to a sphere. As shown in Figure 4.15, the situation changes if heat transfer is considered. The spherical particle features the maximum heat transfer, while for the ellipsoid the Nusselt number is decreased by 3–8%, and for the cube by 13–19%. It should be noted that the Nusselt number is an area-averaged value (see Eq. (4.11)). The total heat transfer for both the cube and the ellipsoid is slightly above that for a volume-equivalent sphere, but the total surface area is higher for nonspherical particles. For that reason, the area-averaged Nusselt number is lower than that for the sphere. 10 9 8 7 Nu

90

6 5 4

Cube Sphere Ellipsoid

3 2

0

20

40

60

80 100 120 140 160 180 200 Re

Figure 4.15 Comparison of numerical data (symbols) and new regression curves (solid lines, see Table 4.5) for Nusselt numbers for different particle shapes. The particles are oriented parallel to the direction of flow.

4.5

Nonspherical Particles

According to the correlations given by Ranz and Marshall [7] and Whitaker [8], a generic form was started with 1

Nu = aPr 3

√

1

2

Re + bPr 3 Re 3 + const

(4.15)

and developed to new correlations for the Nusselt number as a function of Re and Pr. The influence of the Prandtl number was not investigated explicitly, but from 1 the literature it is known that the term Pr 3 is a sufficient approximation for the influence of the Prandtl number [7, 8]. The final correlations are listed in Table 4.5. In contrast to the findings for the drag coefficient, the constant part of the Nusselt number is comparable for all particle shapes. The coefficient a also remains comparable for all particle shapes; however, the coefficient b is negative for nonspherical particles. The given correlations reflect the numerical database very well, with a maximum deviation of 0.6% and an average deviation of 0.2%. This is also shown in Figure 4.15 by comparing the data taken from numerical simulations and the given regression curves. 4.5.2 Flow Characteristics of Particles at Different Angles of Attack

The next task is to study the influence of the particle orientation on heat and fluid flow past nonspherical particles. In the literature, the shape of the particle is often specified via its sphericity, but this value provides no information about the orientation of the particle. In the literature (see e.g., [11]), additional characteristics such as the crosswise and lengthwise sphericity have been introduced in order to overcome this problem. When such values are used, additional computational work is required to track the particles because the projected surfaces need to be calculated each time the drag coefficient and other coefficients are evaluated. As an alternative, the particle orientation relative to the flow field can be taken into account directly. It should be noted that the number of degrees of freedom that describe the particle’s orientation in the flow field depends on the particle shape. In this work, the angle � denotes the angle of attack, which is the angle between the local flow vector (x-direction) at the barycenter of the particle and one axis of the particle (see Figure 4.16). Table 4.5

Regression formulae for the Nusselt number.

Particle shape

Spherical Ellipsoidal Cubic

Correlation 1

√

1

2

Re + 0.0104Pr 3 Re 3 + 1.70 √ 1 1 2 Nuellipsoid = 0.640Pr 3 Re − 0.0390Pr 3 Re 3 + 1.59 1 2 1√ Nucube = 0.781Pr 3 Re − 0.115Pr 3 Re 3 + 1.14 Nusphere = 0.568Pr 3

91

92

4 CD and Nu Closure Relations for Spherical and Nonspherical Particles

z y

φ x ψ

u

u

Figure 4.16 Rotation angles used to adjust the different angles of attack. (Reproduced with permission from [10]. Copyright © 2012 Begell House Publishing.)

Thus the rotation using angle � corresponds to a rotation around the y-axis, which is the rotation axis. Because of the symmetrical properties of an ellipsoid, one angle of attack is sufficient to describe all the possible rotations. If a cube is considered, a second angle � becomes relevant; this angle is defined as the rotation around the z-axis. The two angles � and � are independent of each other. The definition of the angles � and � is illustrated in Figure 4.16. Because of the noncommutative rotation, it is important to rotate first through the angle � and afterwards through the angle �. If � is applied first, then the rotation axis for � is rotated as well. To illustrate the effect of a particle’s orientation, for Re = 100, Figure 4.17 shows the heat and fluid flow past spherical and nonspherical particles at different angles of attack using one rotation �. The temperature field is illustrated via the nondimensionalized temperature Tn using Eq. (4.6). From Figure 4.17, it is obvious that the orientation of the particle in the flow field significantly changes the flow characteristics and hence the heat transfer. It is a well-known fact that at Re = 100, a symmetric recirculation domain exists in the wake of the sphere. Similar wake structures are present for an ellipsoid at � = 0◦ and � = 90◦ , but for 0◦ the recirculation domain is much smaller, and for 90◦ it is significantly enhanced. At 45◦ , the flow field exhibits an asymmetry normal to the flow field. This asymmetry results in an asymmetric pressure distribution along the particle and hence to additional lift forces and torques that act on the particle. By contrast, the flow field past a cube is not axially symmetric, and the flow fields at � = 0◦ and � = 90◦ are identical. The length of the recirculation domain is comparable to that for a rigid sphere, and increases slightly if the angle of attack is increased. It should be noted that, because of the cube’s symmetry, the angle of attack needs to be studied only for 0 ≤ � ≤ 45 and 0 ≤ � ≤ 45. If, now, both angles � and � are used to rotate the cube, Figure 4.18 shows three different cases for the angle of attack resulting in different flow regimes for a fixed Reynolds number. Indeed, a Reynolds number of Re = 10 was not small enough to keep the streamlines attached to the particle interface if the cube was perpendicular to the flow direction (see Figure 4.18). This effect changes when the particle is rotated. It can be seen in Figure 4.18 that a rotation through � = 45 increases the characteristic Reynolds number at which the flow detaches from the cube surface. Thus, the streamlines remain attached to the cube surface at Re = 10. This is explained as due to the change in pressure

4.5

Y

Y

Y

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

(a)

ψ = 0°

(b)

ψ = 0°

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

(c)

Y

ψ = 45°

ψ = 0°

ψ = 90°

X

X

Z

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

(f)

(d) Y

X Z

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Y

X Z

(e)

Z

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Y

X

Z

Z

ψ = 15°

Z

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

(g)

93

Y X

X

X Z

Nonspherical Particles

ψ = 30°

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

(h)

ψ = 45°

Figure 4.17 Streamtraces for (a) spherical and (b–h) nonspherical particles at Re = 100. The color of the streamtraces shows the nondimensional

®

temperature Tn . The commercial CFD software Fluent was used.

94

4 CD and Nu Closure Relations for Spherical and Nonspherical Particles Re = 10

Re = 10

x

z Y

Y

φ = 0, ψ = 0

(a)

φ = 0, ψ = 0

(d) Re = 20

Re = 10

z

φ = 0, ψ = 45

(b)

x

Y

z

φ = 0, ψ = 45

(e)

Re = 10

x

Y

Re = 20

z

x

z

Y

(c)

x

z

φ = 45, ψ = 45

x

Y

(f)

φ = 45, ψ = 45

Figure 4.18 Flow pattern of a flow around a cube for (a–c) Re = 10 and for (d–f ) Re = 20. The color of the streamtraces shows the nondimensional temperature Tn . The IB solver was used.

gradients and shear stresses near the particle surface when rotating the particle. As a result, flow separation at the rear point of the particle is not observed. Hence, the geometry with � = 45 allows the streamlines to follow the particle surface. A further increase in this characteristic Reynolds number can be seen when the second angle � = 45 is also used for rotation. In particular, the Reynolds number then has to be increased to a value Re > 20 to force the streamlines to separate from the body and to form a wake. A second characteristic Reynolds number is governed by the loss of flow symmetry with respect to the original symmetry planes. In contrast, many other works take this state as the first characteristic Reynolds number by treating all steady axisymmetric flows as one single regime. Our simulations showed that for an unrotated cube this critical Reynolds number is around Re ≈ 250, which is slightly lower than the findings of Saha [14], which reported Re = 270 as a critical value. In contrast to the finding on the first characteristic Re, here the corresponding value decreases when the cube is rotated to � = 45 and additionally to � = 45. Figure 4.19 reveals that for a single rotated cube � = 0, � = 45 at Re ≥ 100, the flow consists of four streamwise vortical threads of equal strength. The flow remains axisymmetric until Re ≥ 140, where the flow loses its planar symmetry in one direction. When the cube is rotated twice, � = 45, � = 45, there is only one symmetry plane remaining. For this reason, only one pair of vortices appears above Re ≥ 90.

4.5

Re = 100

Nonspherical Particles

95

Re = 140

Re = 90 Figure 4.19 Flow patterns of a flow around two differently rotated cubes (isosurfaces of �2 criteria) with � = 0, � = 45 for Re = 100 and 140 as well as with � = 45, � = 45 for Re = 90. (Reproduced with permission from [10]. Copyright © 2012 Begell House Publishing.)

4.5.3 Influence of Particle Orientation on Drag Forces and Heat Transfer 4.5.3.1 Drag Forces

The relation between the drag coefficient and the angle of attack is highlighted in Figure 4.20 for different particle shapes and different Reynolds numbers. For all particle shapes and Reynolds numbers, the drag increases along with the angle of attack. The influence of � on cD is at most 80% for the ellipsoid at Re = 200. In this process, the main effect is that the pressure forces rise as a result of the increased projected surface area, while the friction forces remain nearly constant. If a cubic particle is rotated, the projected surface area is changed only slightly, which causes a minor increase in the pressure forces. However, the friction forces rise significantly, which is contrast to the ellipsoidal particle. The maximum impact of � on the drag coefficient for the cube is approximately 32%. A statistical analysis of the flow characteristics showed a distinct relation between the drag coefficient and the squared sine function of � for the ellipsoid and between 2 1.8

2 1.9

Re = 50 Re = 100

1.8 1.7 cD

cD

1.6 1.4

Re = 50 Re = 100

1.6 1.5

1.2

1.4 1

1.3

0.8 −100 −80 −60 −40 −20 0 20 40 60 80 100 ψ (degree)

(a)

Ellipsoid

1.2 −50 −40 −30 −20 −10 0 10 ψ (degree)

(b)

Cube

Figure 4.20 Drag coefficient for (a) ellipsoidal and (b) cubic particles as a function of � and Re.

20

30

40

50

96

4 CD and Nu Closure Relations for Spherical and Nonspherical Particles

cD and the squared sine function of 2� for the cube. The qualitative dependence on sin2 (�) or sin2 (2�), respectively, is independent of the Reynolds number, as shown in Figure 4.20. The next step is to quantify the influence of the angle of attack on the drag coefficient and heat transfer by deriving closure relations, which can also be used in submodels for CFD calculations. Since the symmetry properties are different for ellipsoids and cubes, the most promising approach to take into account the angle of attack is to develop individual models for each particle shape. Because of the symmetry of the particles, it is possible to transform the angle of attack in the form ( ) | | (4.16) �̃ cube = | � + 45 mod 90 − 45| | | ( ) | | �̃ ellipsoid = | � + 90 mod 180 − 90| (4.17) | | In order to include �̃ in the closure relations for cD , the generic form for the drag coefficient (Eq. 4.14) is extended by adding a new term f (�), ̃ so cD =

b a + √ + const + fcD (�) ̃ Re Re

(4.18)

Table 4.6 provides the additional term fcD as a function of the transformed angle of attack �. ̃ These terms can be directly added to the correlations that are summarized in Table 4.4. The mean deviation between the numerical results and the regression model for cD is in the range of 0.4% (cube) to 0.6% (ellipsoid). From Table 4.6, it can be seen that the Reynolds number dependence for cD is comparable for all particle shapes, but the leading coefficient is much lower for the cubic particle. This fact reflects the findings discussed above, that the impact of � on cD is more distinct for an ellipsoid than for a cube. In order to demonstrate the validity of the derived models, Figure 4.21 compares the numerical results against those from the derived correlations, which demonstrates the good prediction of cD for ellipsoidal and cubic particles. 4.5.3.2 Heat Transfer

For the Nusselt number, the influence of the angle of attack is lower compared to that for cD , which is illustrated in Figure 4.22. At a Reynolds number of 200, the influence of � ranges from 7% for the cube to 12% for the ellipsoid. Similar to the findings for the drag coefficient, the Nusselt number for the ellipsoid varies as a function of sin2 �. By contrast, the Nusselt number relation for the cube is not straightforward. Studying Table 4.6

Regression terms for cD as a function of �. ̃

Particle shape Ellipsoidal Cubic

̃ fcd (�) 2.21 Re0.303 0.059 Re0.292

sin2 (�) ̃ sin2 (2�) ̃

5.5 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5

ψ = 0° ψ = 0° ψ = 45° ψ = 45° ψ = 90° ψ = 90°

Nonspherical Particles

5

97

ψ = 0° ψ = 0° ψ = 22.5° ψ = 22.5° ψ = 45° ψ = 45°

4.5 4 3.5 cD

cD

4.5

3 2.5 2 1.5 1

0

0.5

20 40 60 80 100 120 140 160 180 200

(a)

0

(b)

Re

20 40 60 80 100 120 140 160 180 200 Re

(a)

7.2 7 6.8 6.6 6.4 6.2 6 5.8 5.6 5.4 5.2 −100 −80 −60 −40 −20

6.2 6 5.8 5.6 Re = 50 Re = 100

Nu

Nu

Figure 4.21 Comparison between predicted (solid lines) and numerically calculated (symbols) drag coefficients at different Re and �. (a) Ellipsoid. (b) Cube.

5.4

Re = 50 Re = 100

5.2 5 4.8

0

4.6 −50 −40 −30 −20 −10 0

20 40 60 80 100

ψ (degree)

(b)

10 20 30

ψ (degree)

Figure 4.22 Nusselt number as a function of �. The Reynolds number equals 50 and 100. (a) Ellipsoid. (b) Cube.

Figure 4.22b, it becomes evident that the Nusselt number drops slightly for � ≤ 15 ◦ , and rises for larger angles of attack. Although � modifies the Nusselt number significantly at Re = 100, the influence of � on Nu is only minor for Re = 50. The relation for the Nusselt number (Eq. 4.15) was extended in a similar way as for cD , so 2 1√ 1 ̃ (4.19) Nu = a Pr 3 Re + bPr 3 Re 3 + const + fNu (�) The resulting term fNu is listed in Table 4.7 and can also be added to the correlations given in Table 4.5. The mean error of the semiempirical correlation is between 0.2% (ellipsoid) and 1.4% (cube). The relation for the heat transfer is not the same as those for cD . In particular, the heat transfer of a cubic particle depends ̃ but with different signs. Figure 4.23 on two terms, namely sin(2�) ̃ and sin2 (2�), demonstrates that Nu can generally be predicted well for nonspherical particles at different angles. 4.5.3.3 Drag Forces and Nusselt Relations for Two Rotations

Compared to the two previous subsections, now both angles � and � are applied on a cubic particle. The resultant influence on the correlation formulae for cD and

40 50

98

4 CD and Nu Closure Relations for Spherical and Nonspherical Particles

Table 4.7 Orientation-dependent terms for Nusselt number. ̃ fNu (�) 1

Ellipsoidal

̃ 0.0153Pr 3 Re0.807 sin2 (�)

Cubic

̃ + 0.0456Pr 3 Re0.48 sin(2�) ̃ −0.22Pr 3 Re−0.0825 sin2 (2�)

1

1

10

9

9

8

8

7

7 ψ = 0° ψ = 0° ψ = 45° ψ = 45° ψ = 90° ψ = 90°

6 5 4 3

0

20

40

60

Nu

Nu

Particle shape

Re

ψ = 0° ψ = 0° ψ = 22.5° ψ = 22.5° ψ = 45° ψ = 45°

5 4 3 2

80 100 140 160 180 200

(a)

6

0

(b)

20

40

60

80 100 140 160 180 200 Re

Figure 4.23 Nusselt number correlations (solid lines) and numerical results (symbols) for different particle shapes ((a) ellipsoid; (b) cube) and Reynolds numbers.

Nu will be taken into account. Beginning with the drag coefficient, again, Eq. (4.18) is extended by a second term for the angle �. A strong dependence on the crossterm �� was observed. Thus the extended Eq. (4.18) takes the following form: cD =

a b + √ + const + f (Re)�̃ + g(Re)�̃ + h(Re)�̃ �̃ Re Re

(4.20)

According to Eq. (4.16), the second angle might also be converted into a domain of definition, with 0 ≤ �̃ �̃ ≤ 45 due to the symmetry of the cube, using the following equations: �̃ = |(� + 45 mod 90) − 45|

and

�̃ = |(� + 45 mod 90) − 45| (4.21)

which is similar to Eq. (4.16). Hence, the values for the angles are converted to a value between −45◦ and 45◦ and also mirrored at zero so that the direction of rotation is irrelevant. A series of different basic functions were tested in order to formulate the dependence. Using Wolfram Mathematica [48], for Eq. (4.20) this leads to the coefficients √ f (Re) = (4.28 ⋅ 10−4 Re − 9.48 ⋅ 10−9 Re2 ), √ g(Re) = (4.67 ⋅ 10−4 Re − 1.03 ⋅ 10−8 Re2 ), (4.22) √ h(Re) = (8.01 ⋅ 10−6 Re + 1.81 ⋅ 10−10 Re2 )

4.5

Nonspherical Particles

99

When the maximum error between the numerical results und regression model was ascertained, a deviation of 3.4% (Re = 150) was found, with a mean error about 0.7% (see Figure 4.24a). Now the relation for cD is compared with the correlation formulae proposed in the literature [11, 31] (see Figure 4.24b). When rotating the cube using both angles, the deviation increases up to a mean value of 12.4% (see the lower part of Figure 4.24). Furthermore, it is very complicated to find the proper values for the crosswise and lengthwise sphericity parameters as discussed in the beginning of this section. Similar to the previous analysis for the drag coefficient, the influence of the two rotation angles on the surface-averaged Nusselt number is now investigated. Again, this is based on extending Eq. (4.19) by an additional term for the second angle �. 1

Nu = aPr 3

√

1

2

Re + bPr 3 Re 3 + const + f (Re)�̃ + g(Re)�̃

(4.23)

In contrast to the findings for cD , the dependence of the Nusselt number on the cross-term �� is negligible. Using Wolfram Mathematica, the coefficients are calculated as 3.16 ⋅ 10−3 + 1.71 ⋅ 10−7 Re2 , √ Re 2.99 ⋅ 10−3 g(Re) = √ + 1.52 ⋅ 10−7 Re2 . Re f (Re) =

(4.24)

The accuracy of this formula has a maximum relative deviation of 1.6% and an averaged deviation between the regression equation and the numerical results of 0.3%. The coefficients (4.24) used in Eq. (4.23) are presented in Figure 4.25. φ = 0, ψ = 0 num φ = 0, ψ = 0 φ = 0, ψ = 45 num φ = 0, ψ = 45 φ = 45, ψ = 45 num φ = 45, ψ = 45

5 4.5

cD

4 3.5 3

5.5 4.5 4

2.5

(a)

3.5 3 2.5

2

2

1.5

1.5

1

num φ = 45, ψ = 45 Hölzer [11] Haider [31]

5

cD

5.5

0

20

40

60

80 Re

1

100 120 140 160

(b)

0

20

40

60

80

100 120 140 160

Re

Figure 4.24 Absolute values for the drag coefficient as a function of the Reynolds number for a cube rotated with � = 45 and � = 45. Comparison of the regression model with the numerical calculations (a) and with the literature data (b).

4 CD and Nu Closure Relations for Spherical and Nonspherical Particles

cD

100

7.5 7 6.5 6 5.5 5 4.5 4 3.5 3 2.5

φ = 0, ψ = 0 num φ = 0, ψ = 0 φ = 0, ψ = 45 num φ = 0, ψ = 45 φ = 45, ψ = 45 num φ = 45, ψ = 45

0

20

40

60

80 Re

100 120 140 160

Figure 4.25 Absolute values for the Nusselt number as a function of the Reynolds number for different angles of attack. Comparison of the regression formula with the numerical results.

4.5.4 Discussion

Finally, it should be noted that this work covers a much smaller range of Reynolds numbers than in the citations above. Moreover, only two particle shapes with different angles of attack were considered. Hence, this is obviously a considerable simplification of the wide range of parameters considered by Hölzer and Sommerfeld [11] or Haider and Levenspiel [31]. However, it demonstrates the possible improvements and the significant influence of the angle of attack (e.g., see Figures 4.24 and 4.25). 4.5.5 Conclusion

In the first part of this chapter two comprehensive validated solvers were compared and validated against each other and against published data for the flow past a sphere. The validation showed a good agreement of the results. In the following the interphase heat transfer between gas and porous particles with different porosities was studied numerically. Specifically, the porous particles are represented by a regular cluster of small grains. Reynolds numbers investigated range from 20 up to 500 and particle porosities between 0.6 and 0.92 were incorporated. The analysis of numerical simulations showed that for low porosities � < 0.7, the fluid flow past the porous particle behaves like that past a solid one. In contrast for high porosities � > 0.85 and high Reynolds numbers we observed that the formation of a recirculating wake behind the particle is prevented. The influence of different particle surface ratios and porosities on the drag coefficiant and Nusselt number is discussed. Further, the predictions for the surface averaged Nusselt number were compared with data from the literature.

References

In the last section, the flow around nonspherical particles namely spheroids and cubes at subcritical Reynolds numbers was examined and discussed. Additionally, we pointed out that there is a significant influence of the angle of attack on drag coefficient and Nusselt number, respectively. The fluid flow and heat transfer past these particles is discussed. Based on the calculated data correlation formulas for the drag coefficient and Nusselt numbers were proposed in order to include the effect of particle orientation in existing models. In summary it can be said that developing regression formulae for more nonspherical particles under different angles of attack and a wide range of Reynolds numbers is far more challenging and still largely untackled. References 1. van der Hoef, M., van Sint Annaland,

2.

3.

4.

5.

6.

7.

8.

M., Deen, N., and Kuipers, J. (2008) Numerical simulation of dense gas-solid fluidized beds: a multiscale modeling strategy. Annual Review of Fluid Mechanics, 40, 47–70. Zhu, H.P., Zhou, Z.Y., Yang, R.Y., and Yu, A.B. (2008) Discrete particle simulation of particulate systems: a review of major applications and findings. Chemical Engineering Science, 63 (23), 5728–5770. Deen, N.G., van Sint Annaland, M., van der Hoef, M.A., and Kuipers, J.A.M. (2007) Review of discrete particle modeling of fluidized beds. Chemical Engineering Science, 62, 28–44. Dennis, S.C.R., Walker, J.D.A., and Hudson, J.D. (1973) Heat transfer from a sphere at low Reynolds numbers. Journal of Fluid Mechanics, 60, 2736–283. Johnson, T.A. and Patel, V.C. (1999) Flow past a sphere up to a Reynolds number of 300. Journal of Fluid Mechanics, 378, 19–70. Dixon, A.G., Taskin, M.E., Nijemeisland, M., and Stitt, E.H. (2011) Systematic mesh development for 3D CFD simulation of fixed beds: single sphere study. Computers and Chemical Engineering, 35 (7), 1171–1185. Ranz, W.E. and Marshall, W.R. Jr. (1952) Evaporation of drops: part II. Chemical Engineering and Processing: Process Intensification, 48, 173–180–. Whitaker, S. (1972) Forced convection heat transfer correlations for flow in pipes, past flat plates, single cylinders, single spheres, and for flow in packed

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beds and tube bundles. AIChE Journal, 18 (2), 361–371. Richter, A. and Nikrityuk, P.A. (2012) Drag forces and heat transfer coefficients for spherical, cuboidal and ellipsoidal particles in cross flow at sub-critical Reynolds numbers. International Journal of Heat and Mass Transfer, 55, 1343–1354. Wittig, K., Richter, A., and Nikrityuk, P.A. (2012) Numerical study of heat and fluid flow past a cubical particle at subcritical Reynolds numbers. Computational Thermal Sciences, 4 (4), 283–296. Hölzer, A. and Sommerfeld, M. (2008) New simple correlation formula for the drag coefficient of non-spherical particles. Powder Technology, 184, 361–365. Raul, R., Bernard, P.S., and Buckley, F.T. Jr. (1990) An application of the vorticityvector potential method to laminar cube flow. International Journal for Numerical Methods in Fluids, 10, 875–888. Saha, A.K. (2004) Three-dimensional numerical simulations of the transition of flow past a cube. Physics of Fluids, 16 (5), 1630–1646. Saha, A.K. (2006) Three-dimensional numerical study of flow and heat transfer from a cube placed in a uniform flow. International Journal of Heat and Fluid Flow, 26, 80–94. Richter, A. and Nikrityuk, P.A. (2013) New correlations for heat and fluid flow past ellipsoidal and cubic particles at different angles of attack. Powder Technology, 249, 463–474.

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(2012) 3d numerical study on the influence of particle porosity on heat and fluid flow. Progress in Computational Fluid Dynamics, 12 (2/3), 207–219. Nandakumar, K. and Masliyah, J.H. (1982) Laminar flow past a permeable sphere. Canadian Journal of Chemical Engineering, 60, 202–211. Feng, Z.-G. and Michaelides, E.E. (1998) Motion of a permeable sphere at finite but small Reynolds numbers. Physics of Fluids, 10, 1375–1383. Wittig, K., Schmidt, R., Schulze, S., and Nikrityuk, P.A. (2013) 3d numerical simulation of a porous particle heating. Proceeding of the 21st Annual Conference of the CFD Society of Canada, May (6–9), Sherbrooke, Quebec, Canada. Wittig, K. and Nikrityuk, P.A. (2012) Laminar heat and fluid flow past a porous particle of different shaped and sized grains. ECCOMAS 2012 Congress, September (10–14), Vienna, Austria. Khadra, Kh., Angot, P., Parneix, S., and Caltagirone, J.P. (2000) Fictitious domain approach for numerical modelling of Navier-Stokes equations. International Journal for Numerical Methods in Fluids, 34 (8), 651–684. Ferziger, J.H. and Peric, M. (2002) Computational Methods for Fluid Dynamics, 3rd edn, Springer-Verlag, Berlin, Heidelberg, New York. Patankar, S.V. (1980) Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corporation. Rhie, C.M. and Chow, W.L. (1983) Numerical study of the turbulent flow past an airfoil with trailing edge separation. AIAA Journal, 21 (11), 1525–1532. Leister, H.J. and Peric, M. (1994) Vectorized strongly implicit solving procedure for seven-diagonal coefficient matrix. International Journal of Numerical Methods for Heat and Fluid Flow, 4, 159–172. ANSYS, Inc. (2012) ANSYS-FLUENT V 14.0 – Commercially available CFD software package based on the Finite Volume method. Southpointe, 275 Technology Drive, Canonsburg, PA 15317, USA, www.ansys.com.

27. Leonard, B.P. (1979) A stable and accu-

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References 40. Bagchi, P., Ha, M.Y., and Balachandar,

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S. (2001) Direct numerical simulation of flow and heat transfer from a sphere in a uniform cross-flow. Journal of Fluids Engineering, 123, 347–358. VDI Gesellschaft Verfahrenstechnik und Chemieingenieurwesen (2006) VDI-Wärmeatlas, 10th edn, VDI-Buch Springer, Berlin Heidelberg. ANSYS, Inc. (2013) ANSYS-FLUENT V 14.0 – Commercially available CFD software package based on the Finite Volume method. Southpointe, 275 Technology Drive, Canonsburg, PA 15317, U.S.A., www.ansys.com. Bhattacharyya, S., Dhinakaran, S., and Khalili, A. (2006) Fluid motion around and through a porous cylinder. Chemical Engineering Science, 61 (13), 4451–4461. Yu, P., Zeng, Y., Lee, T.S., Chen, X.B., and Low, H.T. (2011) Steady flow around

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and through a permeable circular cylinder. Computers & Fluids, 42, 1–12. Gunn, D.J. (1978) Transfer of heat or mass to particles in fixed and fluidised beds. International Journal of Heat and Mass Transfer, 21, 467–476. Yow, H.N., Pitt, M.J., and Salman, A.D. (2005) Drag correlations for particles of regular shape. Advanced Powder Technology, 16 (4), 363–372. StatPoint Technologies, Inc. (2011) STATGRAPHICS Centurion XVI – Commercially available software package for statistical analysis. Warrenton, Virginia, U.S.A., www.statlets.com. Wolfram Research Inc (2012) Mathematica edition: version 8.0, champaign, illinois, www.wolfram.com. Clift, R. Grace, J. and Weber, M.E. (1978) Bubbles, Drops, and Particles, Dover Publications, Inc., Minaola, New York.

™

103

105

5 Single Particle Heating and Drying Robin Schmidt, Kay Wittig, and Petr A. Nikrityuk

In order to improve existing combustors and to develop new combustion techniques, it is necessary to gain an improved understanding of the complex processes that occur in and around particles during combustion.

I.W. Smith [57]

5.1 Nonporous Spherical Particle Heating in a Stream of Hot Air 5.1.1 State of the Art

Heat and fluid flow past a sphere has wide engineering relevance in various applications, such as gasification in an entrained-flow or fluidized-bed reactor and heterogeneous catalysis in chemical reactors. In particular, considering a gasifier where cold particles are injected into the hot reactor, the particles are heated before drying or pyrolysis starts. Early computational work on heat and fluid flow around a sphere at low to moderate Re numbers was carried out by Dennis et al. [1]. Later, Johnson and Patel [2] carried out both numerical and experimental investigations into fluid flow past a sphere for Re ≤ 300. Different flow regimes were described as a function of Re. Recently, Dixon et al. [3] provided a systematic numerical study of heat and fluid flow past a single sphere up to Re > 20 000. In particular, it was shown numerically that the well-known Ranz and Marshall [4] and Whitaker [5] relations are in acceptable agreement with numerical predictions. All the studies reviewed so far, however, have been focused on isothermal cases or cases with a constant temperature on the particle surface, corresponding to the so-called external problem. In the external problem (�g ≪ �p ), the heat transfer to the particle is limited by the liquid phase around the particle. The alternative Gasification Processes: Modeling and Simulation, First Edition. Edited by Petr A. Nikrityuk and Bernd Meyer. © 2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA.

106

5 Single Particle Heating and Drying

(�g ≫ �p ) is referred to as an internal problem, where the heating of the particle is determined by the conductivity of the solid [6, 7]. One of the first theoretical studies devoted to conjugate heat transfer from a rigid spherical particle was carried out for the creeping flow regime (see the works of Abramzon and Borde [8] or Abramzon and Elata [9]). The conjugate heat transfer problem for moderate Re numbers was solved by Nguyen et al. [6]. In particular, the thermal conductivity ratio was varied between 1∕3 and 3.0, and the volume heat capacity ratio was set at 1 and 0.001. Recently, Juncu [7, 10, 11] carried out numerical investigations, which have been published in a series of papers, for conjugate heat transfer from a circular cylinder and a spherical particle in a laminar cross-flow at low Reynolds numbers. In particular, transport equations in the gas and solid media were solved separately and coupled using boundary conditions on the interface. It was shown that two parameters, namely the volume heat capacity ratio (Henry number, He = �p cp,p ∕�g cp,g ) and the thermal conductivity ratio, play a governing role in accurate prediction of the heat transfer. A proposed equation for estimating the conjugated heat transfer was given by Nguyen et al. [6]: ( Nu =

� 1 + Nuint Nuext

)−1 (5.1)

where � is the ratio of the thermal conductivities �g ∕�p , Nuint is the Nu number of the corresponding internal problem (� ≫ 1), and Nuext is the Nu number of the ’external (� ≪ 1) heat transfer problem. The model given by Eq. (5.1) estimates the Nu number of the conjugated heat transfer problem by adding the internal and external heat transfer resistance. Finally, it should be noted that most studies in the field of conjugate heat transfer for a sphere or cylinder placed in a gas stream have only focused on small Re number flows. So far, however, there has been little discussion about the transient dynamics of particle heating for Re with the order of magnitude (O2 ), though some effort has been made to develop a Nu correlation which includes the contribution of internal heat transfer to the overall heat transfer. Motivated by this fact, the aim of this section is to evaluate the transient heating of a carbon sphere for the Re number range 1–200. For this purpose, a computational fluid dynamics (CFD)based model is used that solves the conservation equations in the gas and the solid phase directly using a fixed Cartesian grid. At this point, it should be noted that the particle volume-averaged temperature T p is used for the calculation of the Nu number instead of the surface temperature T p , which is generally used in heat and mass transfer analysis [12]. The main reason for choosing T p lies in its use in subgrid models, where it is often the principal variable in the energy balance equation describing the heat transfer between the dispersed and gaseous phases (e.g., see [13, 14]).

5.1

Nonporous Spherical Particle Heating in a Stream of Hot Air

5.1.2 Problem and Model Formulation

The system under investigation consists of a single sphere which is placed in a stationary position, with the hot gas passing around it. The principal scheme of the domain is shown in Figure 5.1. The diameter of the particle Dp is 0.04 m, and its initial temperature T0 is 300 K. The inlet temperature of the air around the particle and at the inflow into the system is set at Tin = 400 K. The spherical particle is placed on the symmetry axis at 2∕7 L1 in cylindrical coordinates, where L1 = 70 Dp and L2 = 20 Dp . The inflow velocity, uin , is assumed to be uniform and is determined by means of Reynolds numbers calculated as follows: Rein =

uin Dp �g �g

,

(5.2)

where �g is the dynamic viscosity, and �g is the density of the gas. The properties of the solid phase corresponds to carbon, but in order to investigate the influence of the thermal conductivity �p on the overall heat transfer, �p is varied from 0.0242 to 0.34 W (m K)−1 . However, in the present section, the results for � = 0.17 W (m K)−1 are presented. The analysis of simulations carried out for different � values can be found in Schmidt [15]. To proceed with the governing equations, the following basic assumptions are made:

• • • •

The gas flow is treated as an incompressible medium. The viscous heating effect is neglected. The thermophysical properties are constant, giving a Prandtl number of 0.749. The buoyancy effect is neglected.

With the above assumptions, the conservation equations for mass, momentum, and energy transport written for the gas phase take the following form: ∇ ⋅ u⃗ = 0 (5.3) ( ) ∂⃗u 2 � + � u⃗ ⋅ ∇ u⃗ = −∇p + �∇ u⃗ + FIBM (5.4) ∂t ( ) ∂T �cp + �cp u⃗ ⋅ ∇ T = ∇ ⋅ (�∇T) (5.5) ∂t An inflow boundary condition is set at the left domain boundary (see Figure 5.1). The right side is treated as an outlet boundary and a symmetry boundary condition is set at the top and bottom. At the particle surface, the no-slip boundary condition is enforced using the continuous forcing immersed boundary (IB) method [16, 17]. The heat transfer in the gas and in the particles is calculated directly using Eq. (5.4), where �, cp , and � depend on the value of the particle marker � as follows: { �g , � = 0, gas �= (5.6) �p , � = 1, solid

107

5 Single Particle Heating and Drying

L1 Tin

Outlet 

uin Inlet 

108

r

T0

Dp/2

L2

Symmetry axis

z Figure 5.1 Scheme of the setup: single sphere in a 2D axisymmetric domain. Table 5.1

Physical properties used for the gas and the solid phase.

� kg m3 cp J (kg K) � W (m K) � Pa s � , m2 s−1 �c p

Air

Carbon

Carbon/air ratio

1.225 1006.43 0.0242 1.8 × 10−5 1.96 × 10−5

2000.0 1200.0 0.00807 – 0.17 – 7.1 × 10−8

1633 1.2 0.33 – 7 – 3.6 × 10−3

where � refers to �, cp , and �. Their values for gas and coal are given in Table 5.1. For the calculation of the heat fluxes between the control volumes (CVs), the harmonic averaging procedure is used; for details see [18]. Finally, it should be noted that for the heating problem no specified thermal boundary condition is used at the solid–gas interface for the heat transfer equation (Eq. (5.4)). 5.1.2.1 Linear Model

The model described in the previous section resolves the temperature profile inside the particle. In the following, this modeling is referred to as direct numerical simulation (DNS) because of the direct calculation of the temperature inside the solid particle. In order to investigate the influence of the heat transfer inside the particles on the dynamics of particle heating, we introduce a so-called linear model, where we assume a spatially homogeneous distribution of the temperature inside the particle. In this case, the energy conservation equations for the gas phase is ∂Tg

( ) (5.7) + �g cp,g u⃗ ⋅ ∇ Tg = �g ∇2 Tg . ∂t In the solid phase, the convective energy transport can be neglected (u⃗s = 0), and the resultant energy conservation equation is �g cp, g

∂Tp

= �p ∇2 Tp ∂t with the following boundary conditions at the interface: �p cp,p

�p

∂Tp ∂n

= �g

∂Tg ∂n

;

T p = Tg

(5.8)

(5.9)

5.1

Nonporous Spherical Particle Heating in a Stream of Hot Air

After the volume integration of Eq. (5.8) and application of Gauss’s theorem, this equation can be reformulated as follows: ∫Vp

�p cp,p

mp cp, p

∂Tp ∂t

∂T p ∂t

= �p

= �p

∮

∫Vp

∇2 Tp dVp

∂Tp ∂n

dSp

(5.10)

(5.11)

where mp is the particle mass and T p is the particle-averaged temperature. Inserting the boundary conditions given by Eq. (5.9) into Eq. (5.11) produces the final equation mp cp,p

∂T p ∂t

= �g

∮

∂Tg ∂n

dSp

(5.12)

where the integral on the right-hand side of the equation defines the total heat flux between the particle and the gas phase. Because of this reformulation, the boundary condition for Eq. (5.9) on the particle surface becomes a Dirichlet boundary condition using T p calculated from Eq. (5.12). The Dirichlet boundary condition is enforced using the implicit fictitious boundary method (for details, see [15]). 5.1.3 Illustration of Results and Subgrid Model

Before we explore the results of simulations, some important remarks should be made in respect of the grid resolution we used in the simulations. In particular, the grid contained 260 × 85 CVs in z- and r-direction. Inside the particle, the grid was refined to 50 CV∕Dp . This grid resolution is verified during the test runs carried out for the validation [15]. The time step is set at 5 × 10−2 s and the modeled physical time is 5 × 103 s. When analyzing the particle heating, the results are unified using the following dimensionless temperature: �=

T − T0 Tin − T0

(5.13)

The next parameter used is the Fourier (Fo) number, which defines the dimensionless time: ap t �p t Fo = 2 = (5.14) D cp,p �p D2 with ap as the thermal diffusivity of the solid phase. The nondimensional particle volume-averaged temperature � p is used to study the dynamics of the particle heating process, that is r

�p =

0 2π Vp ∫0 ∫0

z0

� � r dr dz

(5.15)

109

5 Single Particle Heating and Drying

with the volume of the sphere Vp . The heat transfer is described by the Nu number calculated according to the following equation: Nu =

Dp

Sp ΔT ∮

(5.16)

∇T d S.

At this point, it should be repeated that in this work the particle volume-averaged temperature � p is used for the calculation of the Nu number instead of the surface temperature � surf . Next we present the results of numerical investigations for the case of particles with good thermal conductivity (�p ∕�g > 1). The results for particles with low thermal conductivity (�p ∕�g < 1) can be found in Schmidt [15]. Considering a single moving spherical particle made of carbon as an object of investigation, we use the ratio �p ∕�g = 7. The time history of the nondimensional particle volume-averaged temperature � p is shown in Figure 5.2. It can be seen that the particle heating is faster as the Re number increases, which is logical. The linear model with homogeneous temperature distribution inside the particle (dotted line) predicts faster heating than the corresponding DNS simulations. The analysis of Figure 5.2 shows that the sphere is completely heated up at Fo ≈ 1 for Rein ≥ 50. It can be seen that the entire heating of the particle can be divided into three regimes. The first regime is characterized by high heating rates of the particle governed by the maximum temperature difference between the surface and ambient gas due to the initialization setup. This phase is governed by the formation of the flow and temperature boundary layer around and inside the particle. The duration of this regime is in the order of Fo ≈ 10−2 (e.g., see [12]). An analysis of the simulation results shows that during this regime the difference between the initial particle temperature T0 and the minimum temperature inside the particle is < 1%. See Table 5.2, which shows the time when the minimum temperature reaches � = 0.01 and the corresponding mean particle temperature � p for a spherical particle. 1.0

Re = 100

0.8

Re = 50 Re = 10

0.6

Re = 1

θp

110

0.4

Linear DNS

0.2 0.0 0.0

0.4

0.8

1.2

1.6

2.0

Fo Figure 5.2 Time history of � p predicted using DNS and a linear model for different Re values (�p ∕�g = 7).

5.1

Nonporous Spherical Particle Heating in a Stream of Hot Air

Table 5.2 Nondimensional start time of the intrinsic heating phase Fomin and the corresponding nondimensional mean particle temperature � p for a sphere at different Rein numbers (�p ∕�g = 7) Rein

Fomin

�p

1 10 50 100

0.0248 0.0204 0.0204 0.0195

0.0487 0.0617 0.0862 0.1039

The third and last regime corresponds to the complete heating of the solid. For a spherical particle, this regime is achieved at Fo ≈ 2 for 1 ≤ Rein ≤ 10 and at Fo ≈ 1 for Rein ≥ 50 (see Figure 5.2). Finally, the second regime can be defined as intrinsic heating, where the temperature difference between the ambient gas and the minimum temperature inside the particle decreases. To illustrate the intrinsic heating regime, Figure 5.3 shows snapshots of the nondimensional temperature �. It can be seen that the temperature gradient at the front of the particle is higher than at the rear. As a result, the temperature distribution inside the particle is asymmetric. Increasing Re leads to an enhancement of this effect. Because of the intensive heating at the front, the temperature isolines are half-moon-like in this area of the sphere. A cold core forms in the rear part of the particle as a result of the heating from behind. In order to explore the influence of the so-called asymmetry effect caused by the gas flow, in Figure 5.4 we compare the axial nondimensional temperature profiles inside the sphere predicted numerically and analytically. The analytic model [19] is introduced in Section 5.2.4. It can be seen that at small Re numbers, for example, Rein = 1, the temperature profile is in good agreement with the analytic solution, which assumes spherical symmetry. However, at Rein = 100, a deviation between the numerical and the analytical results can be detected. At the same time, the figure shows that, during 0.95

0.85

0.75 0.95 0.85 0.15 (a)

(b)

Figure 5.3 Snapshots of the temperature inside and around the graphite sphere for (a) Rein = 1 and (b) Rein = 100 at Fo = 0.05 (Dp = 0.04 m, �p ∕�g = 7).

111

112

5 Single Particle Heating and Drying

0.8

DNS Analytic

0.8 Fo = 0.2

Fo = 0.4

Fo = 0.2

0.2 0.0

DNS Analytic

0.6

0.4

θ

θ

0.6

Fo = 0.1

0.4 0.2

Fo = 0.01

19.75

19.50

(a)

20.00

20.25

0.0

20.50

Fo = 0.01

19.50

19.75

(b)

z/D

20.00 z/D

20.25

20.50

Figure 5.4 Comparison of the axial nondimensional temperature � profiles near and inside the sphere predicted with DNS and analytically for different times at (a) Rein = 1 and (b) 100 (�p ∕�g = 7).

the first regime (Fo < 0.01), the asymmetry is not fully developed. However, it is interesting to observe that the temperatures predicted using the two models are identical at the center of the particle (r∕Dp = 20). The time histories of the nondimensional particle-averaged temperature predicted using the DNS and analytic model are close to each other (see Figure 5.5). Figure 5.6 depicts the time history of the Nu number predicted numerically using Eq. (5.16). An analysis of this figure reveals that, after the flow starts, the Nu number approaches a constant value within Fo < 0.1. This convergence of the Nu numbers is explained by the buildup of the temperature field inside the particle, which is defined by the thermal diffusivity of the solid phase. Because of the fact that, during the particle heating, the Nu number approaches a constant value after the first phase, a Nu number relation can be used to calculate the time history of the particle-averaged temperature. 1.0 Re = 100

0.8

Re = 10

0.6 θp

Re = 1

0.4

Analytic DNS

0.2 0.0 0.00

0.25

0.50 Fo

0.75

1.00

Figure 5.5 Comparison of the nondimensional mean particle temperature � p inside the sphere predicted with DNS and analytically for different times at (a) Rein = 1 and (b) 100 (�p ∕�g = 7).

5.1

Nonporous Spherical Particle Heating in a Stream of Hot Air

8.0 Re = 100

6.0

Nu

Re = 50

4.0 Re = 10 Re = 1

2.0

0.0 0.00

0.01

0.02

0.03

0.04

0.05

Fo Figure 5.6 Time history of the Nu numbers calculated for a sphere at different Re numbers using the DNS model (�p ∕�g = 7).

Following Sanitjai and Goldstein [20], the Ranz–Marshall model [4] is a standard model for the calculation of the Nu number as a function of Re and Pr, given by Nu = 2.0 + 0.6 Re1∕2 Pr1∕3 . (5.17) Recently, Dixon et al. [3] showed numerically that the Ranz–Marshall relation produces results close to those of the particle-resolved numerical simulations for Re ≤ 20 000. However, the Ranz–Marshall equation describes the external heat transfer, which is defined by �p ≫ �g and modeled using the linear model. But to predict the conjugated heat transfer, the internal heat transfer resistance must be considered, too. This can be done by modifying the Ranz–Marshall equation. In particular, based on the presented numerical simulations, a valid Nu number correction is obtained by adding the term (1∕1 + �) to the Ranz-Marshall equation. The variable � is given by �g Bi = (5.18) �= Nu �p This modification guarantees that, as the thermal diffusion of the solid increases, the modified equation blends into the original equation. Finally, the extended Ranz–Marshall relation for the sphere takes the form ) ( 1 , Re ≥ 1 (5.19) Nu = 1.7 + 0.6 Re1∕2 Pr1∕3 1+�

It should be noted that Eq. (5.19) is valid for Re ≥ 1. However, when Re approaches zero, the Nu number must approach 2. Following the theoretical work by Juncu [7], the combination of the Ranz– Marshall equation with the classical model (Eq. (5.1)) provides a good estimation of the Nu number when Nuint is set to 6.58. Finally, the closure relation for the Nu number given by Eq. (5.19) is tested for the prediction of the mean temperature of a particle Tp heated in a hot stream of

113

5 Single Particle Heating and Drying

air using a 0D energy conservation equation written in the form ∂Tp

Nu �g

Sp (Tin − Tp ) (5.20) ∂t D It should be noted that the use of Tp in Eq.(5.20) assumes that the energy is homogeneously distributed inside the particle. Figure 5.7 shows the time histories of the nondimensional mean particle temperature for a sphere at Rein = 100 predicted using the CFD-based model represented by Eqs (5.3)–(5.4) and the socalled subgrid relation defined by Eqs (5.20) and (5.19). Close agreement can be seen between the two models. In particular, the maximum deviation between the two models is 0.26% at Fo = 0.021. mp cp,p

=

5.1.4 Semiempirical Two-Temperature Subgrid Model

Next, we present a two-temperature subgrid model which is the basic model for transient heat transfer modeling used for prediction of drying (Chapter 5.3) and can be used for pyrolysis and intrinsic char conversion (Chapter 10). The heating subgrid model developed solves temperature equations for the particle surface temperature Tsurf and for the inner particle temperature Tp . The scheme of the model is shown in Figure 5.8. The energy conservation equation for the particle takes following form: dTp

= �p, surf (Tsurf − Tp ) (5.21) dt where �p, surf is the heat transfer coefficient between the particle surface and the inner particle, which is calculated as follows [21]: 4π�p (5.22) �p, surf = 1 − R1 R mp cp, p

T

p

1.0 0.8 0.6 θp

114

0.4

DNS Subgrid

0.2 0.0 0.0

0.1

0.2

0.3

0.4 Fo

0.5

0.6

0.7

0.8

Figure 5.7 Comparison of the mean particle temperature between the subgrid (onetemperature model, no radiation) and the DNS model at Rein = 100 (�p ∕�g = 7).

5.1

Nonporous Spherical Particle Heating in a Stream of Hot Air

Rp

T∞

RT

Uin

Tsurf Tp

Figure 5.8 Scheme of the two-temperature model for particle heating.

The radius RT is the position of Tp inside the particle, that is (5.23)

RT = fT Rp

where fT is a blending factor. The particle surface temperature is calculated based on the energy balance equation 4 4 −�p, surf (Tsurf − Tp ) + �surf, gas Sp (T∞ − Tsurf ) + �� �Sp (T∞ − Tsurf ) = 0 (5.24)

where the third term on the left side describes the radiative heat flux, with �� as the emissivity and � as the Stefan–Boltzmann constant. Using the radiative 4 − T 4 ), the surface heat flux from the last time step or iteration Q̇ rad = �� �Sp (T∞ surf temperature is calculated as follows: Tsurf =

�p, surf Tp + �surf, gas Sp T∞ + Q̇ rad �p, surf + �surf, gas Sp

(5.25)

The heat transfer coefficient to the gas phase �surf, gas is calculated based on the Nu number, as �surf, gas =

Nu �g Dp

(5.26)

The Nu number is calculated in line with Eq. (5.19). Next, the results of the two-temperature subgrid model are compared with the results of the CFD model. In this validation case, the particle is spherical, the radiative heat exchange is neglected (�� = 0), and the ratio �p ∕�g equals 7. The other transport properties are given in Table 5.1. The blending factor fT is set at 0.51∕3 . To be specific, with fT = 0.51∕3 , the volume of the inner particle (R < RT ) and the volume of the outer layer (RT < R < Rp ) are equal. It should be noted that numerous simulations have shown this value to be a good choice as long as �p > �g . Figure 5.9 depicts the time history of the nondimensional mean particle temperature (�p ) and the nondimensional mean surface temperature (�surf ) obtained using the CFD and subgrid models. The surface temperature in the case of the DNS model (� surf ) is obtained as the spatial average of the temperature of the solid CVs at the particle interface. Radiation was neglected in this validation case.

115

116

0.8

1.0 Re = 100 Re = 50

0.8 Re = 10 Re = 1

θp

0.6

θsurf

1.0

5 Single Particle Heating and Drying

0.4

Re = 10 Re = 1

0.6 0.4

Subgrid DNS

0.2 0.0 0.0

Re = 100 Re = 50

Subgrid DNS

0.2 0.3

0.6

(a)

0.9

1.2

0.0 0.0

1.5

0.3

0.6

(b)

Fo

0.9

1.2

1.5

Fo

Figure 5.9 Time history of (a) the nondithe DNS model (� p & � surf ) and the twomensional particle temperature �p and (b) temperature subgrid model (�p ∕�g = 7). The the nondimensional surface temperature �surf radiative heat exchange was neglected. calculated for different Re numbers using

1000 900

T (K)

800 700

Subgrid Tp Subgrid Tsurf

600

DNS Tp DNS Tsurf

500 400 300

0

1

2

3

4

5

t (s) Figure 5.10 Time history of the particle (p) temperature and the surface (surf) temperature calculated using the DNS model and the two-temperature subgrid model. The radiative heat exchange was taken into account.

The results of a validation case, which includes radiation, are presented in Figure 5.10. It can be seen that the two-temperature model predicts the temperature time histories; only minor deviations are observed. In this validation test case, the ambient temperature was set to 1000 K, the gas atmosphere is nitrogen, and the particle is initialized at T0 = 303.15 K. The slip velocity of the coal particle (Dp = 2 mm, �p = 0.1277 W (m K)−1 ) is set at 0.724 m s−1 , which corresponds to Rein = 100. For further details on DNS including a description of implementation of the radiation term, the reader is referred to Schmidt [15]. 5.2 Heating of a Porous Particle

Heating phenomena related to porous particles are of great importance in energy process engineering and chemical engineering. In comparison to the heating

5.2

Heating of a Porous Particle

of nonporous particles, where analytical solution (see Section 5.2.4) describes the heating dynamics close to particle-resolved CFD, the impact of particle porosity on the transient heating is not well understood. Therefore, in this section the transient heating of a porous particle in a stream of hot air is investigated numerically. The distinguishing feature of this work is characterized by using foam-like structures, which are utilized to model the porosity of a spherical particle directly [22]. The main aim of this section is to explain the influence of the pore geometry on the transient heat transfer between a moving porous particle and the gas phase. Hence, parameter studies for different pore sizes and porosities were performed. Finally, the numerical results are compared with those of an analytical model [19] using different expressions to predict the thermal conductivity of porous media [23]. 5.2.1 Problem and Model Formulation

It is assumed that the porous particle consists of pure carbon and is placed in a uniform cross-flow of hot air. Further, it is assumed that no chemical reactions occur. Figure 5.11 shows a schematic view of the computational domain including boundary conditions used in simulations. The inflow velocity u∞ together with the diameter D of the porous particle D = 2 mm defines the Reynolds number characterizing the flow field, that is

Re = �u∞

D �

(5.27) Outflow

Symmetry z y

x u

H L Inflow Projected particle B Figure 5.11 Computational domain scheme with boundary conditions. The dimensions L, B, and H depend on Re to resolve the boundary layer properly, see Table 5.3. The particle is located in the middle of the computational domain 10D up to 75D short of the inflow.

117

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5 Single Particle Heating and Drying

Table 5.3 Sizes of the computational domain for different Re. The grid resolution in each case is about 6 × 106 cells. Range of Re Re < 5 5 ≤ Re < 20 20 ≤ Re < 100 Re ≥ 100

L

B

H

150D 100D 60D 40D

150D 80D 40D 20D

150D 80D 40D 20D

The particle Reynolds number Re is varied between 1 and 500 in this study. All hydrothermal properties of the fluid are treated to be constant and correspond to air at 400 K. This is also the inflow temperature and the initial temperature of the surrounding gas, where the initial temperature of the particle surface takes 300 K. The corresponding Prandtl number is Pr = 0.7. Details on domain size for different Re values are given in Table 5.3. To solve the problem, the Navier–Stokes equations written in incompressible form are solved numerically coupled with the heat transfer equation. No-slip boundary conditions on the particle surface were set using an IB method in the continuous forcing mode [24, 25]. The viscous heating and buoyancy effects are neglected. Finally, the governing equations for mass, Momentum, and energy conservation have the following form: ∇ ⋅ u⃗ = 0, ∂⃗u + �⃗u ⋅ ∇⃗u = −∇p + �Δ⃗u + F⃗IBM , � ∂t ∂T �cp + �cp u⃗ ⋅ ∇T = ∇ (k∇T) ∂t

(5.28) (5.29) (5.30)

More details on the treatment of the IB continuous force represented by the term F⃗IBM including interface reconstruction can be found in [26–28]. The 3D CFD solver was validated against experimental data and benchmark results [26, 28]. Figure 5.12 depicts a zoomed view of the computational grid inside the porous particle (120 cells along the diameter of the particle). The entire grid has 6 × 106 CV cells. 5.2.2 Porosity

The porosity 𝜀 of the particles included in the examination varies between 0.7 and 0.8. It is modeled using a microscale representation of the pores, where the open pore geometry is fully resolved. Specifically, the pores are modeled as an arbitrary cluster of monodisperse solid spheres taken from a packed bed, which had been previously calculated using a discrete element method (DEM) applied to the gravity-driven sedimentation of rigid spheres in a cylindrical cavity [22, 29].

5.2

(a)

Heating of a Porous Particle

(b)

Figure 5.12 Zoomed view of the computational grid near the particle. The resolution in this part of the mesh is the same for all Re. (a) Shows a 2D slice through the particle, while (b) shows how the particle is resolved on this slice.

The spatial distribution of solid spheres is used to represent the pores (void fraction) inside the porous particle, leading to a foam-like structure. It should be noted that, when varying the pore size, the porosity itself does not automatically have to change. Since the pores are monodisperse spheres, they can be described using their diameter d = D∕c, where c is the amount of aligned pores whose diameters add up to D. Once the pores are positioned inside the particle, their diameter d is increased by a small factor 1 < f ≪ 2, which makes the pores intersecting and allows adjusting the porosity 𝜀. Though both the parameters c and f affect the particle porosity, only c has a major influence on the overall pore surface, which increases with an increase of c. 5.2.3 Results of Simulations

The results of the simulations regarding the comparison of the time history of ∗ the particle-averaged temperature T predicted using 3D numerical simulations ∗ and the analytic model are presented in Figure 5.13. There, the values of T are obtained from ∗

T =

T − T0 1 dV Vp ∫Vp T∞ − T0

(5.31)

and Eq. (5.32), respectively. Here, Vp is the volume occupied by a sphere with diameter D. The analysis of the results of the calculations for Re < 100 reveals that the values ∗ of T predicted by the numerical computations are close to those of the analytical model.

119

120

5 Single Particle Heating and Drying

In particular, at very low values of the particle Reynolds number (Re ≈ 1), the particle-averaged temperature calculated numerically corresponds to the predictions obtained from the analytical model using ksequ . If then Re is increased, deviations between numerical results and analytical model using ksequ regarding the ∗

time history of the particle-averaged temperature T are observed. This effect is explained by the flow penetration into the pores of the particle, which can be seen in Figure 5.13. There, the particle-averaged temperature including the pores is always higher than just the mean temperature of the solid phase. Further, for increased Re, the heating time of a porous particle is reduced. However, the analytic model in conjunction with the heat conductivities kgeom and ksequ is able to predict such reduced heating times as well. Especially, for Reynolds numbers around Re = 500, the heating of the porous particle is well described by the analytical model using kpar . When the Reynolds number increases further (Re > 500), the analytic model deviates from the numerical simulations. To overcome this problem, the analytic model has to be improved to recognize the flow within the pores. Figure 5.14 shows temperature contour plots on the surface of porous particles, where the particles have been cut at y = 0. Although the particles are heated by the surrounding gas, even at Re = 500 the gas moving through the particle is insufficient to obtain a linear heating profile from the upstream to the downstream side of the particle. Instead, Figure 5.14 shows that up to Re = 500, generally, there is a higher temperature at the particle surface. When looking at the influence of the pore size on the particle-averaged temperature, Figure 5.15 depicts that this parameter has hardly any effect on the particle heating. At 𝜀 = 0.7, the time evolution of particle-averaged temperature is nearly the same for all pore sizes. Additionally, in Figure 5.14 the upper two particles feature a similar heating profile, by having the same porosity and a different pore 1

1 0.8

0.8

Re = 20

0.6 T*

T*

0.6

0.4

0.4 Solid Solid and pores Analytic (par) Analytic (sequ)

0.2 0 0.01 (a)

Re = 500

0.1

0 0.01

1 Fo

Solid Solid and pores Analytic (par) Analytic (sequ)

0.2

(b)

0.1

1 Fo ∗

Figure 5.13 (a,b) Nondimensional time history of the particle-averaged temperature T . Here, porosity is constant 𝜀 = 0.74. “Solid” corresponds to the solid phase of the particle, and the pores are ignored.

5.2

Heating of a Porous Particle Z

Z Y

Y

X

X

T 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

T 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

Figure 5.14 Cut through a porous particle with � = 0.74 (large pores dlarge ≈ D∕6) at Re = 100 (left), and Re = 500 (right), while Fo ≈ 0.2. The contour coloring on the surface of the solid phase is the dimensionless temperature.

1 0.8

ε = 0.8

T*

0.6 0.4 Large pores Medium pores Small pores Analytic (par) Analytic (sequ)

0.2 0 0.01

0.1

1 Fo

Figure 5.15 Influence of the pore size on the time history of the particle-averaged tem∗ perature T for Re = 100. For the three cases large, medium, and small, the pore sizes take dlarge ≈ D∕6, dmedium ≈ D∕8, and dsmall ≈ D∕10. ∗

structure. However, for � = 0.8, the dependence T from Fo has a greater slope if the pore diameter has a higher value. Because of the higher internal surface of a particle, the Nusselt number normally takes a smaller value. By applying the surface correction by Wittig et al. [25], the Nu values for different pore sizes become normalized with respect to the surface of a spherical particle and thus comparable. Additionally, there is a minor influence of the porosity on the drag coefficient because of the small flow penetration into the pores of the particle (see Figure 5.16). There, the dimensionless flow rate through porous particles is shown when the surrounding flow field is in its developed regime. It is observed that the flow field penetrates into the particle but is

121

5 Single Particle Heating and Drying

0.1 0.08

Large pores, ε = 0.7 Large pores, ε = 0.8 Small pores, ε = 0.7 Small pores, ε = 0.8

Vparticle/Vin

0.06 0.04 0.02 0 −0.02

−0.4

−0.2

0

0.2

0.4

Position on x-axis/D Figure 5.16 Normalized gas flow rates of the axial fluxes calculated at the slices “YZ” (through the porous particle) along the axis of the particle (X-coordinate). The recirculating wake behind the particle penetrates into the particles pores.

slowed down and blocked, when, at the same time, the recirculating wake enters the pores from behind the particle. Hence, the macroscopic flow field is similar to that around a solid sphere. Thus the drag coefficients are nearly the same as those for a solid particle. Thus, for constant Re, the variation between the drag coefficients for different porosities and pore sizes add up to around 1%. Finally, the results of the numerical simulations showed that, in comparison to the heating of a solid sphere, a porous sphere needs less time to be heated in a stream of hot air for a fixed Re. In more detail, the time that is needed to heat the particle decreases when the porosity increases. In particular, Figure 5.17 shows the 1 0.8

Re = 100

0.6 T*

122

0.4 ε = 0.7 ε = 0.72 ε = 0.74 ε = 0.76 ε = 0.78 ε = 0.8

0.2 0 0.01

0.1

1 Fo

Figure 5.17 Nondimensional time history of the particle-averaged temperature (solid + gas phases) predicted for different particle porosities at Re = 100.

5.2

Heating of a Porous Particle

nondimensional time history of the particle-averaged temperature calculated for different porosities at a constant Reynolds number. It can be seen that an increase in � leads to a decrease of the heating time. This fact is explained by the lower mass and increased total surface of the solid phase inside the porous particles. 5.2.3.1 Conclusion

Three-dimensional numerical simulations of porous particle heating under convection–diffusion conditions have shown that for the Reynolds numbers between 1 and 500 and particle porosity between 0.7 and 0.8 the particle-averaged temperatures predicted using 3D numerical simulations and analytic model show good agreement for Re ≤ 100. However, at Re values exceeding 500, the effect of flow penetration into the pores plays a significant role in the intraparticle heat transfer. The analytic model that uses the arithmetic-averaged thermal conductivity produces results close to the numerical simulations for Re ≈ 500. Additional parametric studies have shown that the pore size and geometry have a minor influence on the overall heat transfer for Re < 500. Finally, we found out that an increase in the porosity of the particle (exceeding the value of 0.7) leads to a decrease of the time needed to heat up this porous particle. This effect is explained by the decrease of the particle density. Parallel to that effect, the increase in Re causes a similar effect, namely, the so-called heating time decreases. 5.2.4 Appendix: Analytical Model

The analytical model for the unsteady heating of a sphere moving in a hot gas can be found in Bird et al. [12] and Carslaw et al. [19]. Defining T0 as the initial particle temperature and T∞ as the inflow and initial ambient temperature, the particle-averaged temperature can be calculated as follows: ( ) ∞ ∑ exp(−𝜀2i Fo) ∗ T − T0 2 T = (5.32) = 6Bi T∞ − T0 𝜀2i (𝜀2i + Bi(Bi − 1)) i=1 where 𝜀i is obtained as the ith solution of the equation 𝜀 = 1 − Bi tan(𝜀)

(5.33)

The nondimensional parameters used in Eq. (5.32) may be written as

• Biot number: Bi =

�R kp

• Fourier number: Fo = a t2 = R

k t cp � R2

where the radius is defined as R = 0.5D. The heat transfer coefficient kp � = Nu D was calculated using the Nu correlation from Ranz and Marshall [4]: 1

Nu = 2 + 0.6 Re0.5 Pr 3

(5.34)

(5.35)

123

124

5 Single Particle Heating and Drying

The overall thermal conductivity kp of the porous medium depends in a complex way on the pore geometry [23]. If the porous medium has a hedgehog-like structure, the heat conduction from the particle center to the particle surface occurs on parallel trails. Then the overall conductivity kp = kpar is the weighted arithmetic mean of the solid ks and fluid kf conductivity kp = kpar = (1 − 𝜀)ks + 𝜀kf

(5.36)

where � is the particle porosity (void fraction). In an alternative case, the heat conduction takes place sequentially through the phases. Then the overall conductivity kp = ksequ is the weighted harmonic mean of solid ks and fluid kf conductivity, given by )−1 ( 𝜀 1−𝜀 + (5.37) kT = ksequ = ks kf 5.3 Spherical Particle Drying in a Stream of Hot Air

The problem of solid drying has many technological contexts. For example, drying is an important step in industrial processes such as burning or the partial oxidation of coal or biomass. It is estimated that 12% of the total industrial energy consumption goes toward drying [30]. In some industries, the energy usage for drying can be even higher: for example, up to 70% for the manufacturing of most wood products or 50% in the textile industry [31]. The literature provides a number of articles devoted to the field of drying. To understand the drying of porous materials, it is necessary to describe the heat and mass transfer within the solid and in the boundary layer. Three basic mechanisms are known for water transport in porous media, namely vapor diffusion, capillary flow, and evaporation–condensation [32]. For a review of different mathematical models applied to drying, see Waananen et al. [33]. The works of Luikov [34] and Whitaker [35] can be considered as fundamental to the understanding of drying processes. In particular, Luikov [34] derived the macroscopic transport equations using a volumetric summation of the transport equations for each species in all phases. This was done on the basis of phenomenological flux expressions. The theory has two major drawbacks. The first is that some of the coefficients used are not physical properties and are linked to the process by phenomenological relationships [36]. The second drawback is related to the phase conversion factor, a value that is assumed to be constant [36]. Luikov’s theory assumes that the evaporation front is receding, and two phases were distinguished in porous media: a moist zone and an evaporation zone. This model was utilized in the finite-volume (FV) study by Lamnatou et al. [37]. They simulated the drying of plates to find the optimal placement in a solar dryer. The next “generation” of drying models was introduced by Whitaker [35], who utilized the volume averaging technique to describe drying processes using the

5.3

Spherical Particle Drying in a Stream of Hot Air

continua conservation equations. In particular, many models for the simulation of the drying of porous materials are based on Whitaker’s theory (e.g., [38, 39]). The advantages of this model in comparison to the theory described by Luikov [34] are that the assumptions are very clear, the physics is represented better, and the parameters are all well defined [36]. For a review of simulations devoted to industrial drying applications, the reader is referred to the work of Jamaleddine and Ray [40]. They showed that while CFD modeling is widely used to predict the hydrodynamics and the heat and mass transfer in dryers, the quality of the numerical predictions is limited by the models, approximations and assumptions used. In spite of numerous models devoted to the modeling of drying, there is still a lack of CFD-based semiempirical models applied to coal drying. Motivated by this fact, the following section is devoted to the development of a simple and fast subgrid model that takes into account the necessary physics for drying a particle in air. For this purpose, the receding core-based model (e.g., see [41–44]) is adopted to describe the drying of a single coal particle at different ambient conditions. The distinguishing feature of the model developed is that it is validated and tuned against a detailed CFD-model based on the volume-averaged conservation equations. Additionally, the subgrid model was validated against experimental data published in the literature. 5.3.1 CFD-based Drying Model

One of the most used CFD-based model for drying of particles or any object is the so-called continuum model or volume-averaging model, where the pore structure of particles/objects is not resolved. The properties of each phases inside the porous solid are calculated using volume-weighted techniques. In particular, the volume fractions of the different phases are defined as �s =

Vs , Vav

�l =

Vl , Vav

�g =

Vg Vav

,

∑

𝜀i = 1

(5.38)

The governing equations for this model have the following form [15, 45]:

• Continuity equation ∂�g �g ∂t

( ) �g + ∇ ⋅ (�g ⟨⃗ug ⟩) = ṁ ′′′ 1 − v �l

(5.39)

where ṁ ′′′ v is the volumetric vapor mass flux. • Impulse conservation equation ∂�g ⟨⃗ug ⟩ ∂t

+ (�g ⟨⃗ug ⟩ ⋅ ∇)⟨⃗ug ⟩ = −∇p + �∇2 ⟨⃗ug ⟩ −

�⟨⃗ug ⟩ Kg

(5.40)

Here, the Forchheimer correction term is neglected, and Kg is the permeability coefficient.

125

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5 Single Particle Heating and Drying

• Energy conservation equation ∂(�cp )eff T ∂t

[ ] [ ] + ∇ ⋅ (� cp )g ⟨⃗ug ⟩ T + ∇ ⋅ (� cp )l ⟨⃗ul ⟩ T ( ) ̇ ′′′ = ∇ ⋅ �eff ∇T − ṁ ′′′ v ΔV H + Qrad

(5.41)

It can be seen that we take into account the convective heat transport in the porous media by gas and liquid convection. is only activated in the surface CVs of the particle: The radiative heat flux Q̇ ′′′ rad 4 = fS �� � ⋅ (T∞ − Tp4 ) ⋅ Q̇ ′′′ rad

SCV VCV

(5.42)

with the ambient temperature far away from the particle surface T∞ , the Stefan– Boltzmann constant �, the emission coefficient �� , the CV surface SCV , and volume VCV . The utilization of the surface correction factor fS is governed by numerics. In particular, because of the stair-step approximation of the particle, the numerical surface exceeds the theoretical value, and fS is calculated as follows: Sp,ss (5.43) fS = π D2p where Sp,ss is the total surface of the particle on the numerical grid using the stair-step representation. The parameters �eff , cp,eff , and �eff are the effective density, heat capacity, and thermal conductivity, which are given by (�cp )eff = �s �s cp,s + �l �l cp,l + �g �g cp,g

(5.44)

�eff = �s �s + �l �l + �g �g

(5.45)

�s = 1 − �g,dry ,

�l = X�s �s ∕�l ,

�g = 1 − �s − �l

(5.46)

with �g,dry as the gas fraction of the dry media.

• Species conservation of vapor: ∂ �g �g Y

( ) ( ) + �g ∇ ⋅ ⟨⃗ug ⟩Y = ∇ ⋅ �g Dv ∇Y + ṁ ′′′ (5.47) v ∂t The diffusion coefficient is calculated by assuming that the Lewis (Le) number equals unity: �g ⋅ �2 (5.48) Dv = Le cp,g �g g

• Species conservation of liquid water [46, 47]: ∂ �s �s X + ∇⃗Jl = −ṁ ′′′ (5.49) v ∂t where X is the liquid moisture content in the particle phase based on the mass of the dry solid. For a detailed description of the supplementary closure equations and the numerical scheme for solving the conservation equations written above, the reader is referred to Schmidt [15].

5.3

Spherical Particle Drying in a Stream of Hot Air

5.3.2 Subgrid Models 5.3.2.1 Standard Model

The mainstream submodels for drying particles used in many commercial CFD codes are based on the assumption that the heat and mass transfer inside the particle are infinitely fast. Thus the particle is characterized by one mean temperature Tp and moisture content for the whole particle X. Because of the homogeneous moisture distribution, evaporation occurs solely from the particle surface. The governing equations for the heat and mass flow are [48]. mp cp,p

dTp dt

= �Sp (Tg − Tp ) + 𝜀� �Sdry (Tg4 − Tp4 ) − ṁ v ΔV H

−4π�g Dg dml log = −ṁ v = 1 dt Rp

(

1 − Yp 1 − Y∞

)

Sh 2

(5.50) (5.51)

with the heat transfer coefficient �, vapor mass fraction on the particle surface Yp , and in the gas phase Y∞ . One of the disadvantages of simple subgrid models for drying similar to the equations written above is the fact that they do not take into account the formation of a drying front inside the particle, and thus the heat and mass transfer might be overpredicted. 5.3.2.2 New Model

To overcome this problem, we use a receding core-based model adding more “physics” in the balance equations. In particular, we divide the particle into two shells: the dry outer shell, and the wet inner one (see Figure 5.18). The shells are characterized by their radius Rdry , Rwet ; thermal conductivity �dry , �wet ; heat capacity cp, dry , cp, wet ; and density �dry , �wet . The diffusion coefficient Ddry is defined only for the dry phase. The properties of the wet shell are calculated using the continuum assumption [36, 49, 50]: ( )1 3 ml Rwet = 4 (5.52) π�l �l 3 �cp wet,eff = �s �s cp,s + �l �l cp,l + �g,wet �g,wet cp,g,wet

(5.53)

mwet cp, wet = �cp wet,eff Vwet

(5.54)

�wet = �s �s + �l �l + �g,wet �g,wet

(5.55)

with the volume fraction of the solid matrix �s , the liquid �l , and the gas phase �g,wet . It should be noted that the � values are constant and do not change during drying. In the dry shell, the liquid water fraction �l is zero and the properties are calculated according to [36, 49–52]: �cp dry,eff = �s �s cp,s + �g,dry �g,dry cp,g,dry

(5.56)

127

128

5 Single Particle Heating and Drying

Rp T∞ and Y∞ Dry Rwet Wet

Rdry

  Ywet Twet and

Tsurf and Ysurf

Tdry and Ydry

Figure 5.18 Scheme of the shrinking core model for particle heating and drying.

mdry cp, dry = �cp dry,eff (Vp − Vwet )

(5.57)

�dry = �s �s + �g,dry �g,dry

(5.58)

2 Ddry = D �g,dry

(5.59)

The temperature profile that is formed in reality is approximated using three temperatures in the model: the temperature of the wet core and of the wet–dry interface Twet , the characteristic temperature of the dry shell Tdry , and the surface temperature of the particle Tsurf . To represent the concentration profile within the particle, the vapor mass fraction Ywet and Ysurf are used (see Figure 5.18). The temperature and vapor mass fraction of the ambient gas phase are T∞ and Y∞ . The energy balance for the inner wet shell includes the conductive heat transport from the dry shell and the heat flux due to evaporation: dTwet = �wet, dry (Tdry − Twet ) − ṁ v ΔV H (5.60) dt where �wet, dry is the heat transfer coefficient between the two shells, ṁ v is the evaporating mass flux, and ΔV H is the evaporation enthalpy. Assuming that the wet core radius changes slowly during drying, a fully developed temperature profile is considered in the dry shell [43]. Thus the heat transfer coefficient �wet, dry is calculated as follows [21]: mwet cp,wet

�wet, dry =

4π�dry 1 Rwet

−

1 Rdry

(5.61)

Using the results of Section 5.1.4, the position Rdry is calculated as follows: ) ( 1 (5.62) Rdry = max 0.5(Rp + Rwet ), 0.5 3 Rp

5.3

Spherical Particle Drying in a Stream of Hot Air

and thus the time history of Tdry is similar to the time history of the mean particle temperature in the detailed CFD simulations. The energy balance of the dry shell contains the heat flux to the wet shell, the heat flux from the particle surface, and the energy that is needed to heat up the formerly wet solid from Twet to Tdry : mdry cp,dry

dTdry

= − �wet, dry (Tdry − Twet ) + �dry, surf (Tsurf − Tdry )

dt

−

ṁ v cp, dry X0

(Tdry − Twet )

(5.63)

with �dry, surf =

4π�dry 1 0.5 Rdry

−

1 Rp

.

(5.64)

The energy balance for the particle surface includes the conductive inward heat flux, the convective heat transfer to the surrounding gas phase, and the radiation heat exchanges with a wall. The surface of the particle is assumed to be an infinitesimally thin layer, and thus Tsurf is calculated using the steady-state assumption dTsurf ∕dt = 0: −�dry, surf (Tsurf − Tdry ) + �surf, gas Sp (T∞ − Tsurf ) 4 4 +�� �Sp (T∞ − Tsurf )=0

(5.65)

with the surface of the particle Sp , the heat transfer coefficient to the gas phase �surf gas , the Stefan–Boltzmann constant �, and the emission coefficient �� . After Eq. (5.66) is rearranged using the radiative heat flux from the last time step or iter4 ), the surface temperature is calculated as follows: ation Q̇ rad = �� �Sp (Tg4 − Tsurf Tsurf =

�dry, surf Tdry + �surf, gas Sp T∞ + Q̇ rad �dry, surf + �surf, gas Sp

(5.66)

The influence of the Stefan flow is taken into account in the heat transfer coefficient �surf, gas [53]: �surf, gas = exp

ṁ ′′v cp,g ( ṁ ′′ c ) v

�0

p,g

(5.67) −1

with as the surface-based evaporation rate and � 0 as the heat transfer coefficient for ṁ ′′v = 0: ṁ ′′v

�0 =

Nu�g

(5.68)

2 Rp

The Nu number is calculated using the Ranz–Marshal equation [4]: 1

Nu = 2.0 + 0.6 Re0.5 Pr 3

(5.69)

When the particle drying is complete, only the dry shell with one mean temperature remains.

129

130

5 Single Particle Heating and Drying

In the subgrid model, the evaporation front is sharp and the flow of liquid water is not taken into account. The balance equation for the moisture mass is d ml = −ṁ v (5.70) dt The evaporation of water inside the particle causes a convective flow, the so-called Stefan flow. For the heat transfer inside the particle, the Stefan flow is neglected. This simplification is justified by the high thermal conductivity of the solid, which dominates the heat transport inside the particle. For the mass transfer, on the other hand, the convective transport is not negligible as the diffusion in the porous system is hindered. Taking the Stefan flow into account, the mass flux in a quiescent medium is [54] �D dY (5.71) 1 − Y dr Integrating Eq. (5.71) for the mass transfer from the wet–dry interface to the particle surface using ṁ = −4πR2

B.C.:

Y (Rwet ) = Ywet ,

leads to ṁ i =

−4π�gd Ddry 1 Rwet

−

1 Rp

( log

Y (Rp ) = Ysurf 1 − Ywet 1 − Ysurf

(5.72)

) (5.73)

For the mass flux at the particle surface, it follows that B.C.:

Y (Rp ) = Ysurf ,

Y (∞) = Y∞

which leads to ( ) −4π�g Dg 1 − Ysurf ṁ o = ( ) log 1 1 − Y∞ − ∞1 R

(5.74)

(5.75)

p

This equation is valid only in a quiescent medium. To take the forced convection into account, one can follow the film theory. One result of this theory is that the nondimensional concentration gradient, namely the Sherwood number Sh, connects the film radius of the boundary layer � with the particle radius [54]: Sh � = Rp Sh − Shdiff

(5.76)

were Shdiff is the Sh number for Re = 0 (Shdiff = 2). Thus the boundary condition moves from an infinite radius inward to the film radius �, and the resulting equation is ( ) −4π�g Dg 1 − Ysurf Sh log (5.77) ṁ o = 1 1 − Y∞ 2 Rp

The Sh number is calculated using the assumption that the Le number is unity. Equations (5.73) and (5.77) are two equations with three unknowns: ṁ i , ṁ o , and

5.3

Spherical Particle Drying in a Stream of Hot Air

Ysurf . Using the fact that the mass flux must be the same inside and outside the particle (ṁ v = ṁ i = ṁ o ), the system of equations is solved by calculating the root ṁ i − ṁ o = 0

(5.78)

The vapor pressure and thus the vapor concentration on the wet–dry interface Ywet is calculated using the temperature of the wet core Twet . The properties of the gas phase in the boundary layer were calculated using the mean temperature Tm and mean water vapor concentration Ym : Tsurf + T∞ (5.79) 2 Y + Y∞ (5.80) Ym = surf 2 In the dry shell, the gas properties are calculated at the mean temperature of the dry shell Tdry and the mean vapor concentration Ym, dry : Tm =

Ywet + Ysurf (5.81) 2 Finally, in the wet shell, the temperature Twet and the concentration Ywet are used to estimate the gas properties. The unknown variables Tsurf , Tdry , Twet , and ṁ v are calculated by iteratively solving Eqs (5.60), (5.63), (5.66), and (5.78). Ym, dry =

5.3.3 Illustration and Validation of Models 5.3.3.1 Results: CFD-Based Model

Next we show the results for the convective drying of a single coal particle using CFD simulations. The model described in Section 5.3.1 was implemented into an open-source FV code [55]. The simulations were carried out in two dimensions using cylindrical coordinates. The general setup of the system is shown in Figure 5.1. The sphere with the diameter Dp is placed in a system with the dimensions L1 = 70 Dp × L2 = 10 Dp . The center of the particle is at z = 2∕7 L1 and r = 0. The system is meshed with a Cartesian grid with a total number of 43 050 CVs, which corresponds to 410 CVs for the z-direction and 105 in the r-direction. The grid is locally refined to a resolution of 100 CV∕Dp in the area of the particle. The particle is initiated with the temperature T0 = 303.15 K and the moisture content X0 = 0.4, which equals a pore saturation Φ of 0.89. The convective flow is set in the z-direction with the inlet velocity uin . The ambient gas is mainly nitrogen with a small amount of vapor, Yin = 1 × 10−3 . The particle drying is investigated for two different particle sizes: Dp = 2 and 0.2 mm. The temperature Tin is varied in the range 413.15–1000 K. To determine the effect of the heat transfer on the drying process, different inlet velocities uin are used, and the flow condition is described by the inlet Re number Rein , given by Rein =

uin �in 2Rp �

(5.82)

131

132

5 Single Particle Heating and Drying

Table 5.4

Inlet flow conditions and time steps used

Dp (10−3 m)

Tin (K)

Rein

uin (m∕s−1 )

�in (kgm−3 )

�t (s)

2

413.15 1000

1 1

0.0139 0.0724

0.815 0.337

5.00 × 10−4 5.00 × 10−5

0.2

1000

1

0.7240

0.337

1.25 × 10−6

with the inlet density �in . The transport properties of the gas phase, except the viscosity, are calculated as a function of the temperature and mixture composition. The coal properties are taken from the work of Zhang and You [38], which describes a lignite coal. For the radiative heat transfer, the coal is treated as black body (�� = 1). To avoid high temperature gradients at the simulation start-up, initialization is carried out using the steady-state solution of the flow around the cold particle. The inlet velocity uin for Rein = 1 and the inlet density �in for the different particle sizes and inlet temperatures are given in Table 5.4. To study the dynamics of the process, the mean particle temperature T p , evaporation rate ṁ v′′′ , and moisture content X are used: ∫0 1 ∫0 2 Tlocal � r dz dr L

Tp = π

L

Vp

∫0 1 ∫0 2 ṁ ′′′ v � r dz dr L

′′′ ṁ v

=π

L

Vp

∫0 1 ∫0 2 Xlocal � r dz dr L

X=π

(5.83)

(5.84)

L

Vp

(5.85)

with � as a marker for the particle, which is unity in the particle and zero in the gas phase, and the volume of the particle Vp is Vp = π

∫0

L1

∫0

L2

� r dz dr

(5.86)

To illustrate the dynamics of the drying process, Figure 5.19 shows contour plots of the evaporation rate and the temperature isolines inside the particle at ambient temperatures of 413.15 K (Kelvin) and 1000 K. It can be seen that the drying starts immediately at the particle surface and the inner particle is heated up simultaneously. When the particle surface is dry, an evaporation front is formed inside the particle. In the case of 413 K, the drying front is formed after 4–5 s, and in the other case of 1000 K the drying front is observed after 0.5 s. With ongoing drying, the radius of the wet core decreases. But the drying front thickness depends strongly on the ambient temperature. In detail, at 413 K the drying front thickness increases during the whole process. This is explained by the gradient in the capillary pressure which results in a liquid water flow directed to the surface. This

399

1

= 413 K, t = 15 s

20.2

20 z/Dp

0.4

0

0.6

1

75

387

20 z/Dp

20.2

55.0 49.0 43.0 37.0 31.0 25.0 19.0 13.0 7.0 1.0

987

873

0.4

9

87

8

73

759

0

(f)

9

531

0.2

20.4

75

645

19.6

380

19.8

20 z/Dp

20.2

Tin = 1000 K, t = 3 s

Figure 5.19 Contour plots of the evaporation rate ṁ ′′′ v and isolines of the temperature Tin = 413 K (a,c,e) and at Tin = 1000 K (b,d,f ), Rein = 100, Dp = 2 mm.

reduces the moisture content in the wet core. In the other case, at 1000 K, the wet core is heated up to the boiling point and the drying rate is about 10 times larger. As a result of the high drying rate, a sharp drying front is observed during the whole process (see Figure 5.19). Further, the moisture content of the wet core is not reduced; it remains nearly at the initial value. Figure 5.20 depicts the time histories of the mean particle temperature T p , the minimum temperature inside the particle Tmin , the mean moisture content X, and ′′′ the mean evaporation rate ṁ v for the 2 mm particle at different inlet Re numbers and an inlet temperatures of 1000 K. As expected, the figure shows that the drying rate is a function of the Re number: a higher Re number accelerates the drying process. This effect is more noticeable at low ambient temperatures where radiative heat transfer is weaker (for details, see [15]). The time history of the evaporation rate is influenced by the particle temperature, the position of the evaporation ′′′ front, and the moisture content. An analysis of the ṁ v curves shows that, at the

645

Tin = 413 K, t = 30 s

20.4

531

19.8

20.2

7

19.6

(e)

20 z/Dp

41

0

7

375

39

41

38

375

0.2

19.8

Tin = 1000 K, t = 1.5 s

873

7

9

r/Dp

0.4

417

2000 1800 1600 1400 1200 1000 800 600 400 200 0

399

399

38

19.6

r/Dp

411

380

64 5 53 1

7

(d)

0.6

759

41

0.2

20.4

645 531

645

19.8

9

75

873

417

in

19.6

873

7

98

759

0

20.4

417

375

0.2

55.0 49.0 43.0 37.0 31.0 25.0 19.0 13.0 7.0 1.0

3

375 363

20.2

645

3

20 z/Dp

2000 1800 1600 1400 1200 1000 800 600 400 200 0

Tin = 1000 K, t = 0.1 s

0.6 387

75 36

19.8

531

399387

351

41 1 39 389 7

3

19.6

(b)

35

r/Dp

0.4

0

20.4

7

411

20.2

3

41

87

0.6

20 z/Dp

3

80

645

0.2

98

19.8

9

5

83

7

98

759

3 33 87 9

19.6

315

30

0.4

835 721

531 417

r/Dp

7

5

Tin = 413 K, t = 1 s

(c) T

9

32

31

0

33

55.0 49.0 43.0 37.0 31.0 25.0 19.0 13.0 7.0 1.0

133 873

987

0.6

375 315

0.2

(a)

387 351

r/Dp

1 41 99 3 5 37

0.4

399 375

387 351

r/Dp

0.6

Spherical Particle Drying in a Stream of Hot Air

59 455

5.3

20.4

2000 1800 1600 1400 1200 1000 800 600 400 200 0

5 Single Particle Heating and Drying

1000

1000

900

900

800

800 Tmin (K)

Tp (K)

134

700 600

Re = 1 Re = 10 Re = 50 Re = 100

500 400 0

2

4

(a)

6

8

10

600

Re = 1 Re = 10 Re = 50 Re = 100

500 400 0

12

2

4

(b)

t (s)

6

8

10

12

t (s) 200

0.4

175 −1 −3 m′′′ v , kg (s m )

Re = 1 Re = 10 Re = 50 Re = 100

0.3 X

700

0.2 0.1

Re = 1 Re = 10 Re = 50 Re = 100

150 125 100 75 50 25

0

0 0

(c)

2

4

6 t (s)

8

10

12

0

(d)

2

4

6

8

10

12

t (s)

Figure 5.20 Time history for (a) the mean particle temperature T p , (b) the minimum particle temperature Tmin , (c) the mean moisture content X and (d) the mean evaporation mass ′′′ flux ṁ v for a Dp = 2 mm particle at 1000 K and different Rein numbers. ′′′

beginning of the drying process, ṁ v reaches a maximum value. In the case of 413 K, the maximum is reached before the drying front is formed. In the other case, at 1000 K, the maximum is reached when the drying front is located inside the particle. This is explained by the large temperature gradient occurring between the particle surface, which is heated rapidly, and the evaporation front, which is at boiling temperature, see Figure 5.20b. The high heat transfer overcompensates for the increasing heat and mass transfer resistance as long as the evaporation front is near the particle surface. Finally, it should be noted that the numerical simulations predict for both temperatures no constant-rate drying period, which is in accordance with the results of Zhang and You [38] and Ljung et al. [56]. After the discussion of the drying processes for particles of 2 mm diameter, we now follow the results for small particles with Dp = 0.2 mm briefly. For small particles, the same Re number range is investigated. The time history of the mean particle temperature T p , the minimum temperature Tmin , the mean moisture con′′′

tent X, and the evaporation rate ṁ v for different Re numbers at 1000 K is shown in Figure 5.21. As expected, the small particle dries faster than the large ones. In other aspects, the time histories in Figure 5.21 indicate that the drying process is very similar to the 2 mm particles. Specifically, a detailed analysis of the results showed the same effects already described for the large particles, including the

Spherical Particle Drying in a Stream of Hot Air

1000

1000

900

900

800

800 Tmin (K)

Tp (K)

5.3

700 600

Re = 1 Re = 10 Re = 50 Re= 100

500 400 0

0.05

0.1

(a)

0.15

0.2

0.25

600

Re = 1 Re = 10 Re = 50 Re= 100

400 0

0.3

0.05

0.1

(b)

0.4

0.15

0.2

0.25

0.3

t (s) 16 000

0.2 0.1

14 000 mv′′′ , kg (s−1 m−3)

Re = 1 Re = 10 Re = 50 Re= 100

0.3 X

700

500

t (s)

135

Re = 1 Re = 10 Re = 50 Re = 100

12 000 10 000 8000 6000 4000 2000

0

(c)

0

0.05

0.1

0.15

0.2

t (s)

0.25

0

0.3

(d)

0

0.05

0.1

0.15

0.2

t (s)

Figure 5.21 Time history for the mean particle temperature Tmean , minimum particle tem′′′ perature Tmin , mean moisture content X, and mean evaporation mass flux ṁ v for a Dp = 0.2 mm particle at 1000 K at different Rein numbers.

moisture distribution inside the particle. It should be noted that differences in the drying process arise from the fact that the radiative heat transport is less important for the small particles in comparison to the large ones. This fact can be illustrated by the ratio of the maximum evaporation rates between Rein = 100 and unity. At an inlet temperature of 1000 K, the ratio is 2.5 for the 2 mm particle and 3.8 in the case of the 0.2 mm particle. The numbers confirm that radiation increases the heat transfer more effectively in the case of large particles. 5.3.3.2 Validation of Subgrid Model

In this section, the results of the receding core drying subgrid model, the detailed CFD model, and the standard model described by Eqs (5.50)–(5.51) are compared. The initial conditions in the subgrid model are identical to those of the CFD simulation. The 0.2 mm particle is initialized with a homogeneous temperature of 303.15 K and a moisture content of X = 0.4. The ambient temperature is set at 1000 K, and the inlet Reynolds number Rein ranges from unity to 100. The subgrid model uses an explicit time integration scheme with a time step Δt = 10−6 s. Figure 5.22a shows that for all values of Rein the receding core subgrid model reproduces the time history of the moisture content very well for Re numbers

0.25

0.3

136

5 Single Particle Heating and Drying

0.4

16 000 Re = 1 Re = 10 Re = 50 Re = 100 Subgrid

0.2

14 000 12 000

Re = 1 Re = 100 Re = 1 Subgrid

10 000

Re = 100 Subgrid

mv′′′ , kg (s−1 m−3)

X

0.3

0.1

8000 6000 4000 2000

0 0

0.02

0.04

(a)

0.06

0.08

0.1

0

0.12

0

0.02

0.04

(b)

t (s)

0.06

0.08

0.1

0.12

t (s)

Figure 5.22 Comparison of the time histories of X and ṁ ′′′ v predicted by the subgrid ′′′

and the CFD models (X, ṁ v ) for a 0.2 mm particle at Tin = 1000 K and different Rein numbers.

above 10. For Re = 1, the drying time is slightly underpredicted. The corresponding time histories of the evaporation rates are shown in Figure 5.22b. It can be seen that the drying rate increases more slowly in the subgrid than in the CFD model because the whole wet core must be heated for evaporation to take place. Furthermore, the figure reveals a good agreement in the maximum evaporation rates for low and high Rein numbers. In detail, the deviations are ≈ 10% and ≈ 14% for Rein = 1 and 100, respectively. Next, the validation of the model using experimental results [38] is discussed. Zhang and You [38] investigated the drying of coarse lignite particles (Dp = 30 mm) at 413.15 K. The initial moisture content of the particle is 0.42, and the time step in the subgrid model is set at 2 ⋅ 10−3 s. Figure 5.23 shows the comparison of the time histories of the moisture content X in the experiments and the developed model. The figure shows that the agreement between the experimental and the numerical results is good. It is only at the end of drying that the evaporation rate in the subgrid model is a bit too low, but the curves are 0.4 Subgrid uin = 0.7 m/s Experiment uin = 0.7 m/s

0.3 X

Subgrid uin = 1.5 m/s Experiment uin = 1.5 m/s

0.2

0.1

0

0

2000

4000

6000

8000

10000

t (s) Figure 5.23 Comparison of experimental [38] and simulated time histories of the moisture content for Hailaer lignite.

5.3

Spherical Particle Drying in a Stream of Hot Air

0.4

9000

Re = 100 simple subgrid −1 −3 m′′′, v kg (s m )

X

Re = 1 simple subgrid

7500

Re = 1 subgrid

0.3

137

Re = 1 CFD

0.2 0.1

Re = 1 subgrid Re = 1 CFD

6000 4500 3000 1500

0

0

0.02 0.04 0.06 0.08

(a)

0.1

0

0.12 0.14 0.16

0.4

0.02 0.04 0.06 0.08

0.1

0.12 0.14

t (s) 20 000

Re = 100 simple subgrid

17 500 −1 −3 m′′′, v kg (s m )

Re = 100 subgrid

0.3 X

0

(b)

t (s)

Re = 100 CFD

0.2 0.1

Re = 100 simple subgrid

15 000

Re = 100 subgrid

12 500

Re = 100 CFD

10 000 7 500 5000 2500

0

(c)

0

0.01

0.02

0.03

0.04

t (s)

0.05

0.06

0

0.07

(d)

0

0.01 0.02 0.03 0.04 0.05 0.06 0.07 t (s)

Figure 5.24 Comparison of the time histories of ṁ ′′′ v (a, c) and X (b, d) of the subgrid and the simpler subgrid model at inlet Re numbers 1 and 100 (Tin = 1000 K).

very similar for uin = 0.7 m/s and 1.5 m/s, where the experiments show a greater difference. Finally, we compare the results of the receding core subgrid model and the standard model, see Figure 5.24. The figure shows that the drying time is underpredicted by the standard model. Specifically, in the standard subgrid model, the drying time at Rein = 100 is approximately only half of that obtained with the new subgrid model, see Figure 5.24c. The difference between the two models decreases as the Re number decreases. At Rein = 1, the difference in the drying time is ≈ 40%. This trend can be explained using the drying rates presented in Figure 5.24b, d. Figure 5.24d shows that neglecting the intraparticle heat transfer in the standard subgrid model at Rein = 100 results in a large overprediction of the evaporation rate. The maximum evaporation rate of the simple model is 30% higher than that of the developed subgrid model. On analyzing Figure 5.24b, it can be seen that the importance of the intraparticle heat transfer decreases for low Re numbers. In particular, at Rein = 1, the difference between the evaporation rates of the two subgrid models is only ≈ 18%. Hence, the main reason for the short drying time in the standard subgrid model is that the drying rate is constant until the end of drying.

138

5 Single Particle Heating and Drying

5.4 Conclusions

In this chapter, the unsteady heating of a single particle was investigated numerically using the IB method. The gas properties corresponded to air, and the particle was made of carbon corresponding to a Henry number of 2000. Numerical investigations were performed for the Reynolds number range 1 < Re < 200 and for temperature loadings of 1.33. Based on the numerical simulations, an extension of the Ranz–Marshall equation was derived taking into account thermal conductivity of the particle. The new relation allowed the prediction of the transient dynamics of particle heating using particle-averaged temperature only. Finally, a two-temperature semiempirical subgrid model was presented that predicted the mean particle temperature and the mean particle surface temperature of thermally conducting particles (�p > �g ). The model was validated using the results of the DNS model. In the second part of the chapter three-dimensional numerical simulations of porous particle heating under convection-diffusion conditions were carried out for the Reynolds numbers between 1 ≤ Re ≤ 500, where particle porosity was changed between 0.7 and 0.8. The comparison of the time histories of the particle-averaged temperatures predicted using 3D numerical simulations and analytic model showed good agreement for Re ≤ 100. However, at Re values exceeding 500 the effect of flow penetration into the pores plays a significant role in the intra-particle heat transfer. The analytic model which uses the arithmetic-averaged thermal conductivity produces results close to the numerical simulations for Re ≈ 500 . A parameter study has shown, that the pore size and geometry has a minor influence on the overall heat transfer for Re < 500. We found out that an increase in the porosity of the particle (exceeding the value of 0.7) leads to a decrease of the time needed to heat up this porous particle. This effect is explained by the decrease of the particle density. Parallel to that effect, the increase in Re causes a similar effect, namely, the so called heating time decreases. In the second part of the chapter, we presented a CFD-based model for the drying of a single coal particle moving in hot ambient gas. The model was based on the volume-averaged equations for the heat flux and the gas and the liquid flow in the porous particle. Based on the CFD simulations, a subgrid model was developed utilizing the receding core assumption. The temperature field in the particle was approximated with three temperatures, and the mass transfer was treated using an analytical expression which took the Stefan flow into account. The subgrid model was validated against experimental data published in the literature. Test runs showed that the subgrid model was able to reproduce the time histories of the moisture content and the drying rates observed in the CFD simulations with minor deviations. Furthermore, the predicted mean particle temperature of the dry shell at the end of the drying process was close to the mean particle temperature in the CFD simulation.

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Proceeding of the 21st Annual Conference of the CFD Society of Canada, Sherbrooke, Quebec, Canada, May 6–9, 2013. Nield, D.A. and Bejan, A. (2006) Convection in Porous Media, 3rd edn, Springer Science, New York. Khadra, K., Angot, P., Parneix, S., and Caltagirone, J. (2000) Fictitious domain approach for numerical modelling of Navier-Stokes equations. International Journal for Numerical Methods in Fluids, 34 (8), 651–684. Wittig, K., Golia, A., and Nikrityuk, P.A. (2012) 3d numerical study on the influence of particle porosity on heat and fluid flow. Progress in Computational Fluid Dynamics, 12 (2/3), 207–219. Wittig, K., Richter, A., and Nikrityuk, P.A. (2012) Numerical study of heat and fluid flow past a cubical particle at subcritical reynolds numbers. Computational Thermal Sciences, 4 (4), 283–296. Dierich, F. and Nikrityuk, P.A. (2013) A numerical study of the impact of surface roughness on heat and fluid flow past a cylindrical particle. International Journal of Thermal Sciences, 65, 92–103. Wittig, K. (2014) Three dimensional numerical modelling of laminar and turbulent heat and fluid flows around and in immersed objects. PhD thesis, Technische Universität Bergakademie Freiberg, Germany, Freiberg, Schulze, S., Nikrityuk, P., and Meyer, B. (2013) Proceeding of the International Conference Powder, Granule and Bulk Solid: Innovations and Applications– (PGBSIA 2013), Patiala, India, November 28–30, 2013. Strumillo, C., Jones, P.L., and Romuald, Z. (1995) Energy aspects of drying, in Handbook of Industrial Drying, 2nd edn (ed. A.S. Mujumdar), Marcel Dekker, pp. 1241–1276. Kudra, T. (2004) Energy aspects in drying. Drying Technology, 22 (5), 917–932. Huang, C.L.D., Siang, H.H., and Best, C.H. (1979) Heat and moisture transfer in concrete slabs. International Journal of Heat and Mass Transfer, 22 (2), 257–266.

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141

143

6 Unsteady Char Gasification/Combustion Dmitry Safronov

In selecting any models for the description of gas–solid reaction systems care should be taken that the sophistication of the model is consistent with the accuracy of the information available on the behavior of the system. J. Szekely, J. W. Evans and H. Y. Sohn [31]

6.1 Introduction

In the design of novel combustors or gasifiers working on solid carbonaceous fuels (particles), the important issue is the prediction of the heating and burning rates of such fuels. Considering coal in the role of a solid carbonaceous fuel, chemically reacting coal particles have been extensively studied over the last 100 years because of the practical importance of coal in the production of energy and chemicals. In 1924, Nusselt [1] proposed the one-film model, and in 1931 Burke and Schuman [2] developed the two-film model assuming infinitely fast CO oxidation in the gas phase. Both models are now typically used as a starting point in modeling a chemically reacting coal particle and for the validation of computer codes. For a detailed description of both models, see [3]. The dynamic modeling and simulation of the oxidation of a single char particle has for long attracted much attention because of its high practical significance. In the early 1980s, Amundson and coworkers [4–7] performed a large number of numerical studies devoted to the modeling of combustion and gasification for a single coal particle in quiescent and nonquiescent environments. In the case of a nonquiescent environment, the so-called stagnant boundary layer approximation was used. In particular, it was shown that the two-film model is capable of adequately predicting the combustion of a coal particle at higher temperatures in a steady-state regime of combustion. An extended review of the work carried out before 1980 can be found in the work by Sundaresan and Amundson [7].

Gasification Processes: Modeling and Simulation, First Edition. Edited by Petr A. Nikrityuk and Bernd Meyer. © 2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA.

144

6 Unsteady Char Gasification/Combustion

As time passed, more advanced models and numerical simulations appeared in the literature devoted to the investigation of partial oxidation of a single coal particle in different environments (e.g., see [8–15]). Analysis of these works shows that the so-called moving-grid-based models are generally used where conservation equations are solved in solid and gas phases using separate grids conforming to the shape of the particle interface. In the case of the shrinking particle model, which assumes that the particle diameter decreases as a result of carbon conversion, the grids have to be adapted at every time step as the interface moves. The location of the solid–gas interface is defined explicitly using interfacial boundary conditions for the heat and species conservation equations. In the case of the shrinking core model, which assumes that a particle diameter is constant, the particle density change is defined using interfacial heat and species concentration balance. In particular, recent works [13] and [15] are examples of the two approaches, respectively. The disadvantage of the moving-grid methods is their complexity in 2D and 3D implementations. The main challenge in reconstructing the moving interface is the permanent changing of grid points at the interface when it is shrunk or stretched. In fixed-grid models, the governing conservation equations are solved for the entire domain including both the solid and gas phases. The interfacial boundary conditions are incorporated into the governing equations using special source terms. The interface in this class of methods is not explicitly tracked but is reconstructed using an appropriate field variable. With regard to the modeling of the sharp interface problem using a fixed-grid approach, the interface thickness is generally about one cell of the computational mesh and often is treated as a pseudo-porous medium depending on the type of the model. The major advantage of this class of methods is their relative simplicity during implementation and their robustness. For instance, recent work by Kassebaum and Chelliah [14] is an example of the fixed-grid method’s concept. In particular, Kassebaum and Chelliah developed a mathematical model and numerical solution approach to model the oxidation of an isolated porous carbon particle in a quiescent atmosphere under quasi-steady conditions. The volume-averaged conservation equations were integrated numerically from the center of the particle to the edge of the gas-phase thermal mixing layer with the specification of porosity as a function of the radial distance. Recently Safronov et al. [16] modeled transient partial oxidation of a coal particle using the fixed-grid method. Influence of gas temperature as well as water vapor concentration on oxidation process was studied. However, the particle was considered as nonporous, allowing the heterogeneous reactions to occur only on the external particle surface and homogeneous reactions in the ambient gas phase. Compared to the oxidation process of nonporous carbon, that of char is more complex especially because of its porous structure. While in unreacted core models like the shrinking particle model mentioned above particle core is impervious to the gas species and heterogeneous reactions occur at the particle surface, in the case of porous particles chemical reaction and species transport within the pore structure can play a critical role. The so-called reactive core models, such as

6.2

Modeling Approach

shrinking core model, consider mass transport and chemical reactions also inside porous solids. Reaction kinetic models based on overall physical properties such as porosity, internal surface area, and pore size distribution have been developed and successfully validated for conversion of porous chars with low ash content. In the attempt to describe the pore structure accurately, the volumetric model, grain model, random pore model (RPM), and other models were developed [17]. All these models have in common that char reactivity is defined as a function of the degree of carbon conversion. The RPM initially proposed by Bhatia and Perlmutter [18] has been used to model reactions in porous solids and represents porous media as a system of growing and collapsing pores. The model accounts for a reactive surface area variation during reaction based on (experimentally measurable) initial char structural properties and was generally accepted as a successful model to describe the change of char reactivity in the kinetically controlled regime. The model is widely used for the prediction of char structure evolution during combustion and gasification (e.g., see the recent works [13, 19–23]). The modeling approach discussed below is based on the fixed-grid sharp interface tracking model presented in [16]. The model was modified in order to take into account the intraparticle diffusion and oxidation taking place in the pores of a solid char matrix. The proposed model considers energy and multicomponent mass transfer with heterogeneous and homogeneous reaction inside the char particle and in the surrounding boundary layer. Variable physical properties are included to account for the changes in temperature and gas composition as well as changes in the pore structure and particle size due to the char conversion.

6.2 Modeling Approach Problem Formulation To illustrate the fixed-grid method, a single spherical coal particle placed in a quiescent gaseous environment was considered. The initial diameter of the particle ds = ds, 0 varied in the range 200 μm to 2 mm. The ambient gas phase consists of O2 , CO2 , CO, H2 O, H2 , and N2 , as shown in Figure 6.1. Flame sheet Gas phase species: O2, CO2, CO, H2O, H2, N2

Particle surface C 0

rs

Figure 6.1 Problem formulation.

rf

r=∞

145

146

6 Unsteady Char Gasification/Combustion

The chemistry is modeled using semi-global homogeneous and heterogeneous reactions, written as follows [24, 25]: Heterogeneous (surface) reactions: ΔR H = −32.73 MJ kg−1 C

C + O2 −−−−→ CO2 ,

−1

(R1)

2 C + O2 −−−−→ 2 CO ,

ΔR H = −9.2 MJ kg

C

(R2)

C + CO2 −−−−→ 2 CO ,

ΔR H = 14.4 MJ kg−1 C

(R3)

C + H2 O −−−−→ CO + H2 ,

ΔR H = 10.9 MJ kg−1 C

(R4)

Homogeneous (gas phase) reactions: ΔR H = −10.1 MJ kg−1 CO

CO + 0.5 O2 −−−−→ CO2 , CO + H2 O −−−−→ CO2 + H2 ,

ΔR H = −1.47 MJ kg

−1

−1

CO2 + H2 −−−−→ CO + H2 O ,

ΔR H = 1.47 MJ kg

CO

CO

(R5) (R6) (R7)

The particle is porous, allowing diffusion and chemical reactions to occur simultaneously through the solid phase. Therefore, the heterogeneous reactions occur in a diffusive zone inside the particle rather than at the sharp boundary representing the external particle surface. The homogeneous reactions take place in voids of solid phase and in the ambient gas phase. Assumptions To proceed with the governing equations, the following basic

assumptions have been made: 1) 2) 3) 4)

The particle is spherical. The particle consists of coal char only (no ash was considered). All voids inside the particle are opened for diffusion and chemical reaction. The effects of convection and gas radiation are ignored. Thus, the Stefan flow effect is neglected. 5) The ambient gas phase is assumed to be isobaric with p = 105 Pa . 6) The Lewis number Le = �∕� cp D is about unity. 6.2.1 Governing Equations

In the case of the proposed spherical geometry, the mass conservation equation can be written in the general form ( ) ( ) ∂ � Yi ∂Yi 1 ∂ 2 = 2 r � Di ∂� ∂r r ∂r ∑ ∑ ∑ G (6.1) �i, j R̂ Sj �i, j R̂ Pj + �� �i, j R̂ j + (1 − �) +�� j

j

j

⏟⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏟

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

⏟⏞⏞⏞⏞⏟⏞⏞⏞⏞⏟

Gas

Porous char

Surface

6.2

Char particle, ε = 0, δε = 0, 0 < < 1 i −1 0

Modeling Approach

Gas phase, ε = 1, δε = 0, = 1 i i+1 rs

r

Interface cell, 0 < ε < 1, δε = 1, 0 < < 1 Figure 6.2 Discretization scheme.

where the first term on the right-hand side is the diffusion term. i stands for the participating reactants O2 , CO2 , CO, H2 O, H2 , and N2 , and j stands for the reaction number (R1)–(R7). The rate of chemical reaction j is R̂ j , and the stoichiometric coefficient for the specie i in this reaction is �i, j . The first source ∑ term on the right-hand side of the equation � � �i, j R̂ G is responsible for the j

j

homogeneous reactions in the gas phase and in pores; the second source term ∑ (1 − �) j �i, j R̂ Pj is responsible for the heterogeneous reactions in pores; and the ∑ last term �� j �i, j R̂ Sj takes into account heterogeneous reactions on the particle surface. The volume fraction of gas � takes the following values according to the discretization scheme (Figure 6.2): ⎧1, ⎪ � = ⎨0, ⎪0 … 1, ⎩

for the gas phase for the particle

(6.2)

for the interface cell

The porosity of the particle as volume of voids inside the particle related to the total volume takes the following values: ⎧0, ⎪ � = ⎨0 … 1, ⎪0 … 1, ⎩

for the gas phase for the particle

(6.3)

for the interface cell

The interface-marker function �� is calculated as follows: ⎧0, for the gas phase (� = 1) ⎪ �� = ⎨0, for the particle (� = 0) ⎪1, for the interface cell (0 < � < 1) ⎩

(6.4)

147

148

6 Unsteady Char Gasification/Combustion

The heat conservation equation can be written in the general form ( ) ( ) ∂ � cp T 1 ∂ 2 ∂T = 2 r � ∂� r ∂r ∑ ∂r ∑ + �� ΔR Hj R̂ Pj ΔR Hj R̂ G j + (1 − �) j

j

⏟⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏟ ( + ��

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

Gas

∑

Porous char

)

ΔR Hj R̂ Sj + Q̇ rad

(6.5)

j

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ Surface

where the first term on the right-hand side represents the heat transfer through the gas/solid phase, as defined by the heat conductivity �. The second three terms on the right are the source terms which involve temperature changes due to the enthalpy of chemical reactions ΔR Hj and the radiation term ) ( 4 (6.6) Q̇ rad = av �� s � Ts4 − T∞ which describes surface radiation. Here, �� s denotes the emissivity factor of the particle surface. 6.2.2 Initial Conditions and Boundary Conditions

The values of all parameters are initially constant along the particle radius and the particle surroundings; temperatures of the particle and gas are equal, which are the initial conditions for Eqs (6.1) and (6.5). Boundary conditions have to be specified at the center of the particle as well as at the outlet boundary. Additionally, interface equations have to be specified to account for the processes on the surface of the particle. The proposed spherical symmetry implies the boundary conditions for the particle center (r = 0) as ∂Yi || ∂T || = 0, =0 (6.7) ∂r ||r=0 ∂r ||r=0 At the outlet boundary, far away from the particle surface (r = ∞), Dirichlet boundary conditions are assumed for the ambient gas temperature T and the species mass fractions of the ambient gas phase Yi : Yi ||r=∞ = Yi, ∞ ,

T|r=∞ = T∞

(6.8)

Chemical reactions on the interface as well as in the ambient gas phase and inside the porous particle are taken into account by means of appropriate source term formulations. The rates of all heterogeneous reactions depend, except for the species mass fraction at the surface, on the interface area between the gas and solid available for reaction. For reactions on the particle surface, this would be the specific external

6.2

Modeling Approach

surface area of particle av = ΔS∕ΔV = 1∕Δr (ΔS is the change in external surface area and ΔV the change in elemental volume for a given discretization). For reac′′′ tions in pores, the specific internal surface of open pores S is important. With the definitions above, all source terms can now be formulated for each specie i (except N2 as an inertial specie). Oxygen O2 :

∑

1.75 �1, j R̂ G j = −�

j

∑

(

0.5 MO2

kc 5 YCO

MCO

′′′

)0.25 (

YO2 M O2

YH 2 O

)0.5 (6.9)

MH2 O

′′′

�1, j R̂ Pj = −S � kc 1 YO2 − S � kc 2 YO2

(6.10)

�1, j R̂ Sj = −av � kc 1 YO2 , s − av � kc 2 YO2 , s

(6.11)

j

∑ j

Carbon dioxide CO2 :

∑

1.75 �2, j R̂ G j =�

j

+� ∑

2

MCO2

MCO2

′′′

�2, j R̂ Pj = S � �2, j R̂ Sj = av �

MCO MCO

M O2

j

∑

MCO2

MCO2

j

MO2

( kc 5 YCO ( kc 6 YCO

Y O2

)0.25 (

MO2

)0.5

MH 2 O

)

YH2 O

YH2 O

(

2

MH2 O

− � kc 7 YCO2

YH2

) (6.12)

MH2

kc 1 YO2 − S � kc 3 YCO2

′′′

(6.13)

kc 1 YO2 , s − av � kc 3 YCO2 , s

(6.14)

Carbon monoxide CO:

∑

( 1.75 �3, j R̂ G kc 5 YCO j = −�

j

( 2

− � kc 6 YCO ∑

′′′ �3, j R̂ Pj = S �

j ′′′

Y O2 MO2

YH2 O

)0.25 ( )

MH2 O

YH2 O

)0.5

MH 2 O MCO k Y +� MCO2 c 7 CO2 2

(

YH2 MH2

) (6.15)

2 MCO 2 MCO ′′′ k Y +S � k Y MO2 c 2 O2 MCO2 c 3 CO2

+S �

MCO k Y MH2 O c 4 H2 O

(6.16)

149

150

6 Unsteady Char Gasification/Combustion

∑

�3, j R̂ Sj = av �

j

2 MCO 2 MCO kc 2 YO2 , s + av � k Y MO2 MCO2 c 3 CO2 , s

+ av �

MCO k Y MH2 O c 4 H2 O, s

(6.17)

Water vapor H2 O:

∑

2 �4, j R̂ G j = −�

kc 6 YCO

MCO

j

+ �2 ∑

(

MH2 O MH2 O MCO2

kc 7 YCO2

YH 2 O

)

MH2 O ( ) YH2

(6.18)

MH 2

′′′ �4, j R̂ Pj = −S � kc 4 YH2 O

(6.19)

�4, j R̂ Sj = −av � kc 4 YH2 O, s

(6.20)

j

∑ j

Hydrogen H2 :

∑ j

2 �5, j R̂ G j =�

MH2 MCO

− �2

∑

′′′ �5, j R̂ Pj = S �

j

∑ j

�5, j R̂ Sj = av �

( kc 6 YCO

MH2 MCO2 MH2 MH2 O MH 2

MH 2 O

YH2 O

)

MH2 O ( ) YH2

kc 7 YCO2

(6.21)

MH 2

kc 4 YH2 O

(6.22)

kc 4 YH2 O, s

(6.23) ′′′

In can be seen that, in the gas phase (r > rs , � = 1, � = 1, S = 0), the internal specific surface area is zero and therefore according to Eqs (6.9)–(6.23) only homogeneous reactions (R5)–(R7) take place. Directly on the particle surface (r = rs , �� = 1), only heterogeneous reactions (R1)–(R4) take place. Inside the ′′′ porous carbon particle (0 ≤ r < rs , � = 0, 0 < � < 1, S > 0) following Eqs (6.1) and (6.5), both homogeneous and heterogeneous reactions will be considered.

6.2

Modeling Approach

6.2.3 Reaction Kinetics and Transport Properties

The kinetic coefficients kc j of chemical reactions (R1)–(R7) are calculated using the extended Arrhenius expression ) ( −Ea j (6.24) kc j = Aj T nj exp RT where Aj is the pre-exponential factor, nj is the temperature exponent, and Ea j is the activation energy. The values for Aj , nj , and Ea j with the corresponding units are given in Table 6.1. For reaction (R1), the additional equation for the higher temperature range T > 1650 K ( ) (6.25) kc 1 = 2.632 × 10−5 Ts − 0.03353 T was suggested in [24]. However, numeric studies have shown that the speed of the reaction R1 can be neglected for surface temperatures over ≈ 1500 K, as explained in Section 6.3.1. The heat capacity cp and the thermal conductivity � of the gas mixture were calculated using the mixing rule ∑ ∑ cp i Yi , � = �i Y i (6.26) cp = i

i

where the specific heat capacities and the thermal conductivities of single species are cp i = Ai + Bi T + Ci T 2 + Di T 3 + Ei T 4 (6.27) C B (6.28) ln �i = Ai ln T + i + 2i + Di T T The corresponding polynomial coefficients are taken from [28]. The density of the gas phase is calculated by means of the ideal gas law as p ̃ = 1 ̃ �=M , M (6.29) RT ∑ Yi i

Table 6.1

Mi

Kinetic coefficients for chemical reactions.

Equations

Aj

nj

Ea j

References

R1 R2 R3 R4 R5 R6 R7

593.83 m s−1 K−1 1.500 × 105 m s−1 4.605 m s−1 K−1 11.25 m s−1 K−1 1.25 × 1011 m2.25 mol−0.75 s−1 2.74 × 106 m3 mol−1 s−1 1.00 × 108 m3 mol−1 s−1

1 0 1 1 0 0 0

1.496 × 105 J mol−1 1.494 × 105 J mol−1 1.751 × 105 J mol−1 1.751 × 105 J mol−1 1.673 × 105 J mol−1 8.368 × 104 J mol−1 1.205 × 105 J mol−1

[24] [24] [26] [26] [3] [27] [27]

151

152

6 Unsteady Char Gasification/Combustion

The ambient gas phase is assumed to be isobaric with uniform pressure p = 1 × 105 Pa . The diffusion coefficients Di were calculated from the assumption Le ≈ 1 as Di = � 2

�i Le cp i �

(6.30)

6.2.4 Evolution of Pore Structure and Interface Tracking

For the calculation of the structure parameters of the porous char particle during gasification, the most widely accepted RPM of Bhatia and Perlmutter was used ′′′ [18]. According to this approach, the variations of the internal surface area S and particle porosity � are both functions of carbon conversion degree XC : ′′′

S =S

′′′ 0

( ) ( )√ 1 − � ln 1 − XC 1 − XC

) ( � = � 0 + 1 − � 0 XC

(6.31) (6.32)

′′′

2

Here, � = 4 π L0 (1 − � 0 )∕(S 0 ) is the dimensionless pore structural parameter of carbon matrix, characterizing initial char structure; those value depends on ′′′ the nature of the coal (see e.g., [18]). S 0 and � 0 are the initial values of the specific internal surface area and porosity of the char particle, respectively. The parameter � may be treated as a fitting parameter and may be obtained with ′′′ the curve-fitting of experimental values of S (XC ) at different carbon conversion ′′′ 0 degrees. Values of S , � 0 , and � depend on the nature of the coal char (see e.g., [29]). The carbon conversion degree itself is defined as the relative carbon consumption in porous char volume:

XC =

m0C, p − mC, p

(6.33)

m0C, p

The total carbon consumption rate, as a mass of solid carbon reacted over a period of time, consists of volume-related ṁ C, p and surface-related ṁ C, s parts: ṁ C =

d�p dVp dmC = Vp + �p d� d� d� ⏟⏟⏟ ⏟⏟⏟ ṁ C, p

(6.34)

ṁ C, s

where the apparent density of a porous char particle is given by �p = � � + (1 − �) �C

(6.35)

6.3

Numerics and Code Validation

The volume-related carbon consumption rate ṁ C, p considers the reactions inside the porous particle and is given by ( 2 MC M ′′′ k Y � C kc 1 YO2 + � ṁ C, p = Vp S M O2 MO2 c 2 O2 ) MC MC k Y +� (6.36) k Y +� MCO2 c 3 CO2 MH2 O c 4 H2 O The external-surface-related consumption of carbon depends on reactions on the particle’s external surface: ( M 2 MC 2 k Y ṁ C, s = 4 π rs � C kc 1 YO2 , s + � M O2 MO2 c 2 O2 , s ) MC MC k Y +� k Y (6.37) +� MCO2 c 3 CO2 , s MH2 O c 4 H2 O, s Consequently, the interface movement can be calculated from dVp d�

= 4 π rs2

ṁ C, s drs = d� 4 π rs2 �p

(6.38)

Equations (6.38) and (6.35) together with Eqs (6.32) and (6.33) provide the coupling between the volume and surface related consumption rates of carbon in the interface cell and the interface movement. The volume fraction of gas � in the interface cell can be calculated according to Eq. (6.2) and Figure 6.2 as ( ) Vgas rs �= (6.39) Vcell where Vgas and Vcell are the volume of the gas phase in the interface cell and the total volume of the interface cell, respectively.

6.3 Numerics and Code Validation

Numerical simulations are performed to investigate the influence of different ambient conditions on the partial oxidation behavior of carbon particles. The finite-volume numerical method is used for solving the system of partial differential equations (6.1) and (6.5), that is, for obtaining the values of temperature T and concentrations Yi along the radial coordinate r and their change over time. The values of the source terms for the mass and heat balance are obtained via the relations for chemical reaction kinetics (Eqs (6.9)–(6.23)). Changing of the particle size is taken into account by solving Eqs (6.37) and (6.38). For numerical purposes, the whole computational domain was subdivided into multiple control volumes (usually more than 1 × 105 ) of equal size. For each finite volume, the fraction of

153

6 Unsteady Char Gasification/Combustion

gas phase � was calculated as a function of the current particle size according to Eqs (6.2) and (6.39). The interface cell changes its location depending on the current particle radius rs (see the discretization scheme in Figure 6.2). That allows the particle interface to be tracked relatively simply. For spatial discretization, the central difference scheme (CDS) approach was used. Time derivatives were discretized using the three-time-level scheme. The tridiagonal matrix algorithm (TDMA) was utilized to solve the final matrix. The governing equations for both the particle and the gas phase were solved in a fully coupled way. Transport properties for the interface cell i were calculated using the harmonic average between corresponding properties in the i − 1 (solid) and i + 1 (gas) cells (see Figure 6.2). Since the model utilizes a continuum approach, that is, pores inside the particle are not resolved, some limitations exist for the spatial discretization: values for the control volume should not exceed the value of a representative elemental volume for the given pore size distribution at any point of time. The computer program was validated against analytical solutions for the diffusive pseudo-steady-state combustion of a solid coal particle (in other words, with a fixed particle radius) in dry air (YH2 O ≈ 0) based on one- and two-film models (Figure 6.3). Only a short description of these models is given in Section 6.5. An extended description of both models can be found in [3]. The validation studies showed that the domain radius equal to 250 ds, 0 corresponds to the minimum domain size, when it does not influence the final solution. While one- and two-film models allow the validation of processes in the gas phase and at the particle external surface and their coupling, for the validation of diffusion and reaction inside the particle an analytical model for steady-state diffusion and reaction in a porous particle given in [30] was utilized. A short description of the model can be found in Section 6.5. Figure 6.4 provides a comparison between the analytical and numerical solutions for the dimensionless CO2 concentration inside the particle and molar flow of CO2 at the particle surface. 10−1

4500

10−2

4000

3000

10−4 10

−5

10−6 10−7 10−8

(a)

3500

10−3 Tf (K)

· mC (kg m−2 s−1)

154

Two-ilm One-ilm

2000 1500 1000

Δ r/rs = 0.01 Δ r/rs = 0.1 Δ r/rs = 0.2

500 0

1000 1500 2000 2500 3000 T∞ (K)

2500

(b)

Two-ilm One-ilm

Δ r/rs = 0.01 Δ r/rs = 0.1 Δ r/rs = 0.2 1000 1500 2000 2500 3000 T∞ (K)

Figure 6.3 (a) Burning rate of carbon and (b) flame temperature as a function of the gas temperature for rs = 1 mm and different discretizations.

1.0

Analytic 900 K Numeric 900 K Analytic 1000 K Numeric 1000 K Analytic 1100 K Numeric 1100 K

YCO2/YCO2, s

0.8 0.6 0.4

Numerics and Code Validation

−1 ˙ W CO2 (mol s )

6.3

0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 (a)

r/rs

(b)

10−3 10−4 10−5 10−6 10−7 10−8 10−9 10−10 10−11 10−12 10−13 10−14

Analytic Numeric 600 800 1000 1200 1400 T(K)

Figure 6.4 (a) Dimensionless CO2 concentration as a function of the radial coordinate r∕rs for rs = 1 mm and different temperatures and (b) molar flow of CO2 as a function of the temperature for rs = 1 mm (b).

6.3.1 Results and Discussion

Before the analysis of the results, we give a short description of the “physics” of the processes taking place on the particle surface as well as in the gas phase. Figure 6.5 provides a schematic illustration of the process of coal oxidation for the given conditions. Following the analytical modeling for a nonmovable spherical particle in a quiescent dry-gas environment, the combustion of a carbon particle involves the diffusion of O2 through a stagnant film and its reaction at the particle surface to produce CO2 and CO according to the reactions (R1) and (R2), respectively. The CO produced at the surface diffuses outward and is consumed in the flame sheet, producing CO2 according to the reaction (R5). At the same time, the resulting CO2 reacts with the carbon surface and produces additional CO according to the reaction (R3). Since CO oxidation occurs very fast, both CO and O2 are zero H2O CO2

O2 CO2

O2

H2

H2O CO

C Particle

Flame sheet H2

O2

CO2

CO O2

H2O

CO2

H2O Figure 6.5 Schematic representation of coal oxidation.

155

156

6 Unsteady Char Gasification/Combustion

Table 6.2

Gas compositions (BC).

Dry air atmosphere Reducing atmosphere

Table 6.3 Parameter ′′′ 0

S �0 �

YO2 , ∞

YCO2 , ∞

YCO, ∞

YH2 O, ∞

YH2 , ∞

0.233 0.11

0 0

0 0

0.001 0.1

0 0

Char parameters. Value

References

1 × 107 0.4 10

[29] [29] [29]

at the flame sheet. There is no CO leakage through the flame sheet into the gas phase. The existence of the gas-phase flame also cuts off most of the supply of oxygen to the surface such that reactions (R1) and (R2) are now suppressed. The temperature T and CO2 concentration profiles have their peaks at the flame sheet (see, e.g., [16]). Owing to the radiative heat loss, the surface temperature Ts is only slightly higher than the ambient temperature T∞ . Next, the results obtained for two different atmospheres (Table 6.2) are described. The first atmosphere corresponds to the case of “dry” air, referring to the ambient concentration of water vapor YH2 O = 0.001. The second atmosphere corresponds to a reducing atmosphere with significant amount of water vapor YH2 O = 0.1 and reduced oxygen content YO2 = 0.11. Char parameters describing the pore structure such as the structural parameter �, the initial porosity � 0 , and ′′′ the specific internal surface area S 0 used in calculations are listed in Table 6.3. 6.3.1.1 Oxidation Behavior of Porous Particles

First, the general phenomena of the oxidation process will be described. Figures 6.6 and 6.7 give an overview of the oxidation process of a porous char particle in the dry air atmosphere. During the oxidation, a narrow reaction zone is formed inside the particle near the surface. The partially reacted core recedes inwards with conversion. The oxidation inside the particle is mainly diffusioncontrolled. The existence of this partially reacted core leads to significant temperature differences between the center and the external surface of the particles (see Figures 6.6b and 6.7). The heterogeneous reactions inside the particle lead to much more extensive heat consumption on the particle surface because of the highly developed pore structure. The heating effect of the gas flame outweighs the cooling effect of the endothermic surface reactions, and ΔTs is positive. The specific inner surface has its maximum (Figure 6.6c arising from the two opposite effects: the growth and collapse of pores according to RPM.

6.3 1000 m′′C,s m′′C,p

10 2

∆T (K)

10

0

ΔTs ΔTr = 0

400 200 0

10−1

−200 0.2

0.4

0.6

0.8

1.0

1.6 1.4 1.2 1.0 0.8 0.6 rs/rs, 0 0.4 S′′′/S0′′′ 0.2 0.0 0.0 0.2 0.4

−400 0.0

1.2

τ/10−4 (s)

(a)

rs/rs, 0; S′′′/S′′′; 0

600

10 1

10−2 0.0

(c)

800

0.6 0.8 τ/10−3 (s)

0.2

0.4

1.0

1.2 (d)

0.6

0.8

1.0

1.2

1.0

1.2

τ/10−3 (s)

(b)

XC; rs/rs, 0

· m′′C (kg m−2 s−1)

10 3

Numerics and Code Validation

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 XC 0.1 rs/rs, 0 0.0 0.0 0.2

0.4

0.6 0.8 τ/10−3 (s)

Figure 6.6 (a) Consumption rate of carbon, (b) temperature differences ΔTs = Ts − T∞ and ΔTr=0 = Tr=0 − T∞ , (c) structure parameters, and (d) conversion rate as functions of the burning time for rs, 0 = 100 μm, T∞ = 1500 K, and dry air atmosphere.

6.3.1.2 Details Inside the Particle

Figure 6.8 shows the distribution of single gas species, temperature, porosity, and conversion rate inside the particle at different times. The above-mentioned reaction zone and partially reacted particle core can be clearly recognized. Because of the fact that porosity is linearly proportional to the carbon conversion degree according to Eq. (6.32), porosity distribution profiles follow the same pattern as those of conversion degree (see Figure 6.8e,f ). 6.3.1.3 Effect of Ambient Gas Composition

Compared to the dry air atmosphere, the reducing one is characterized by lower oxygen concentration combined with high water vapor content (Table 6.2). The influence of the different gas compositions can be simply explained by the stoichiometry and thermodynamics of chemical reactions. The lower oxygen concentration YO2 in case of reducing atmosphere leads to lower gas and external surface temperature according to the heterogeneous reactions (R1) and (R2) as well as the homogeneous reaction (R5). At the same time, the increased concentration of water vapor YH2 O causes an increase in the speed of the reactions (R4) and (R6). All these cause an overall decrease in temperature in the gas phase as well as inside the particle, leading to lower carbon consumption rates ṁ C, s and ṁ C, p and, hence, to prolonged burn-out times, as illustrated in Figures 6.6, 6.9, and 6.10.

157

6 Unsteady Char Gasification/Combustion

τ = 2.5 × 10−4 s 0.5

O2 CO2 CO H 2O H2

0.4 Y

0.3 0.2 0.1

2.0 T/T∞

158

1.5 1.0 0.5 0

1

2

3

4

5

6

7

8

9

10

r/rs, 0 Figure 6.7 Snapshots of concentration and temperature distribution for rs, 0 = 100 μm, T∞ = 1500 K and dry air atmosphere.

The higher concentration of CO2 inside the particle and especially near the particle surface in case of dry air condition (compare Figure 6.7 and Figure 6.10) is explained by reaction (R1), prevailing only at the beginning of the oxidation process. After a short start-up phase, there is no essential difference in CO2 concentrations for both atmospheres. 6.3.1.4 Effect of Initial Particle Size and Ambient Gas Temperature on the Oxidation Regime

The diameter of the particle under investigation was set equal to 2 mm for the “large” particle and 200 μm for the “small” one. Small particles have more uniform distribution of species concentration compared to the large ones and heterogeneous reactions occur throughout the particle volume resulting in higher specific oxidation rates. The oxidation regime inside the particle is shifted toward kinetically controlled. This difference in oxidation regimes between small and large particles grows with the increase of the temperature. It is a well-known fact that for nonporous particles in diffusion-controlled oxidation regime the burn-out time is proportional to the particle diameter squared (see e.g., [3]). In the case of porous particles, the difference in burning times is slightly larger as predicted by the above-mentioned ds2 law for 1500 K (compare Figure 6.9 and Figure 6.11) and increases further with temperature increase (Figures 6.12 and 6.13). The more uniform species distribution in case of small particles leads to more uniform heat consumption due to heterogeneous reactions inside the particle compared to the large particles. However, smaller particles have smaller outer

6.3

0.20 YCO2

0.25

τ=0s τ = 5.0 × 10−5 s τ = 1.0 × 10−4 s τ = 5.0 × 10−4 s

0.25

YCO

0.10

159

τ=0s τ = 5.0 × 10−5 s τ = 1.0 × 10−4 s τ = 5.0 × 10−4 s

0.20

0.15

0.15 0.10 0.05

0.05 0.00 0.0

Numerics and Code Validation

0.5

1.0

1.5

2.0

2.5

3.0

r/rs,0

(a) 0.20

0.00 0.0

3.5

1.0

1.5 2.0 r/rs, 0

2.5

3.0

3.5

1.3

τ=0s τ = 5.0 × 10−5 s τ = 1.0 × 10−4 s τ = 5.0 × 10−4 s

0.15

0.5

(b)

1.2 1.1

YO2

1.0 T (K)

0.10

0.9 τ= 0s τ = 5.0 × 10−5 s τ = 1.0 × 10−4 s τ = 5.0 × 10−4 s

0.8

0.05

0.7 0.00 0.0

0.5

1.0

(c)

1.5 2.0 r/rs,0

2.5

3.0

0.6

3.5

0.8

0.8

0.6

0.6

XC

1.0

0.0 0.0 (e)

0.5

1.0

1.5 2.0 r/rs,0

1.0 1.5 2.0 2.5 3.0 3.5

0.4

τ=0s τ = 5.0 × 10−5 s τ = 1.0 × 10−4 s τ = 5.0 × 10−4 s

0.2

0.5

r/rs, 0

1.0

0.4

0.0

(d)

2.5

τ=0s τ = 5.0 × 10−5 s τ = 1.0 × 10−4 s τ = 5.0 × 10−4 s

0.2 3.0

0 0.0

3.5 (f)

0.5

1.0

1.5 2.0 r/rs,0

2.5

Figure 6.8 Mass fractions of (a) CO2 , (b) CO, (c) O2 , (d) temperature, (e) porosity, and (f ) conversion degree as functions of radial location for rs, 0 = 100 μm, T∞ = 1500 K, and different times.

surface and hence lower surface radiation, so that temperature differences between the center and surface of the particle are of the same magnitude for both particle sizes. As mentioned above, temperature increase leads to the transition of the intraparticle oxidation from kinetic-controlled to diffusion-controlled regime. It can be easily shown that for pure kinetically controlled regime with species uniformly

3.0

3.5

160

6 Unsteady Char Gasification/Combustion

103

m′′C,s m′′C,p

101 ΔT (K)

· m′′C (kg m−2 s−1)

102

100 10−1 10−2 10−3 0.0

0.5

1.8 1.6 1.4 1.2 1.0 0.8 0.6 rs/rs, 0 0.4 S′′′/S′′′ 0 0.2 0.0 0.0 0.5

−3

2.0

2.5

(s)

1.0

1.5

τ/10

ΔTs ΔTr = 0

0.5

2.0

2.5

−3 (s)

(d)

1.0

1.5

2.0

2.5

2.0

2.5

τ/10−3 (s)

(b)

XC; rs/rs, 0

rs/rs, 0; S′′′/S′′′; 0

1.5

τ/10

(a)

(c)

1.0

600 500 400 300 200 100 0 −100 −200 −300 −400 −500 0.0

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 X 0.1 r /r C s s, 0 0.0 0.0 0.5

1.0

1.5

τ/10−3 (s)

Figure 6.9 (a) Consumption rate of carbon, (b) temperature differences ΔTs = Ts − T∞ and ΔTr=0 = Tr=0 − T∞ , (c) structure parameters, and (d) conversion rate as functions of the burning time for rs, 0 = 100 μm, T∞ = 1500 K, and reducing atmosphere.

distributed along the particle radius ṁ C, p ṁ C, s

′′′

= S rs

(6.40)

Figure 6.9a shows the results for small particles at 1500 K, which confirms this tendency. Figure 6.13, however, shows clearly that with temperature increase the ratio between ṁ C, p and ṁ C, s becomes smaller because of the transition to the diffusion-controlled regime. 6.4 Advice for Beginners

To perform practical computations, the mathematical model has to be implemented in a computer program using numerical methods as discussed in Section 6.3. It takes dedicated effort to produce an efficient and error-free program. The following advice and suggestions are offered for the benefit of those readers who wish to implement the model equations into his/her own computer program.

6.4

Advice for Beginners

τ = 2.5 × 10−4 s 0.5

O2 CO2 CO H2O H2

0.4 Y

0.3 0.2 0.1

T/T∞

2.0 1.5 1.0 0.5 0

1

2

3

4

5

6

7

8

9

10

r/rs, 0 Figure 6.10 Snapshots of concentration and temperature distribution for rs, 0 = 100 μm, T∞ = 1500 K and reducing atmosphere.

• The mathematical model should not be implemented in its full complexity

• • • •

in one step. It is advisable to implement first only the main features such as transport equations and reaction mechanisms, leaving, for example, transport properties constant (i.e., not dependent on temperature and phase composition). Furthermore, interface tracking and pore structure evolution can be implemented independently and after the pseudo-steady-state model has been thoroughly tested and successfully validated. In general, one should go from a simple model to a more advanced one after performing testing and validation in between. It is helpful to test separate parts of the program independently. For example, the subroutine for solving the discretization equations can be tested by supplying arbitrary values for the matrix coefficients and the right-hand side vector. Initial testing can be performed on coarse grids in order to save computational time. Solutions are expected to be physically realistic even for the coarse grids and/or large time steps. The internal consistency of the program can be confirmed by a number of special tests. For example, it should be tested that the converged solution is independent of the relaxation factors and any initial guessed values. Quantitative checks have to be performed in order to indicate the accuracy obtainable with a certain grid fineness. Comparison with available analytical solutions given in Section 6.5 and, for example, [3] provides a useful way of testing the accuracy of the numerical solution. It should be verified that, as the grid is refined, the error in the computed solution diminishes (as example, see Figures 6.3 and 6.14).

161

162

6 Unsteady Char Gasification/Combustion

10

m′′C,s m′′C,p

1

100 10−1 10−2 10−3

(c)

rs/rs, 0 S′′′/S′′′ 0 X C; rs/rs, 0

rs/rs, 0; S′′′/S′′′; 0

10−4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 τ (s) (a) 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.0

300 ΔTs 200 ΔTr = 0 100 0 −100 −200 −300 −400 −500 −600 −700 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 (b) τ (s) ΔT( K)

· m′′C (kg m−2 s−1)

102

0.5

1.0

1.5 2.0 τ (s)

2.5

3.0

3.5 (d)

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 XC 0.1 rs/rs, 0 0.0 0.0 0.5 1.0

1.5 2.0 τ (s)

2.5

3.0

3.5

Figure 6.11 (a) Consumption rate of carbon, (b) temperature differences ΔTs = Ts − T∞ and ΔTr=0 = Tr=0 − T∞ , (c) structure parameters, and (d) conversion rate as functions of the burning time for rs, 0 = 1 mm, T∞ = 1500 K, and reducing atmosphere.

6.5 Analytical Models 6.5.1 One-Film Model

The main assumption in the one-film model is that at the particle surface the carbon reacts kinetically with oxygen to produce carbon dioxide (CO2 ) rather than carbon monoxide (CO), that is, reaction (R1) prevails. Therefore, the flame sheet is located at the particle surface. With these assumptions, a set of algebraic equations can be established for the burning rate of carbon ṁ C [3]: ] [ 1 + YO2 , ∞ ∕�O2 (6.41) ṁ C = 4 π rs � D ln 1 + YO2 , s ∕�O2 ] [ 1 + YCO2 , s ∕�CO2 , (6.42) ṁ C = 4 π rs � D ln 1 + YCO2 , ∞ ∕�CO2

6.5

103 10

0

101 ∆T (K)

· m′′C (kg m−2 s−1)

500

m′′C,s m′′C,p

2

100 10−1

0.2

0.4

(a)

0.6

0.8

ΔTs ΔTr = 0

−500 −1000

0.6

0.8

1.0

τ (s)

(c)

0.2

0.4

(b)

τ (s) 1.8 1.6 1.4 1.2 1.0 0.8 0.6 rs/rs, 0 0.4 S′′′/S 0′′′ 0.2 0.0 0.0 0.2 0.4

−2000 0.0

1.0 1.2

XC; rs/rs, 0

rs/rs, 0; S′′′/S′′′; 0

163

−1500

10−2 10−3 0.0

Analytical Models

1.2

0.6

0.8

1.0

1.2

0.8

1.0

1.2

τ (s) 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 XC 0.1 rs/rs, 0 0.0 0.0 0.2

0.4

(d)

0.6 τ (s)

Figure 6.12 (a) Consumption rate of carbon, (b) temperature differences ΔTs = Ts − T∞ and ΔTr=0 = Tr=0 − T∞ , (c) structure parameters, and (d) conversion rate as functions of the burning time for rs, 0 = 1 mm, T∞ = 2500 K, and reducing atmosphere.

where the stoichiometric coefficients MCO2 MO2 �O2 = , �CO2 = MC MC

(6.43)

The system is closed by means of a kinetic expression for the reaction (R1) involving ṁ C and YO2 [3]: ṁ C = 4 π rs2 kc 1

MC � YO2 , s M O2

Energy conservation equation at the particle surface [3] gives ( ) − ṁ C cp exp 4 π � rs ) ( ṁ C ΔR H1 = ṁ C cp ( ) Ts − T∞ + Q̇ rad − ṁ C cp 1 − exp 4 π � rs where the radiation heat flux Q̇ rad is calculated as ) ( 4 Q̇ rad = 4 π rs2 �� s � Ts4 − T∞

(6.44)

(6.45)

(6.46)

164

6 Unsteady Char Gasification/Combustion

104 · m′′C (kg m−2 s−1)

200

m′′C,s m′′C,p

103

−200

102

−400

ΔT (K)

101

−600 −800

100

−1000

10−1

−1200

10−2

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 τ/10−3 (s)

(a)

−1400 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 (b)

1.2

0.8

XC; rs/rs, 0

rs/rs, 0; S′′′/S′′′; 0

1.0

0.6 0.4 0.2

ΔTs ΔTr = 0

0

rs/rs, 0 S′′′/S0′′′

0.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 τ/10−3 (s)

(c)

τ/10−3 (s) 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 XC 0.1 rs/rs, 0 0.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 τ/10−3 (s)

(d)

1.0

6 × 10−2

0.8

10−2

5×

0.6 0.4 Δr = 10 μm Δr = 100 μm Δr = 200 μm

0.2 0.0 (a)

0

5 10 15 20 25 30 35 40 45 τ (s)

· m′′C (kg m−2 s−1)

rs/rs, 0

Figure 6.13 (a) Consumption rate of carbon, (b) temperature differences ΔTs = Ts − T∞ and ΔTr=0 = Tr=0 − T∞ , (c) structure parameters, and (d) conversion rate as functions of the burning time for rs, 0 = 100 μm, T∞ = 2500 K, and reducing atmosphere.

Δr = 10 μm Δr = 100 μm Δr = 200 μm

4 × 10−2 3 × 10−2 2 × 10−2 1 × 10−2 0 × 100

(b)

0

5 10 15 20 25 30 35 40 45 τ (s)

Figure 6.14 (a) Radius change and (b) burning rate of carbon for T∞ = 1800 K and different spatial steps Δr as a function of the burning time. The visible peaks in the ṁ C curve and the salient points in the rs curve for coarse discretization emerge as a result of changing the interface cell.

6.5

Analytical Models

and � is thermal conductivity of the gas phase. Equations (6.41)–(6.46) were solved iteratively until the convergence of the solution was reached. The FORTRAN77 code of this analytical model is given in the supplementary CD. 6.5.2 Two-Film Model

In the case of the two-film model, two gas films are considered: one at the particle surface, and the other at the detached flame sheet. This model has carbon oxidize to carbon monoxide (CO) rather than carbon dioxide (CO2 ); that is, at the particle surface only reaction (R3) is considered. A trace amount of water vapor is implicitly assumed to exist to facilitate the wet CO oxidation in the gas phase according to reaction (R5). The homogeneous reaction (R5) at the flame sheet is assumed to be infinitely fast. Thus Y O2 , s = 0 ,

Y O2 , f = 0 ,

YCO, f = 0

(6.47)

The following algebraic equations describe the combustion of a coal particle in the case of the two-film model [3]: [ ] ) ( 1 + YCO2 , f ∕�CO2 r s rf � D ln (6.48) ṁ C = 4 π rf − rs 1 + YCO2 , s ∕�CO2 ) [ ] ( 1 + YCO, s ∕�CO r s rf � D ln (6.49) ṁ C = 4 π rf − rs 1 + YCO, f ∕�CO ) ( 1 − YCO2 , f ṁ C = −4 π rf � D ln (6.50) �CO2 ) ( −ṁ C YN2 , f = YN2 , ∞ exp (6.51) 4 π rf � D where �CO = From

∑ i

2 MCO MC

(6.52)

Yi = 1, it is also known that

YCO2 , f = 1 − YN2 , f

(6.53)

The system is closed by means of a kinetic expression for the reaction (R3): ṁ C = 4 π rs2 kc 3

MC � YCO2 , s MCO2

(6.54)

Energy conservation equation at the particle surface gives ( ) ( ) −ZT ṁ C Ts − Tf ZT ṁ C exp rs ṁ C ΔR H3 − Q̇ rad = 4 π rs2 � [ ( ) ( )] ̇ ̇C −Z m −Z T C T m − exp rs2 exp r r s

f

(6.55)

165

166

6 Unsteady Char Gasification/Combustion

where the radiation heat flux Q̇ rad is calculated as ) ( 4 Q̇ rad = 4 π rs2 �� s � Ts4 − T∞

(6.56)

and ZT =

cp

(6.57)

4π�

Energy conservation equation at the flame sheet gives ( ) ( ) −ZT ṁ C Ts − Tf ZT ṁ C exp rf �CO ṁ C ΔR H5 = 4 π rf2 � [ ( ) ( )] ̇ ̇C m −Z −Z T C T m rf2 exp − exp rs r ( f ) ) ( −ZT ṁ C T∞ − Tf ZT ṁ C exp rf + 4 π rf2 � [ ( )] −ZT ṁ C 2 rf 1 − exp r

(6.58)

f

Similar to the one-film case, Eqs (6.48)–(6.58) were solved iteratively until convergence of the solution was reached. The FORTRAN77 code of this analytical model is given in the supplementary CD. 6.5.3 Chemically Reacting Porous Particle

In this analytical steady-state model, diffusion and chemical reaction only inside a porous spherical particle according to reaction (R3) are considered. For the case of constant diffusivity, the balance equation can be written in the following form [30]: ( ) dCCO2 1 d r2 = kc 3 S′′′ CCO2 (6.59) DCO2 2 dr r dr This equation is to be solved with the boundary conditions that CCO2 = CCO2 , s at r = rs and that CCO2 is finite at r = 0. Application of boundary conditions gives the dimensionless concentration of CO2 inside the particle as √ kc 3 S′′′ r sinh DCO2 YCO2 CCO2 r = = s (6.60) √ YCO2 , s CCO2 , s r kc 3 S′′′ sinh rs D CO2

and the molar flow of CO2 at the particle surface r = rs √ (√ [ )] kc 3 S′′′ kc 3 S′′′ r coth r WCO2 = 4 π rs DCO2 CCO2 , s × 1 − (6.61) DCO2 s DCO2 s

6.5

Nomenclature Latin

B cp C D d kc KB L M ̃ M ṁ C p r R R̂ S ′′′ S av T V XC Y

transfer number (–) heat capacity (J kg−1 K−1 ) molar concentration (mol m−3 ) diffusion coefficient (m2 s−1 ) diameter (m) kinetic constant (app. unit) burning rate constant (m2 s−1 ) total pore length per unit volume (m−2 ) molar weight (kg mol−1 ) average molar weight (kg mol−1 ) burning rate of carbon (kg s−1 ) pressure (Pa) radial location (m) universal gas constant (J mol−1 K−1 ) mass production rate (kg s−1 ) external surface area (m2 ) volume-specific internal surface area (m−1 ) volume-specific external surface area (m−1 ) temperature (K) volume (m3 ) conversion degree of carbon (–) mass fraction (–)

Greek

�� ΔR H � � �� s � � � � � 𝜙 �

interface marker function (–) enthalpy of reaction (J kg−1 ) volume fraction of gas (–) porosity (–) emissivity of particle surface (–) effectiveness factor (–) thermal conductivity (J m−1 K−1 ) stoichiometric coefficient (–) density (kg m−3 ) time (s) Thiele modulus (–) structural parameter (–)

Analytical Models

167

168

6 Unsteady Char Gasification/Combustion

Indices

0 f i j p s ∞

(Superscript) Referring initial value (� = 0) Referring to the flame sheet Referring species i Referring reaction j Referring to the particle core Referring to the particle external surface Referring to the infinite remote point (r = ∞)

References 1. Nusselt, W. (1924) Der Verbren-

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nungsvorgang in der Kohlenstaubfeuerung. Zeitschrift des Vereins Deutscher Ingenieure, 68, 124–. Burke, S.P. and Schumann, T.E.W. (1931) Kinetics of a type of heterogeneous reaction: the mechanism of combustion of pulverized fuel. Industrial and Engineering Chemistry, 23 (4), 406–413. Turns, S.R. (2006) An Introduction to Combustion, 2nd edn, McGraw-Hill. Caram, H.S. and Amundson, N.R. (1977) Diffusion and reaction in a stagnant boundary layer about a carbon particle. Industrial and Engineering Chemistry Fundamentals, 16, 171–181. Zygourakls, K., Arri, L.E., and Amundson, N.R. (1978) An analytical study of single particle char gasification. AIChE Journal, 24 (1), 72–87. Mon, E. and Amundson, N.R. (1978) Diffusion and reaction in a stagnant boundary layer about a carbon particle. 2. An extension. Industrial Engineering Chemistry Fundamentals, 17 (4), 313–321. Sundaresan, S. and Amundson, N.R. (1980) Diffusion and reaction in a stagnant boundary layer about a carbon particle. 5. Pseudo-steady-state structure and parameter sensitivity. Industrial Engineering Chemistry Fundamentals, 19, 344–351. Chelliah, H.K. (1995) Numerical modelling of graphite combustion using elementary, reduced and semi-global heterogeneous reaction mechanisms,

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in Modeling in Combustion Science, Lecture Notes in Physics (eds J. Buckmaster and T. Takeno), Springer, Berlin, pp. 130–147. Chelliah, H.K. (1996) The influence of heterogeneous kinetics and thermal radiation on the oxidation of graphite particles. Combustion and Flame, 104, 81–94. Žajdlík, R., Jelemenský, L., Remiarová, B., and Markoš, J. (2001) Experimental and modelling investigations of single coal particle combustion. Chemical Engineering Science, 56, 1355–1361. Manovic, V., Grubor, B., and Loncarevic, D. (2006) Modelling of inherent SO2 capture in coal particles during combustion in fluidized bed. Chemical Engineering Science, 61, 1676–1685. Wang, S., Lu, H., Zhao, Y., Mostofi, R., Kim, H.Y., and Yin, L. (2007) Numerical study of coal particle cluster combustion under quiescent conditions. Chemical Engineering Science, 62, 4336–4347. Manovic, V., Komatina, M., and Oka, S. (2008) Modelling the temperature in coal char particle during fluidized bed combustion. Fuel, 87, 905–914. Kassebaum, J.L. and Chelliah, H.K. (2009) Oxidation of isolated porous carbon particles: comprehensive numerical model. Combustion Theory and Modelling, 13, 143–166. Stauch, R. and Maas, U. (2009) Transient detailed numerical simulation of the combustion of carbon particles.

References

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International Journal of Heat and Mass Transfer, 52 (19–20), 4584–4591. Safronov, D., Nikrityuk, P., and Meyer, B. (2012) Fixed-grid method for the modelling of unsteady partial oxidation of a spherical coal particle. Combustion Theory and Modelling, 16 (4), 589–610. Gómez-Barea, A. and Ollero, P. (2006) An approximate method for solving gas–solid non-catalytic reactions. Chemical Engineering Science, 11 (61), 3725–3735. Bhatia, S.K. and Perlmutter, D.D. (1980) A random pore model for fluid-solid reactions: I Isothermal, kinetic control. AIChE Journal, 26 (3), 379–385. Everson, R., Neomagus, H., and Kaitano, R. (2005) The modeling of the combustion of high-ash coal-char particles suitable for pressurised fluidized bed combustion: Shrinking reacted core model. Fuel, 84, 1126–1143. Everson, R.C., Neomagus, H., Kaitano, R., Falcon, R., Alpen, C., and du Cann, V.M. (2008) Properties of high ash char particles derived from inertiniterich coal: 1. Chemical, structural and petrographic characteristics. Fuel, 87, 3082–3090. Sadhukhan, A.K., Gupta, P., and Saha, R.K. (2010) Modelling of combustion characteristics of high ash coal char particles at high pressure: Shrinking reactive core model. Fuel, 89, 162–169. Everson, R.C., Neomagus, H.W.J.P., and Kaitano, R. (2011) The random pore model with intraparticle diffusion for the description of combustion of char particles derived from mineraland inertinite rich coal. Fuel, 90 (7), 2347–2352.

23. Wang, X., Zeng, X., Yang, H., and

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Zhao, D. (2012) General modeling and numerical simulation of the burning characteristics of porous char. Combustion and Flame, 159, 2457–2465. Anil, W. (2005) Introduction to Computational Fluid Dynamics, Cambridge University Press, New York. Higman, C. and van der Burgt, M. (2008) Gasification, 2nd edn, Elsevier GPP, Gulf Professional Publ, Amsterdam u.a. Libby, P.A. and Blake, T.R. (1981) Burning carbon particles in the presence of water vapor. Combustion and Flame, 41 (0), 123–147. Jones, W.P. and Lindstedt, R.P. (1988) Global reaction schemes for hydrocarbon combustion. Combustion and Flame, 73, 233–249. McBride, B.J., Gordon, S., and Reno, M.A. (1993) Coefficients for calculating thermodynamic and transport properties of individual species. NASA Technical Memorandum. NASA Report No. TM-4513, October 1993. Sadhukhan, A.K., Gupta, P., and Saha, R.K. (2009) Characterization of porous structure of coal char from a single devolatilized coal particle: coal combustion in a fluidized bed. Fuel Processing Technology, 90, 692–700. Bird, R.B., Stewart, W.E., and Lightfoot, E.N. (2007) Transport Phenomena, 2nd edn, John Wiley & Sons, Inc. Szekely, J., Evans, J.W. and Sohn, H.Y. (1976) Gas-Solid Reactions. Academic Press Inc.

169

171

7 Interface Tracking During Char Particle Gasification Frank Dierich and Kay Wittig

Nature is complex, and the process of modeling it is an effort to represent it as closely as possible Marcio L. de Souza-Santos [39]

7.1 Interface and Porosity Tracking for a Moving Char Particle 7.1.1 Introduction

A detailed understanding of the processes inside and around a char particle undergoing combustion/gasification inside a gasification reactor is important in order to develop novel submodels that reflect real processes more closely. It is a wellknown fact that during gasification the char particle’s radius and the carbon content decrease. In view of this, it is very important to have a model accounting for these two effects simultaneously. In particular, the so-called direct numerical simulations (DNSs) can help to “look” at the particle surface and inside the particle during its conversion. One of the problems in DNSs of chemically reacting particles is the complexity1) of interface tracking algorithms, which must be coupled with the momentum, heat transfer, and chemical species conservation equations. Applied to CFD-based simulations of solid/gas, liquid/gas, or liquid/sold interface dynamics under the influence of convection, there are three basic methods: the volume-of-fluid (VOF) [1], the level set [2], and the sharp interface methods [3, 4]. Each method has its own advantages and disadvantages. In the case of reacting interfaces, the sharp interface method is the method most used in the literature devoted to chemically reacting particles (e.g., see [5–7]). In particular, Stauch and Maas [5] studied a single burning carbon particle in a transient numerical 1) By complexity we mean not only the mathematical elements but also the programming issue, which is a nontrivial task when implementing an interface tracking algorithm in a computational fluid dynamics (CFD) code. Gasification Processes: Modeling and Simulation, First Edition. Edited by Petr A. Nikrityuk and Bernd Meyer. © 2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA.

172

7 Interface Tracking During Char Particle Gasification

simulation. They assumed that there was spherical symmetry and a quiescent flow around the particle. Mehrabian et al. [6] developed a layer-based one-dimensional model for the thermal conversion of biomass particles. The particle model was linked to a CFD solver, and coupled simulations were performed. Sadhukhan et al. [8] analyzed the combustion of a coal particle with a spherically symmetrical model. A shrinking reactive core model was used in this 1D model. Sato and Niˇceno [7] developed a sharp interface tracking method for the phase change at a liquid–vapor interface. Zhou et al. [9] developed a 2D combustion code with interface tracking of the reacting surface. The surface is represented by a singlevalued function. This method allows the interface to move only in one direction. Massa et al. [10] used this method to study the combustion of a heterogeneous propellant in a rocket motor chamber. Singer and Ghoniem [11] introduced the adaptive random pore model, which allows different developments for different pore sizes. This model was used in a 1D, spherically symmetrical model to simulate a reacting porous char particle. Everson et al. [12] developed a random pore model with intraparticle diffusion. However, the existing models do not take into account the simultaneous change of the solid-interface location and the porosity of the solid during heterogeneous chemical reactions. Motivated by this fact, we present a sharp-interface-based model of a chemically reacting char particle taking into account both particle porosity change and particle interface movement due to intrinsic and surfacial heterogeneous reactions. In our first step, we consider the Boudouard reaction only. 7.1.2 Model and Governing Equations 7.1.2.1 Setup

The unsteady modeling of a single cylindrical particle as it gasifies is the aim of this section. A single cylindrical particle is placed in a gas flow. The coordinates are transformed to the particle coordinates. The gas moves past the particle and the particle stands still. This is equivalent to moving a particle in a quiescent gas with the same velocity. A 2D model is developed to reduce the computational effort. The setup of the modeling domain is shown in Figure 7.1. The flow passes the particle from left to right. The initial particle diameter is 0.2 mm. The particle porosity is modeled. The initial porosity of the particle is 0.4. The CO2 mass fraction of the ambient atmosphere is 0.9 and the particle consists of carbon only. Thus the Boudouard reaction is considered as a global heterogeneous reaction C + CO2 −−−−→ 2 CO

(7.1)

This reaction consumes carbon from the surface as well as from the inside of the particle. Both effects are taken into account in the unsteady simulation. Radiation is also considered. Particle–particle interactions and particle rotation are not part of the simulation.

4 mm

7.1

Interface and Porosity Tracking for a Moving Char Particle

0.2 mm

4 mm

Char particle uin

4 mm

10 mm

Figure 7.1 Simulation setup.

7.1.2.2 Governing Equations in the Gas Phase

Modeling one chemically reacting carbon particle in a CO2 atmosphere involves different chemical and physical phenomena. The surrounding gas flow is modeled with the incompressible Navier–Stokes equations [13], which consist of the mass conservation equation and the momentum conservation equation, that is ∂� + ∇ ⋅ (� u) = 0, ∂t [ ] ∂u � + � u ⋅ ∇u = −∇p + ∇ ⋅ �(∇u + ∇uT ) ∂t

(7.2) (7.3)

where u is the velocity vector, � is the density, t is the time, p is the pressure, and � is the dynamic viscosity. [ ] Assuming that the dynamic viscosity is constant, the term ∇ ⋅ �(∇u + ∇uT ) reduces to ∇ ⋅ (�∇u). The energy conservation in the surrounding gas flow is described by the following temperature-based equation [14]: � cp

∂T + � cp u ⋅ ∇T = ∇ ⋅ (�∇T) ∂t

(7.4)

where T is the temperature, cp is the specific heat capacity, and � is the thermal conductivity. For the chemical species CO and CO2 , two species conservation equations are used of the form [14] �

∂Yi + � u ⋅ ∇Yi = ∇ ⋅ (� D∇Yi ) ∂t

(7.5)

where i denotes one of the species CO or CO2 , D is the diffusion coefficient, and Yi is the mass fraction of species i. The mass fraction of N2 is calculated with the balance equation Y N2 = 1 −

∑ i

Yi

(7.6)

173

174

7 Interface Tracking During Char Particle Gasification

7.1.2.3 Governing Equations in Porous Particles

In a porous particle, the gas flow is affected by the solid structure. To model this, we use the Brinkman–Forchheimer equation [15] C � � � � ∂u + 2 u ⋅ ∇u = −∇p + ∇2 u − u − F1 � |u|u � ∂t � K � K2 and the mass conservation equation

(7.7)

∂� + ∇ ⋅ (�u) = RVC (7.8) ∂t where � is the porosity, K is the permeability, CF is the Forchheimer coefficient, and RVC is the carbon consumption rate due to the intrinsic chemical reactions inside the porous particle. The permeability K is given by the Carman–Kozeny equation [16, 17] �

K=

dp2 �3 150 ⋅ (1 − �)2

,

(7.9)

where dp is the average pore diameter. The average pore diameter can be calculated with the expression given by Szekely et al. [18], Qixiang et al. [19]: 4� (7.10) S′′′ where S′′′ is the specific surface area. The Forchheimer coefficient CF can be calculated with [17] 1.75 (7.11) Cf = √ 3 150 � 2 dp =

The energy conservation inside the porous particle takes nearly the same form as that in the surrounding gas phase in Eq. (7.4). An additional source term is added for the presence of the intrinsic chemical reaction. The equation takes the form [16] ∂T �P cp,P + � cp u ⋅ ∇T = ∇ ⋅ (�P ∇T) + ΔHC RVC (7.12) ∂t where ΔHC = 14.4 MJ kg−1 is the reaction enthalpy of the Boudouard reaction. The transport properties �P , cp,P , and �P are the mean values of the transport properties of the solid and the gas phase. They are given by �P = � � + (1 − �)�S

(7.13)

where � stands for the transport properties �, �, and cp . The term � denotes the properties of the gas phase, and �S denotes the properties of the solid phase. The species conservation inside the porous particle is governed by (Nield and Bejan [16]) ��

∂Yi + � u ⋅ ∇Yi = ∇ ⋅ (� Deff ∇Yi ) + RVi ∂t

(7.14)

where RVi is the species mass source term of the chemical reaction, and Deff is the effective diffusion coefficient.

7.1

Interface and Porosity Tracking for a Moving Char Particle

7.1.2.4 Boundary Conditions at the Particle Surface

The boundary conditions at the particle surface are affected by the heterogeneous reaction and the heat exchange through radiation. The energy balance consists of the terms for the heat conduction towards the gas and towards the solid particle and the terms for the reaction and the radiative heat transfer. The balance is given by the following equations [20]: ∂T || ∂T || 4 − �P = ΔHC RSC + �S �(T∞ − TS4 ) (7.15) � | ∂n |g ∂n ||p where RSC is the surface carbon consumption rate, n is the outward-pointing surface normal, �S is the emissivity, and � = 5.670 ⋅ 10−8 W m−2 K−4 is the Stefan–Boltzmann constant. The emissivity is assumed to be equal to 1. The mass fluxes at the particle surface are balanced by the diffusive and convective fluxes and the production and consumption rates of the gas species during the surface reaction [20]: ∂Y | ∂Yi || �D i || − �Deff − ṁ ′′C Yi,S = RSi (7.16) ∂n |g ∂n ||p ṁ ′′C = RSC

(7.17) RSi

is the production or where i denotes the chemical species CO and CO2 , and consumption rate of these species during the surface chemical reaction. 7.1.2.5 Reaction Kinetics

The species production and consumption rates RSi are given by Turns [21] RSi = �

�i Mi k Y MCO2 r CO2

(7.18)

where �i is the stoichiometric coefficient in the chemical equation. The stoichiometric coefficient of the reactants is negative and that of the product is positive. The rate constant kr is given by the Arrhenius equation Er

kr = Ar T nr e− RT

(7.19)

where Ar is the pre-exponential factor, nr is the temperature exponent, Er is the activation energy, and R is the universal gas constant. The values of these parameters are given in [22], with Ar = 4.605 m K−1 s−1 , nr = 1, and Er = 1.751 × 105 J mol−1 . The production and consumption rates of the intrinsic reaction inside the particle can be calculated from the values of RSi with [18, 19] RVi = S′′′ RSi

(7.20)

The volume-specific surface area S′′′ can be calculated with the random pore model [23] and depends on the carbon content XC : √ (7.21) S′′′ (XC ) = S0′′′ (1 − XC ) 1 − � ln(1 − XC )

175

7 Interface Tracking During Char Particle Gasification

120 000 100 000 S′′′ (m2 m–3)

176

80 000 60 000 40 000 20 000 0

S′′′ = S′′′(XC) 0

0.2

0.4 0.6 XC (–)

0.8

1

Figure 7.2 Plot of the volume-specific surface area S′′′ (XC ) with the parameters � = 5 and S′′′ = 105 m2 m−3 . 0

The volume-specific surface area and the structural parameter are set to � = 5 and S0′′′ = 105 m2 m−3 . A plot of Eq. (7.21) with the above parameters is given in Figure 7.2. It shows that the volume-specific surface area increases to a maximum at about XC = 0.26, and then it starts to decrease. 7.1.2.6 Change of Porous Structure and Particle Shape

The carbon consumption changes the shape and structure of the particle. Two effects occur during this process. On one hand, carbon is consumed at the particle surface; on the other, carbon is consumed inside the particle by the intrinsic reaction. The consumption at the particle surface leads to a movement of the surface in the normal direction. This can be described by the surface velocity vS , which is given by vS = n

RSC �S (1 − XC )

(7.22)

The vector n is the outward-pointing normal vector, �S is the density of the solid face, and XC is the carbon content. It should be noted that the term RSC is negative as a result of the carbon consumption. The carbon consumption inside the particle is expressed by the change in the carbon content XC , the value of which lies between 0 for no consumption and 1 for total consumption. The change in the carbon content can be described by Qixiang et al. [19] RV ∂XC = C (7.23) ∂t �S,0 where �S,0 is the initial particle density. The intrinsic carbon consumption leads to an increase in the local particle porosity, which can be calculated with � = XC + (1 − XC ) �0 where �0 is the initial porosity.

(7.24)

7.1

Interface and Porosity Tracking for a Moving Char Particle

7.1.2.7 Transport Properties

The dynamic viscosity �, the specific heat capacity cp , and the thermal conductivity � are assumed to be temperature dependent. For the pure gas species nitrogen N2 , carbon monoxide CO, and carbon dioxide CO2 , the corresponding temperaturedependent properties are taken from Bride et al. [24]. The dynamic viscosity � of the gas mixture is calculated according to Wilke [25]: ∑ Xi �i �= (7.25) ∑ j Xj �ij i and [ ( ) 1 ( M ) 1 ]2 2 4 � j 1 + �i M j

�ij =

√ ( 2 2 1+

i

Mi Mj

)1

(7.26)

2

where Xi is the mole fraction of species i, Mi is the molar mass of species i, and �i is the viscosity of the pure species i. The specific heat capacity cp and the thermal conductivity � of the gas mixture are calculated with ∑ cp = Yi cp,i (7.27) i

and �=

∑

Y i �i

(7.28)

i

where cp,i and �i are the specific heat capacity and the thermal conductivity of the pure species i, respectively, and Yi is the mass fraction of species i. The density of the gas mixture is calculated with the ideal gas law [21] p (7.29) �= RT M where p is the pressure, R is the gas constant, T is the temperature, and M is the mean molar weight of the mixture. To calculate the diffusion coefficient D, it is assumed that the Lewis number Le = �∕�cp D is unity [21]. Thus the diffusion coefficient D can be calculated as � D= (7.30) cp � where � is the thermal conductivity, cp is the specific heat capacity, and � is the density of the gas mixture. The diffusion coefficient is therefore the same for every gas species. In the porous matrix, the effective diffusion coefficient is calculated as [18] � Deff = D (7.31) � where � is the tortuosity factor of the solid. A good assumption for the tortuosity factor is [26] � = �−1 . It should be noted that Kundsen diffusion is not taken into account.

177

178

7 Interface Tracking During Char Particle Gasification

7.1.3 Numerics

The governing equations of the gas flow surrounding the particle and those inside the particle are very similar. In the surrounding gas flow, the porosity � = 1 and there is no source term from the chemical reactions. Under such circumstances, Eqs (7.7), (7.8), (7.12), and (7.14) reduce to their free gas flow equivalents (7.2)–(7.5), respectively. This means that only one set of equations needs to be discretized. The finite volume method is used to discretize the governing equations, and the central difference scheme is used to discretize the diffusive terms. The convective terms are discretized with a combination of the upwind and the central difference scheme. For time discretization, the implicit Euler method is used. The Semi-Implicit Method for Pressure Linked Equations (SIMPLE) algorithm is used for coupling the velocity and pressure in a collocated variables setting. This setting is stabilized with the Rhie and Chow [27] method; for details see Ferziger and Peri´c [13]. To solve the matrices, the Strongly Implicit Procedure (SIP) algorithm by Stone [28] is used in the parallelized form given in Dierich and Nikrityuk [29]. The finite volume discretization is carried out on a local refined Cartesian grid. The full grid is shown in Figure 7.3a. This means that the particle surface and the grid do not coincide, as shown in Figure 7.3b. The surface of the particle is represented by marker points, which form a polygon. The marker points should have a uniform distribution on the interface, with a similar distance between two adjacent points. If the distance is too large, the resolution of the interface and its movement will be too low. By contrast, if the distance is too close, the management of the marker points becomes more complicated. The distance between the marker points is therefore chosen to take values between 0.25 and 0.75 of the grid width inside the particle. The advantage of this method is that the particle can change its shape during the conversion process. It also brings the challenge

4 0.1 Y (mm)

Y (mm)

2 0

0

–2 –0.1 –4 –4 (a)

–2

0

2

4

X (mm)

6

8

10

–0.1 (b)

0

0.1

X (mm)

Figure 7.3 (a) Computational grid with 300 × 250 cells and (b) zoomed view of the particle with about 50 cells on the diameter.

7.1

Interface and Porosity Tracking for a Moving Char Particle

of dealing with finite-volume cells that belong to both the porous particle and the surrounding gas phase. To quantify this segmentation of the cells, the volume fraction of gas � is introduced. It represents the ratio between the surrounding gas phase and the particle in each cell, with � = 1 corresponding to a gas-phase-only cell and � = 0 corresponding to a particle-only cell. To calculate the volume fraction of gas � in each cell, the Sutherland–Hodgman clipping algorithm and the shoelace algorithm are used [30, 31]. These are well-known algorithms from computer graphics which clip a polygon against a rectangle. In our case, the polygon is the particle and the rectangle is the finite-volume cell. With this algorithm, we can also calculate the surface of the particle in each cell. This will be important for the source terms on the particle interface. The interface cells with 0 < � < 1 need special treatment. The main idea is to treat the interface cells like particle cells but with a higher porosity. The mean porosity of an interface cell can be calculated with � = � + (1 − �) ⋅ (XC + (1 − XC ) �0 )

(7.32)

This has an influence on all equations that depend on the porosity. They are calculated according to the porosity given above. The interface cells are not fully covered with the particle. For this reason, the chemical source terms RVC in the mass conservation equation (7.8), ΔHC RVC in the energy conservation equation (7.12), and Ri in the species conservation equation (7.14) are weighted with a factor of 1 − � to take this into account. The integration of the boundary conditions (7.15) and (7.16) leads to special source terms in the interface cells. The area AS of the interface between the gas phase and the particle is already known from the Sutherland–Hodgman clipping algorithm. The source terms added to the integrated energy equation (7.12) are 4 − TS4 )) AS (ΔHC RSC + �S �(T∞

(7.33)

There are also source terms added to the integrated species equation (7.14). They take the form AS RSi

(7.34)

The chemical reaction also leads to an additional mass source term for the gas species. This source term is given by AS ṁ ′′C = AS RSC

(7.35)

These source terms are incorporated into the corresponding conservation equations. The motion of the particle interface and the carbon consumption inside the particle are calculated explicitly. After the convergence of the conservation equations, the new particle interface is calculated. To do so, the positions of the marker points are located on the computational grid. Each marker point is located in one interface cell. Multiple marker points can also be located in one interface cell. The local carbon conversion rate of the interface RSC in this cell is used to calculate the interface velocity with Eq. (7.22). The interface normal of one marker point

179

180

7 Interface Tracking During Char Particle Gasification

is calculated with the positions of the neighboring marker points. The normals of the adjacent edges are weighted equally. During the conversion process, the particle size shrinks and the marker points move closer together. If the distance between two marker points becomes smaller than 0.25 times the local grid width, one marker point is removed from the interface. This can lead to the situation where two marker points have a distance larger than 0.75 times the local grid width. In this case, an additional point is added in the center between them. The removal of a grid point leads to a convex particle losing its mass. The particle stays convex in all simulations. The mass loss depends on the number of marker points and increases as the particle size decreases. The overall mass loss through this effect is 1. In this case, the mass transport limits the conversion process and the reaction is said to be diffusioncontrolled. In the opposite case, if the mass transport rate is fast compared to the chemical reaction rate, the Damköhler number DaII < 1. In this case, the chemical reaction rate limits the conversion process and the reaction is said to be kinetically controlled. The second dimensionless number we like to introduce is the Thiele modulus Th [32]. The Thiele modulus describes the ratio of the chemical reaction

7.1

Interface and Porosity Tracking for a Moving Char Particle

rate and the diffusive mass transport inside a porous particle: Th =

Reaction rate Diffusive mass transport rate

It can be calculated with √ kr S′′′ Th = r Deff

(7.38)

(7.39)

where r is the radius of the particle, k r is the rate constant, S′′′ is the volumespecific surface area, and Deff is the effective diffusion coefficient inside the porous particle. If the chemical reaction rate is fast compared to the diffusive mass transport rate, the Thiele modulus Th > 1. In this case, the diffusive mass transport is the limiting factor. In the opposite case, if the diffusive mass transport is fast in comparison to the chemical reaction rate, the Thiele modulus Th < 1. In this case, the chemical reaction rate is the limiting factor. From the Thiele modulus, the effectiveness factor �A can be derived, which is given by 3 (Th coth Th − 1) (7.40) �A = Th2 It describes the ratio of the conversion rate with finite diffusion mass transport and infinitely fast diffusion mass transport. Thus it measures the limitation due to the diffusive mass transport. A plot of the Damköhler number DaII and the Thiele modulus Th of the particles considered for Re = 5 and Re = 40 is given in Figure 7.4. Some values of the Damköhler numbers DaII , the Thiele modulus Th, and the effectiveness factor �A are also listed in Table 7.1. It can be seen that, in the temperature region studied, from T∞ = 2000 to T∞ = 3000, and the Reynolds number region Re = 1 to Re = 40, the Damköhler number is between 0.01 and 10. So we have a transition from a kinetically controlled regime to a diffusion-controlled regime. This

101

DaII (−) Th (−)

100 10−1 10−2 Th Re = 5, DaII Re = 40, DaII

10−3

10−4 1400 1600 1800 2000 2200 2400 2600 2800 3000 T (K) Figure 7.4 Plot of the Thiele modulus Th and the second Damköhler number DaII for Re = 5.40.

181

182

7 Interface Tracking During Char Particle Gasification

Table 7.1 Values of Damköhler numbers DaII , Thiele modulus Th, and effectiveness factor �A for different Reynolds numbers and ambient temperatures. Re

T∞ (K)

DaII

Th

�A

5 5 40 40

2000 2600 2000 2600

0.18 7.00 0.06 0.71

0.95 3.61 0.95 3.61

0.95 0.60 0.95 0.60

transition is not complete because all these values are still close to 1. The Thiele modulus is very close to 1. Thus the chemical reactions inside the particle are controlled equally by the chemical reaction rate and the diffusive mass transfer. To analyze the carbon consumption, we introduce three integral values: the total carbon consumption rate on the particle surface ṁ C,S ; the total carbon consumption rate inside the particle ṁ C,V ; and the total carbon consumption rate ṁ C . They are defined by ṁ C,S =

∫Γ

RSC dS,

ṁ C,V =

∫ G1

RVC dV

(7.41)

(a)

20 18 16 14 12 10 8 6 4 2 0

60

Re = 5, mC,S Re = 5, mC,V Re = 40, mC,S Re = 40, mC,V

Re = 5, mC,S Re = 5, mC,V Re = 40, mC,S Re = 40, mC,V

50 mC (mg s−1)

mC (mg s−1)

and ṁ C = ṁ C,S + ṁ C,V , where Γ is the surface of the particle and G1 is the particle. Figure 7.5 shows ṁ C,S and ṁ C,V for Re = 5 and Re = 40 for an ambient temperature T∞ = 2000 K in Figure 7.5a and T∞ = 2600 K in Figure 7.5b. In these cases, the carbon consumption rate inside the particle ṁ C,V is larger than that on the particle surface ṁ C,S . At the end of the conversion process, the situation changes and the carbon consumption on the particle surface ṁ C,S becomes larger. As shown in Table 7.1, the effectiveness factor �A is still high and the Damköhler number DaII is not larger than 7. For this reason, the reaction surface plays an important role in the reaction rate. The surface inside the particle is larger than the outer surface of the particle. Thus the carbon consumption inside the particle ṁ C,V is greater. Both

40 30 20 10

0

1

2

3 t (s)

4

0

5 (b)

0

0.2 0.4 0.6 0.8

1

1.2 1.4 1.6 1.8

t (s)

Figure 7.5 Carbon mass rate ṁ C,S from the surface and carbon mass rate ṁ C,V from the intrinsic reaction for Re = 5 and Re = 40 at (a) T∞ = 2000 K and (b) T∞ = 2600 K.

7.1

Interface and Porosity Tracking for a Moving Char Particle

183

consumption rates ṁ C,V and ṁ C,V start at a maximum, and the rates ṁ C,V reach a second maximum. The reason for the first maximum is the initial conditions. The temperature of the particle starts with the ambient, and the endothermic chemical reaction lowers the particle temperature in the beginning. This also decreases the chemical reaction rate and explains the first maximum. The second maximum in ṁ C,V is caused by the random pore model. The volume-specific surface S′′′ reaches a maximum at about XC = 0.26. This leads also to a maximum in the reaction rate. The carbon consumption on the particle surface ṁ C,S stays nearly constant over a long period of time. This is caused by the slow decrease in the particle radius and the particle surface at the beginning of the reaction. Figures 7.8 and 7.9 show that the decrease is very slow at the beginning. The total carbon consumption rate ṁ C for different cases is plotted in Figures 7.6 and 7.7. The differences between the rates in Figure 7.6 are less than in Figure 7.7. Thus the consumption rate ṁ C depends more on the temperature than on the Reynolds number. This is in accordance with the kinetically controlled or transition regime in the simulations. The total carbon consumption rates in Figure 7.6a have a second maximum. The reason for this is the second maximum in the carbon

mC (mg s−1)

15

mC (mg s−1)

Re = 1 Re = 5 Re = 20 Re = 40

20

10 5 0

0

1

2

3

(a)

4

5

6

7

t (s)

90 80 70 60 50 40 30 20 10 0

Re = 1 Re = 5 Re = 20 Re = 40

0

0.5

1

1.5

(b)

2

2.5

t (s)

Figure 7.6 Total carbon mass rate ṁ C at (a) T∞ = 2000 K and (b) T∞ = 2600 K for the Reynolds numbers Re = 1, 5, 20, and 40.

80

50 40 30 20

(a)

100 80 60 40 20

10 0

T∞ = 2000 K T∞ = 2300 K T∞ = 2600 K T∞ = 3000 K

120 mC (mg s−1)

60 mC (mg s−1)

140

T∞ = 2000 K T∞ = 2300 K T∞ = 2600 K T∞ = 3000 K

70

0

1

2

3 t (s)

4

5

0

6

0

0.5

1

1.5

(b)

Figure 7.7 Total carbon mass rate ṁ C for (a) Re = 5 and (b) Re = 40 at T∞ = 2000, 2300, 2600, and 3000 K.

2

2.5 t (s)

3

3.5

4

4.5

184

7 Interface Tracking During Char Particle Gasification

1.2 1

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

1

2

3

(a)

4

5

Re = 1 Re = 5 Re = 20 Re = 40

1

r ∗ (–)

0.8 r ∗ (–)

1.2

Re = 1 Re = 5 Re = 20 Re = 40

6

0

7

t (s)

0

0.5

1

(b)

1.5

2

t (s)

Figure 7.8 Dimensionless particle radius r∗ = r∕r0 at (a) T∞ = 2000 K and (b) T∞ = 2600 K for the Reynolds numbers Re = 1, 5, 20, and 40.

1.2 1

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

1

2

3 t (s)

4

5

T∞ = 2000 K T∞ = 2300 K T∞ = 2600 K T∞ = 3000 K

1

r ∗ (–)

r ∗ (–)

0.8

(a)

1.2

T∞ = 2000 K T∞ = 2300 K T∞ = 2600 K T∞ = 3000 K

0

6 (b)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

t (s)

Figure 7.9 Dimensionless particle radius r∗ = rr for the Reynolds numbers (a) Re = 5 and 0 (b) Re = 40 at T∞ = 2000, 2300, 2600, and 3000 K.

consumption rate inside the particle ṁ C,V as described above. However, this effect is not visible for all cases in ṁ C , as Figure 7.6b shows. Figures 7.8 and 7.9 show the development of the dimensionless particle radius r∗ = r∕r0 , where r0 is the initial radius and r is the radius of the cylinder which is volume-equivalent to the particle. This definition is necessary because the particle does not stay cylindrical during the conversion. If the particle’s radius reaches zero, the particle has reacted totally. The total reaction time is linked to the reaction rate during the conversion process. In accordance with the results of the reaction rate, the total reaction time depends more on the temperature than on the Reynolds number. Both an increase in the Reynolds number and an increase in the ambient temperature lead to a shorter total reaction time. The profile of the radius decrease depends mainly on the ambient temperature. Figure 7.9 shows that the radius decreases slowly in the beginning and quickly at the end of the conversion process for lower temperatures. For higher temperatures, the decrease is faster at the start compared to the total reaction time.

Interface and Porosity Tracking for a Moving Char Particle

1

1

0.9

0.9

0.8

0.8

ϕ (–)

ϕ (–)

7.1

0.7 0.6

0.4 0

1

2

3

4

5

6

Re = 1 Re = 5 Re = 20 Re = 40

0.5 0.4 0

7

t (s)

(a)

0.7 0.6

Re = 1 Re = 5 Re = 20 Re = 40

0.5

185

0.5

1

1.5

2

t (s)

(b)

1

1

0.9

0.9

0.8

0.8

ϕ (–)

ϕ (–)

Figure 7.10 Volume-averaged particle porosity � at (a) T∞ = 2000 K and (b) T∞ = 2600 K for the Reynolds numbers Re = 1, 5, 20, and 40.

0.7 0.6

0.4 0 (a)

0.6

T∞ = 2000 K T∞ = 2300 K T∞ = 2600 K T∞ = 3000 K

0.5

1

2

3 t (s)

4

5

0.7 T∞ = 2000 K T∞ = 2300 K T∞ = 2600 K T∞ = 3000 K

0.5 0.4 6

0 (b)

0.5

1

1.5

2

2.5

3

t (s)

Figure 7.11 Volume-averaged particle porosity � for the Reynolds numbers (a) Re = 5 and (b) Re = 40 at T∞ = 2000, 2300, 2600, and 3000 K.

The volume-averaged particle porosity � is plotted in Figures 7.10 and 7.11. The development of the volume-averaged particle porosity starts with the initial porosity at � = 0.4 and ends when the porosity reaches � = 1. Both figures show that for low temperatures the porosity increase is nearly linear. For higher temperatures, the porosity increase starts with a linear profile and accelerates during the conversion process. The flow structure inside and around the particle is shown in Figures 7.12 and 7.13. The vector plots are normalized to the ambient flow velocity u∞ . Figure 7.12 shows a detailed view of the flow structure inside the particle. For the Reynolds number Re = 1 and the ambient temperature T∞ = 2600 K in Figure 7.12d, the Stefan flow from inside the particle and from the particle surface is clearly visible. For lower temperatures, as in Figure 7.12a, the Stefan flow is much smaller because of the slower reaction rate. For higher Reynolds numbers, the reaction rate does not increase at the same speed as the Reynolds number. Thus, in Figure 7.12c,f, the Stefan flow increases in terms of the absolute value compared to Figure 7.12a,d, respectively. Relative to the ambient flow velocity, the Stefan flow decreases. The surrounding flow is also affected by the Stefan flow from inside the particle and

3.5

4

4.5

186

7 Interface Tracking During Char Particle Gasification

0

–0.1

0

u∞

Re = 1, T∞ = 2000 K

0.1

–0.1 u∞

Re = 5, T∞ = 2000 K

–0.1

0

0

X (mm)

u∞

Re = 1, T∞ = 2600 K

0

–0.1

0.1

X (mm)

(e)

0

–0.1 –0.1

0.1

u∞

0.1

–0.1

–0.1

0.1

Re = 40, T∞ = 2000 K

(c)

Y (mm)

Y (mm)

0

0 X (mm)

0.1

0.1 Y (mm)

0 X (mm)

(b)

0

–0.1 –0.1

0.1

X (mm)

(d)

0

–0.1

–0.1

(a)

0.1 Y (mm)

0.1 Y (mm)

Y (mm)

0.1

Re = 5, T∞ = 2600 K

u∞

0 X (mm)

(f)

0.1 u∞

Re = 40, T∞= 2600 K

Figure 7.12 Vector plot of the velocity field at the particle for different temperatures and Reynolds numbers at t = 0.2 s.

from the particle surface as shown in Figure 7.13. The influence strongly depends on the value of the Stefan flow compared to the ambient flow velocity. Figure 7.13d shows a case with a relatively high Stefan flow. A relatively small Stefan flow as in Figures 7.13c,f has nearly no effect on the surrounding flow field. Figures 7.14–7.17 show the contour plots of the CO2 mass fraction YCO2 inside and around the particle and the porosity � inside the particle. Let us first focus on the CO2 mass fraction. The CO2 mass fraction inside the particle is smaller than that in the ambient gas phase. This is caused by the consumption of CO2 due to the chemical reaction. In all cases, the CO2 mass fraction inside the particle increases from 𝜏P = 0.1 to 𝜏P = 0.8. This is caused by the decreasing carbon consumption rate, which is equivalent to a decrease in the CO2 consumption rate during the conversion process, as shown above. The increase in the Reynolds number from Figure 7.14 to Figure 7.16 and from Figure 7.15 to Figure 7.17 leads to a smaller boundary layer. Because of the increased mass transport, the CO2 mass fraction inside the particle is also higher. The increase in the ambient temperatures from 2000 to 2600 K decreases the CO2 mass fraction inside the particle because of the higher reaction rate. This can be seen by comparing Figures 7.14 and 7.15 or Figures 7.16 and 7.17. The porosity distribution inside the particle mainly depends on the ambient temperature. Comparison of Figure 7.14 with Figure 7.15 shows

0.4

0.2

0.2

0.2

0

0

0.2

X (mm)

u∞

Re = 1, T∞ = 2000 K

0

0.2

0

–0.4 –0.4 –0.2

0.4 u∞

X (mm) Re = 5, T∞ = 2000 K

(b)

0.4

0.2

0.2

0.2 Y (mm)

0.4

–0.4 –0.4 –0.2

0

0.2

X (mm)

–0.4 –0.4 –0.2

0.4 u∞

Re = 1, T∞ = 2600 K

0

0.2

u∞

0

–0.4 –0.4 –0.2

0.4

X (mm)

(e)

0.4

–0.2

–0.2

–0.2

0.2

Re = 40, T∞ = 2000 K

(c)

0

0

X (mm)

0.4

0

187

–0.2

–0.4 –0.4 –0.2

0.4

Y (mm)

Y (mm)

0 –0.2

–0.4 –0.4 –0.2

(a)

Y (mm)

0.4

–0.2

(d)

Interface and Porosity Tracking for a Moving Char Particle

0.4

Y (mm)

Y (mm)

7.1

Re = 5, T∞ = 2600 K

u∞

0

0.2

X (mm)

(f)

0.4 u∞

Re = 40, T∞ = 2600 K

Figure 7.13 Vector plot of the velocity field around the particle for different temperatures and Reynolds numbers at t = 0.2 s.

that the porosity differences inside the particle are higher for higher temperatures. The differences for different Reynolds numbers between Figures 7.14 and 7.16 are very low. To analyze the temperature profile inside the particle, we introduce two average temperatures: the average surface temperature TS , and the average particle temperature TP . Figure 7.18 shows that the average surface temperature TS is always higher than the average particle temperature TP . This effect is caused by the endothermic reaction inside the particle. The particle cools down as a result of the chemical reaction and heat is transported to the particle by radiation and diffusive transport to the surface. Thus the surface is the hottest part of the particle. The difference between the ambient temperature and the average particle temperature increases as the ambient temperature increases, as shown in Figure 7.18. The increase in the Reynolds number only has a minor effect on the average surface temperature and the average particle temperature. Figures 7.19 and 7.20 show a very detailed development of different parameters along the centerline of the particle. Figures 7.19b and 7.20b show that the lowest porosity is at the center of the particle. This does not mean that the reaction rate always reaches a minimum at the center of the particle. Figure 7.19c shows that, in this case, the reaction rate maximum moves from the boundary of the particle to the center of the particle. The reason for this is the specific surface area S′′′ and

7 Interface Tracking During Char Particle Gasification 0.4

0.4

0.2 0.

0.2 0.8

7

0.

0 –0.2

–0.2 –0.4 –0.4 –0.2

0

(a)

0.2

0.2

–0.4 –0.4 –0.2

0.4

(c)

X (mm)

0

0.2

0.4

X (mm) 0.1

0.1

0.9 0.8

0.

5

0.7

0

(b)

X (mm) 0.1

–0.2

–0.4 –0.4 –0.2

0.4

0

85 0.

0

Y (mm)

8

7 0. 0 .6

Y (mm)

0.2 Y (mm)

0.4

0.8

188

0.05

5

0.

5 0.

0.05

5 0.8

Y (mm) 8

(e)

X (mm)

–0.05

0.

–0.1 –0.1 –0.05

0.1

0.

9

–0.05

9

0

(d)

0. 9

0

0.9

Y (mm) 0.7

7

0.6

0.8

0 0.

5

0.4

–0.05

7

0

–0.1 –0.1 –0.05

0.05 0.

0.4

0.9

0. 6

0.05 Y (mm)

8 0.

0

0.05

–0.1 –0.1 –0.05

0.1

(f)

X (mm)

0

0.05

0.1

X (mm)

Figure 7.14 Contour plots of (a)–(c) the CO2 mass fraction YCO2 and (d)–(f ) the particle porosity � at 𝜏P = 0.1, 0.5, and 0.8 for Re = 5 and T∞ = 2000 K. 0.4

0.4

0.8

0.

8

0.4

0.8

0.7

0.2

0.5

–0.2

0.6

0.7

0.2

0

0.3 0.6

0.5

–0.2

0.7

0 –0.2

0.7

0.5

0.6

0.4

0.3

0

Y (mm)

0.5

7 0.

Y (mm)

0.2

0.6

0.4

Y (mm)

0.2

0.6

0.7

0.8

8

0. 0.

8

–0.4 –0.4 –0.2

0.2

0

(a)

0.4

9

0.4

X (mm)

0.

0. 5

9 0 .7

Y (mm)

0

7 0.

Y (mm)

7

0.8

0.05

0.6

0.8

–0.1 –0.1 –0.05

0.2

9

0.7

–0.05

0.

9

0.9

0.8

0

0.6

–0.05

0.

0.4

5

0

0.1 0.8

0.05

0.9

0.8

0.8

0.6

(d)

(c)

X (mm)

0.45

0 –0.05

–0.4 –0.4 –0.2

0.4

0.

0.05

0.2

0.1

0.6

0.7

0.

0

(b)

X (mm) 0.1

Y (mm)

–0.4 –0.4 –0.2

0.5

0 X (mm)

0.05

–0.1 –0.1 –0.05

0.1

(e)

0 X (mm)

0.05

–0.1 –0.1 –0.05

0.1

(f)

0

0.05

X (mm)

Figure 7.15 Contour plots of (a)–(c) the CO2 mass fraction YCO2 and (d)–(f ) the particle porosity � at 𝜏P = 0.1, 0.5, and 0.8 for Re = 5 and T∞ = 2600 K.

0.1

7.1

0.6

–0.2

Y (mm)

0.8

0

0.

85

0.8

7

0.

Y (mm)

0.85

0

–0.4 –0.4 –0.2

0.4

(b)

X (mm)

(c)

X (mm)

0

0.2

0.4

X (mm)

8

0.

7

0.1

0.1

0.90.6

0.8

0

0.05

0

0.9

0.

Y (mm)

0.05

45

0.9 0.8 5

0 0.8

5

0.9

–0.05

0. 5

0.

7

–0.05

Y (mm)

45

0.05

–0.05

–0.4 –0.4 –0.2

0.4

0.7

0.5

0. 8

0.7

0.2

0.9

0.1

0

0.8

(a)

0.2

7

0

0.9 0.

–0.4 –0.4 –0.2

0.89

0.8

0 –0.2

–0.2

0.

Y (mm)

0.2

0.2

0.2

189

0.4

0.4

0.4

Y (mm)

Interface and Porosity Tracking for a Moving Char Particle

0.6

–0.1 –0.1 –0.05

(d)

0

0.05

–0.1 –0.1 –0.05

0.1

0

(e)

X (mm)

0.05

–0.1 –0.1 –0.05

0.1

(f)

X (mm)

0

0.05

0.1

X (mm)

Figure 7.16 Contour plots of (a)–(c) the CO2 mass fraction YCO2 and (d)–(f ) the particle porosity � at 𝜏P = 0.1, 0.5, and 0.8 for Re = 40 and T∞ = 2000 K.

0.2 Y (mm)

0.2

0.5 0.6

0

0.7

0.8

0

0.8

–0.4 –0.4 –0.2

(a)

0

0.2

–0.4 –0.4 –0.2

0.4

0.1 9

0.

Y (mm) X (mm)

0 –0.05

0. 9 0.6

0

0.7

0.4

X (mm)

0.9

0.05

6

Y (mm)

5

0.8

0.4

5

0.2

0.9

0.

0.

0.8

Y (mm)

0.7

0.05

8

(d)

(c)

0

0.1 0.8

0.

–0.1 –0.1 –0.05

–0.4 –0.4 –0.2

0.4

0.1

0.7 0.5

0 –0.05

0.2

X (mm)

6

0.

0

(b)

X (mm)

0.8

0 –0.2

–0.2

–0.2

0.05

0.2

0.8 0. 7

0.5

0.7

0.3

Y (mm)

0.8

0.7

0.6

0.2 Y (mm)

0.4

0.4

0.4

0.8 0.7

0 0.7

6

0.9

0.7

–0.1 –0.1 –0.05

0

8 0.9 0.

–0.05

0.

0.05

0.1

(e)

X (mm)

0.05

–0.1 –0.1 –0.05

0.1

(f)

Figure 7.17 Contour plots of (a)–(c) the CO2 mass fraction YCO2 and (d)–(f ) the particle porosity � at 𝜏P = 0.1, 0.5, and 0.8 for Re = 40 and T∞ = 2600 K.

0 X (mm)

0.05

0.1

190

7 Interface Tracking During Char Particle Gasification

2600

2300 2200

2400 2300 2200

2100

2100

2000

2000

1900

1900

1800

0

1

2

(a)

3

4

TS, T∞ = 2000 K TP, T∞ = 2000 K TS, T∞ = 2300 K TP, T∞ = 2300 K TS, T∞ = 2600 K TP, T∞ = 2600 K

2500

T (K)

2400 T (K)

2600

TS, T∞ = 2000 K TP, T∞ = 2000 K TS, T∞ = 2300 K TP, T∞ = 2300 K TS, T∞ = 2600 K TP, T∞ = 2600 K

2500

5

1800

6

0

1

2

(b)

t (s)

3

4

5

6

t (s)

Figure 7.18 Plot of the average particle temperature TP and the average particle surface temperature TS for (a) Re = 5 (b) Re = 40 at T∞ = 2000, 2300, and 2600 K. 2000 1 0.9

1900

1800

ϕ (–)

1850

τP = 0.1 τP = 0.3 τP = 0.5 τP = 0.7 τP = 0.8 τP = 0.9

0.5 0.4 0.1

0.2

–0.10 −0.05

0.3

0

(b)

0.05 0.10 0.15 0.20 X (mm)

120 000

700 mV (kg s−1 m−3)

0.7 0.6

1750 –0.4 –0.3 –0.2 –0.1 0 (a) X (mm)

τP = 0.1 τP = 0.3 τP = 0.5 τP = 0.7 τP = 0.8 τP = 0.9

600 500 400 300 200

τP = 0.1 τP = 0.3 τP = 0.5 τP = 0.7 τP = 0.8 τP = 0.9

100 000 80 000 60 000 40 000 20 000

100 0

0 −0.10 −0.05 0

(c)

τP = 0.1 τP = 0.3 τP = 0.5 τP = 0.7 τP = 0.8 τP = 0.9

0.8

S (m2 m−3)

T (K)

1950

−0.10 −0.05

0.05 0.10 0.15 0.20 0.25 X (mm)

(d)

0

0.05 0.10 0.15 0.20 X (mm)

Figure 7.19 Snapshots of (a) the temperature T, (b) the porosity �, (c) the intrinsic reaction rate ṁ ′′′ , and (d) the specific reactive surface area S′′′ along the centerline of the particle V and the numerical grid for Re = 5 and T∞ = 2000 K at different time points 𝜏P = t∕tP .

the temperature inside the particle. Figure 7.19d shows that the specific surface area S′′′ reaches a maximum at the center of the particle and Figure 7.19a shows that the temperature at the center of the particle increases during the conversion. Both effects lead to an increase of the reaction rate at the center of the particle. Figure 7.20 shows a case which is shifted more toward the diffusion-controlled

7.1

Interface and Porosity Tracking for a Moving Char Particle

191

2600 1

2500

0.9 ϕ (–)

2300

τP = 0.1 τP = 0.3 τP = 0.5 τP = 0.7 τP = 0.8 τP = 0.9

2200 2100

0.7 0.6 0.5

2000 –0.4 –0.3 –0.2 –0.1 0 (a) X (mm)

0.4 0.1

0.2

–0.10 −0.05

0.3

0

(b)

0.05 0.10 0.15 0.20 X (mm)

120 000

6000 mV (kg s−1 m−3)

τP = 0.1 τP = 0.3 τP = 0.5 τP = 0.7 τP = 0.8 τP = 0.9

0.8

τP = 0.1 τP = 0.3 τP = 0.5 τP = 0.7 τP = 0.8 τP = 0.9

5000 4000 3000

τP = 0.1 τP = 0.3 τP = 0.5 τP = 0.7 τP = 0.8 τP = 0.9

100 000 S (m2 m−3)

T (K)

2400

2000

80 000 60 000 40 000 20 000

1000 0

0 −0.10 −0.05 0

(c)

0.05 0.10 0.15 0.20 0.25

−0.10 −0.05

(d)

X (mm)

0

0.05 0.10 0.15 0.20 X (mm)

Figure 7.20 Snapshots of (a) the temperature T, (b) the porosity �, (c) the intrinsic reaction rate ṁ ′′′ , and (d) the specific reactive surface area S′′′ along the centerline of the particle V and the numerical grid for Re = 40 and T∞ = 2600 K at different time points 𝜏P = t∕tP .

35

Unsteady, T∞ = 2000 K Steady, T∞ = 2000 K Unsteady, T∞ = 2600 K Steady, T∞ = 2600 K

mC (mg s–1)

30 25 20 15 10 5 0

0

2

4

6

8

10

t (s) Figure 7.21 Comparison of the total carbon mass rate ṁ C between unsteady and steady calculations for Re = 5.

regime. Figure 7.20c shows that the reaction rates at the center of the particle are much lower than that at the front of the particle. Figure 7.21 shows a comparison of the carbon mass rate between the unsteady and steady simulations. To study the influence of unsteady effects, steady simulations with a constant particle diameter were also performed. The reaction rates of the steady simulations are plotted at the same place where the unsteady

192

7 Interface Tracking During Char Particle Gasification

simulations reach this diameter. The comparison shows that the unsteady development effect on the simulation is negligible.

7.2 3D Interface Tracking for a Porous Char Particle in the Kinetic Regime 7.2.1 Problem Description

Char is a highly amorphous and porous structure. During the partial oxidation of a char particle, gas–solid reactions occur inside the pores. In order to describe the arising dynamics of the porosity, pore models have to be applied [33, 34]. However, the pore models used today were developed over three decades ago, and the development of the specific particle surface during the reaction is still not completely understood. To overcome this problem, we aim to track the growth of spherical pores inside a solid char particle due to chemical reactions on the pore surface. Further, the development of the specific surface area will be visualized. The porous particle is assumed to consist of carbon only and is located in a CO2 atmosphere (YCO2 = 0.9) with some inert nitrogen. The reactions on the particle surface occur in the kinetically controlled regime with a constant surface temperature of Ts = 1200 K, which is simultaneously the constant temperature of the surrounding gas and the particle. Additionally, the concentration of YCO2 at the particle surface is assumed to be constant because of the very high diffusion into the gas phase. Further, there is no fluid flow around the particle and in the pores. Thus, the change in the char mass is attributed to the change in the pore radius due to reactions on the surface of each pore inside the particle and on the outer particle surface:

ṁ ′′ =

dm′′ ∑ �s dVp = dt Ap dt p

(7.42)

Here, �s = 1.2 kg m−3 is the density of carbon, Vp is the volume, and Ap the surface of the pore p, where, as a special case, the outer particle surface can also be treated the same way. Generally, in the following, the subscript p denotes the specified variable for a single pore p. Thus in Eq. (7.42), the sum is calculated over all the pores. The following list summarizes all the assumptions made so far:

• • • • • • •

Kinetically controlled regime One reaction CO2 + C ⇌ 2CO The particle consists of pure carbon The pores and the particle are of spherical shape The atmosphere consists of CO2 and the inert gas N2 YCO2 = const, Ts = const at the particle surface Fluid motion through and around the particle is neglected

7.2

3D Interface Tracking for a Porous Char Particle in the Kinetic Regime

Discretizing Eq. (7.42), the volume change for each pore is written as

ΔVp = Δt

ṁ ′′p ⋅ Ap

(7.43)

�s

where Δt is the time step during which ΔVp is consumed and ṁ ′′p is the surfacial char-mass flux rate at each pore surface Ap . The surfacial char-mass flux is calculated as follows: ′′ ṁ ′′p = kCO

2

MC �Y MCO2 g CO2

(7.44)

Here, YCO2 is the particle-averaged mass fraction of species CO2 at the particle surface, which is constant. The density of the surrounding atmosphere �g can be determined according to the ideal gas law using the ambient pressure p = 101.325 kPa and the mixed molar mass of the surrounding gas Mg = 39.191 g mol−1 . MC and MCO2 are the molar mass of carbon and carbon dioxide, respectively. The kinetics of this surface-based heterogeneous reaction is taken from Turns [21, 35]: ′′ kCO = 4.016 ⋅ 108 e

− 29T790 s

2

m s−1

(7.45)

All hydrothermal properties are summarized in Table 7.2. Because of the assumptions made, Eq. (7.44) is constant. Hence, evaluating all the known terms, Eq. (7.44) yields a constant carbon mass flux per unit area: ′′ ṁ ′′p = kCO

2

MC �Y MCO2 g CO2

= 6.4952 ⋅ 10−4 kg m−2 s−1

(7.46)

Now, Ap is the only unknown remaining in Eq. (7.43). The calculation of Ap requires a reconstruction of the pore surfaces, which is done in Section 7.2.3. Finally, the complete right side of Eq. (7.43) allows the change in the pore volume ΔVp to be calculated for small time steps Δt for each single pore p. Similarly, the Table 7.2 phase.

Hydrothermal properties of a porous carbon particle and surrounding gas

Property

Value

Molar mass MC Molar mass MCO2 Molar mass Mg Ambient pressure p Surface temperature Ts CO2 concentration YCO2 Solid density �s

12.0107 g mol−1 44.0095 g mol−1 39.191 g mol−1 101.325 kPa 1200 K 0.9 1200 kg m−3

193

194

7 Interface Tracking During Char Particle Gasification

change in the pore volume can be described by the change in the pore radius rpi at time step i to a new radius rpi+1 at the subsequent time step i + 1. In Eq. (7.47), for each new time step, the pore radius rpi+1 is determined from the preceding pore radius rpi , the volume change ΔV that is calculated using Eq. (7.43), and the actual pore surface Aip . ( rpi+1 =

(rpi )3 +

3ΔVp (rpi )2

(

)1 3

Eq. (7.43)

(rpi )3 +

=

4π(rpi )2 − ΔAip

3Δt ⋅ ṁ ′′p (rpi )2

)1

3

(7.47)

�s Aip

In order to include the coalescence and intersection of neighboring pores, ΔAip is included in Eq. (7.47). There, ΔAip specifies the portion of the pore surface that disappeared as a result of its intersection with neighboring pores. More precisely, ΔAip is the difference in the case of a sphere with a diameter rpi and pore surface Aip for a pore p at the time step i; ΔAip = 4π(rpi )2 − Aip . It should be noted that the previously discussed model can also be applied to the porous particle surface. The particle shrinks, while at the same time the pores grow. At each new time step i + 1, the porous particle radius has to be calculated using ( rpi+1

=

(rpi )3

−

3ΔVp (rpi )2 4π(rpi )2 − ΔAip

(

)1 3

Eq. (7.43)

=

(rpi )3

−

3Δt ⋅ ṁ ′′p (rpi )2 �s Aip

)1

3

(7.48)

7.2.2 Porous Particle Description

In the previous section, both the porous particle and the pores themselves were assumed to be spherical in shape. Specifically, this corresponds to a microscale pore representation, where the open pore geometry is fully resolved. The distribution of pores inside the porous particle is defined using a packed bed. This packed bed was previously calculated using a DEM applied to the gravity-driven sedimentation of rigid spheres in a cylindrical cavity [36] (for details, see Chapter 3). Then, a sphere is cut from the packed bed and inverted as shown in Figure 7.22. Thus, the spatial distribution of the solid spheres from the packed bed is used to represent the pores (void fraction) inside the porous particle, leading to a foamlike structure. Figure 7.22 illustrates a sample of two particles with different initial pore structures. Different initial pore sizes can be obtained by changing the dispersity of the spheres applied in the DEM code. Further, the initial pore structure of the particle can be modified by stretching or neglecting individual pores (see Figure 7.22). The stretching, in particular, is necessary to obtain an open pore configuration without isolated pores. The relative pore number density for the polydisperse pores examined here is shown in Figure 7.23.

7.2

3D Interface Tracking for a Porous Char Particle in the Kinetic Regime

(a)

Initial porosity ε = 0.7

(b)

Initial porosity ε = 0.3

Figure 7.22 Illustration of porous char particles produced using a monodisperse and polydisperse packed bed. After a sphere is cut from this packed bed, the solid and fluid phases are inverted. Here, (a) and (b) represent different porosities.

Relative pore number density

1.4 1.3 1.2 1.1 1 0.9 0.8 0.7

0.92

0.96

1

1.04

1.08

Pore size in d Figure 7.23 Distribution of different pore sizes in the polydisperse pore configuration used. d represents the reference diameter used for the monodisperse pores.

195

196

7 Interface Tracking During Char Particle Gasification

7.2.3 Internal Surface Reconstruction

Solving Eq. (7.43) requires the internal pore surface Ap , and measurement of the error of Eqs (7.47) and (7.48) calls for the pore volume Vp . Generally, these quantities can be calculated using an analytical method. However, when multiple pores superimpose on each other, the analytical solution of the problem requires a comprehensive distinction in special cases. Otherwise, the quantities Ap and Vp can be reconstructed from a numerical grid. Using this method is more time consuming, but places no limitations on the amount of pores that are allowed to intersect with each other. Thus, in the following, an interface reconstruction method and a volume integration method are presented on a Cartesian grid. It should be noted that Eq. (7.43) requires only the size of the pore surface Ap and not its orientation. Hence, the algorithm developed has to focus only on the two values Ap and Vp . Initially, consider the volume integration for every pore p. Since the numerical grid is Cartesian, all cells are of the same size ΔV . Further, a local weight function �(c, p) is required, allocating to every cell c its weight 0 ≤ �(c, p) ≤ 1, which defines what percentage of cell c lies inside pore p. Using these definitions, the volume of each pore can be calculated by Vp =

∑

�(c, p)ΔV

(7.49)

c

This weight function �(c, p) has to be determined. The first step is to study each cell on the Cartesian grid to determine whether that cell c is mostly covered by a pore p. If it is, then this fluid cell is given a weight �(c, p) = 1. And if this cell c is located mostly outside the previously mentioned pore p, this solid cell gets the weight �(c, p) = 0. It should be noted that the implementation of the sum (7.49) can be accelerated if mostly cells c inside the corresponding pore p are tested, because for cells outside a pore, it is �(c, p) = 0. This allocation produces a stair-step or block-like structure on the grid, which leads to some errors when calculating Vp due to the rounding in the interface cells. Although a simple grid refinement would reduce these rounding errors, a more time-saving approach could concentrate on only refining the interface cells. Here, an adaptive spatial supersampling method is applied. Every interface cell is split into several subcells (see Figure 7.24), and a sample is taken from the center of each subcell. �(c, p) is then the mean of all samples taken from the subcells of a cell c. When evaluating Eq. (7.49), the volume Vp of a pore, resolved using 103 cells and 125 subcells at the interface, deviates from the actual volume by 0.9. This is explained by the small difference in initial pore sizes even for the polydisperse case. As stated above, the particle porosity has a major influence on the development of the specific surface area. Generally, a lower porosity leads to a smaller initial specific surface area S0′′′ . As seen in Figure 7.26b, for a char particle with initial porosity � = 0.21, the pore surface growth due to reactions dominates over the specific surface area development until a consumed carbon rate of X > 0.5. Then, at a particular value of X, the reaction rate of the particle reaches a maximum,

7.2

3D Interface Tracking for a Porous Char Particle in the Kinetic Regime

followed by a decrease in the specific surface area. This decrease in S′′′ ∕S0′′′ is explained by the large number of pore wall intersections due to the movement of the reacting pore surfaces, which reduces the surface area of the corresponding pores. For particles with a high initial porosity, there is no maximum reaction rate. Rather, the specific surface area for such particles decreases monotonically. Next, a minor influence of the pore structure on the specific particle surface development can be seen in Figure 7.26. While this influence is small when looking at the relative value S′′′ ∕S0′′′ shown there, the specific initial surface area S0′′′ generally increases for a larger number of pores n. The question now is whether the results in Figure 7.26 can be predicted by existing pore models. For this purpose, the recent predictions on the specific char surface development are compared with those of the grain pore model and the random pore model. The grain pore model [18] describes uniformly reacting porous pellets by S′′′ = S0′′′ (1 − X)m ,

m=

2 for spheres 3

(7.52)

The random pore model [38] assumes cylindrical pores within the solid phase: √ S′′′ = S0′′′ (1 − X) 1 − � ln(1 − X),

�=

4πL0 (1 − �0 ) (S0′′′ )2

(7.53)

In contrast to the grain pore model, the random pore model considers real pores of different sizes, where new intersections due to the pore growth are allowed. In Figure 7.27, both models are compared against the predicted specific surface development of a char particle with spherical pores. There, the structural parameter � in the random pore model is determined by a least-squares fit. A comparison of the recent predictions for the specific surface 1.6 1.4 1.2 S′″ / S′″ 0

1 0.8 0.6

Poly, ε = 0.7, n = 480 poly, ε = 0.3, n = 169 poly, ε = 0.2, n = 88 Bhatia, ψ = 1.42 Bhatia, ψ = 3.44 Bhatia, ψ = 8.5 Grain model

0.4 0.2 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Consumed carbon rate X Figure 7.27 Comparison of the specific surface area development with the random pore model [38]. � was estimated by a least-squares fit.

199

200

7 Interface Tracking During Char Particle Gasification

1.6 1.4

1 0.8 0.6

Poly, ε = 0.7, n = 480 Poly, ε = 0.3, n = 169 Poly, ε = 0.2, n = 88 Bhatia, ψ = 1.42 Bhatia, ψ = 3.44 Bhatia, ψ = 8.5

0.4 0.2 0

(a)

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Consumed carbon rate X

S′″ / S′″ 0

S′″ / S′″ 0

1.2

(b)

1.1 1 0.9 0.8 0.7 0.6 0.4 0.4 0.3 0.2 0.1

Poly, ε = 0.7, n = 480 Poly, ε = 0.3, n = 169 Poly, ε = 0.2, n = 88 Bhatia, ψ = 1.31 Bhatia, ψ = 2.14 Bhatia, ψ = 3.91

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Consumed carbon rate X

Figure 7.28 Influence of particle shrinking (reactions on external particle surface) on the specific surface area development (a). In (b), a correction factor was applied on �.

development with the random pore model shows good agreement for high particle porosities. For lower particle porosities �, there is a deviation between the two models. Especially for the case of � = 0.2, the specific surface area grows more slowly initially and reaches its maximum at a higher consumed carbon rate X than predicted by the random pore model. The grain pore model does not otherwise seem capable of describing the recent results. It should be noted that, when speaking about the specific surface area, until now only the interior particle surface was considered. In Figure 7.28a, the external particle surface has additionally been incorporated into the value of S′′′ ∕S0′′′ . It can be seen that this has a significant influence on the specific surface area development. Indeed, this additional feature could be included in the random pore model by rewriting it using a so-called adopted initial specific surface, where the outer particle surface is taken into account. Analytically, the specific outer particle ′′′ surface can be described by S0,out = 3∕r(1 − �0 ). Thus, the random pore model rewrites as √ 4πL0 (1 − �0 ) ′′′ S′′′ = S0,tot (1 − X) 1 − �tot ln(1 − X), �tot = (7.54) ′′′ (S0 + 3r (1 − �0 ))2 Finally, in Figure 7.28b, the improved random pore model has been applied. However, the curvature of the random pore model, which was developed for cylindrical pores, does not exactly match the polydisperse spherical pores modeled here. Here, further model investigations have to be performed, and a more universal pore model has to be formulated.

7.3 Conclusions

In the first part of this chapter, a 2D tracking algorithm was developed for the interface and the porosity of a moving and gasifying char particle. The model developed included the modeling of the flow inside the porous particle. Transient simulations

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8 Pseudo-Steady-State Approach for Carbon Particle Combustion/Gasification Matthias Kestel, Dmitry Safronov, Andreas Richter, and Petr A. Nikrityuk

... the agreement of experiment with a theory developed mathematically from a certain combination of assumptions does not validate the assumptions since, because of the complexity of the phenomenon, other combinations of assumptions may lead to the same conclusion. Tu, Davis, and Hottel [50]

8.1 Particle-Resolved CFD Simulations: Spherical Particles 8.1.1 Review of the Literature

The modeling of chemically reacting coal particles has a long history in comparison to modern computational modeling science. In particular, Nusselt [1] proposed an analytical one-film model (OFM) in 1924, and Burke and Schumann [2] presented a two-film model (TFM) in 1931. Nowadays, both of these models are used basically for the validation of computer codes or by students as a starting point in the modeling of chemically reacting coal particles [3]. Recently, some modifications of OFM and TMF (e.g., moving-flame-front model [4]) are used as subgrid models in complex simulations of gasifiers or combustors using computational fluid dynamics (CFD) (e.g., see [5, 6]). One of the first comprehensive numerical studies on chemically reacting coal (carbon) particles was carried out by Amundson and coworkers in the early 1980s (e.g., see the fundamental works [7, 8]). An extended review of the works carried out before 1980 can be found in the work by Sundaresan and Amundson [9]. Basically, the so-called pseudo-steady-state (PSS) approach was used by assuming that the particle radius does not vary significantly with time during the combustion process. In other words, the consumption time of a carbon particles is always large compared to the convective and diffusive timescales for the gas phase. An acceptance of this assumption was first discussed in experimental works by Smith Gasification Processes: Modeling and Simulation, First Edition. Edited by Petr A. Nikrityuk and Bernd Meyer. © 2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA.

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8 Pseudo-Steady-State Approach for Carbon Particle Combustion/Gasification

and Gudmundsen [10]. Recently, Stauch and Maas [11] carried out detailed transient numerical simulations of the combustion of carbon particles, which showed that “the characteristic values of the burning process are only dependent on the current particle diameter, independent of its previous evolution” [11]. Generally, most of the works cited above used the so-called quiescent ambient condition for the modeling of chemically reacting coal/carbon particles. However, applied to industrial gasification technology, the gas flow and relative particle velocity play significant roles in the partial oxidation of coal and char particles. For an example of a discussion on the importance of gas flow phenomena in heterogeneous oxidation, we refer the reader to the work by Matsui et al. [12], who found that the experimental combustion rate can be correlated effectively by expressing the combined chemical and fluid mechanical effects in terms of the surface Damköhler number and by reducing the combustion rate to a nondimensional one. It should be noted that, in spite of the numerous works carried out in the 1980s on the modeling of particle combustion and gasification, until now there have been few numerical studies on the influence of convection in such processes. An overview of the first efforts to take into account the impact of particle velocity on the carbon consumption rates can be found in [13]. This work presents general discussions on the importance of gas flow effects during the oxidation of solid carbonaceous materials including model developments for carbon consumption and particle burnout. One of the first attempts to study the oxidation of carbon particles in a convective environment numerically was presented by Ha and Choi [14]. They presented a 2D model in which the particle’s mass, momentum, and energy conservation equations were solved simultaneously with the gas phase equations in order to study the influence of particle entrainment on heat and mass transfer and combustion around a single spherical carbon particle. The effect of particle size, carbon reactivity, initial relative velocity, and oxygen concentration in a free stream was also explored. The paper provided, however, no information on model validation and the grid resolution used. The impressive development of personal computers and computational software including commercial CFD software has made it possible to perform sophisticated scientific computations on a standard PC. As a result, recently, a series of works [15–21] devoted to particle-resolved simulations of chemically reacting coal particles have been published. In particular, convection phenomena around chemically reacting coal particles were investigated by Raghavan et al. [15]. They performed a numerical and an experimental study of the burning of isolated spherical particles in a mixed convective environment at different free-stream velocities and ambient temperatures and developed correlations for the critical Reynolds number at which transition from the envelope flame to the wake flame occurs and also for the mass burning rates at the subcritical and supercritical Reynolds number regimes. However, only large millimeter-sized particles were investigated. The critical Reynolds number was estimated as a function of the ratio between the fuel evaporation velocity and

8.1

Particle-Resolved CFD Simulations: Spherical Particles

the free-stream gas velocity. This approach, while being physically correct, lacks detailed coupling between heterogeneous kinetics and transport phenomena. Recently, Higuera [16] carried out a numerical study based on a simple chemistry, investigating the influence of the size and velocity of a coal char particle as well as the effect of the temperature and the gas composition on the burning rate, the particle temperature, and the extinction of the flame. It was shown that the Reynolds number of the particle plays a significant role in the establishment of the combustion or gasification regimes. In spite of the large number of parameters and the wide range investigated, for example, the particle diameter 20 × 10−6 m ≤ ds ≤ 5 × 10−3 m, the Reynolds number related to the ambient temperature 0.5 < Rein < 500, the ambient temperature, 1000 K ≤ Tin ≤ 3000 K, no systematic analysis was carried out regarding the influence of the inflow temperature on the behavior of the particle with a fixed Reynolds number. Furthermore, no information was provided in the paper about the grid resolution or the size of the computational domain, and there was no code validation. A three-dimensional simulation of the unsteady combustion of a carbon particle at different particle sizes, ambient temperatures, and gas velocities was performed by Yi et al. [17]. This work confirmed the significant influence of the gas velocity on combustion regimes. Although a numerical procedure was used for the gas flow calculation, the particle–gas interaction was taken into account using the semianalytical one-film subgrid model. The influence of convection on the combustion regimes of a single coal particle in dry air was also studied by Kestel et al. [18]. The existence of three basic regimes, namely the gasification, transitional, and combustion regimes, was shown as a function of the inflow temperature at a fixed Reynolds number. However, the transport properties of the gas were set to constant values, leading to inconsistent Lewis numbers. The numerical study of the influence of the heterogeneous kinetics on the carbon consumption rate under the influence of convection has been carried out by Nikrityuk et al. [20] and Safronov et al. [22] for dry air and by Richter et al. [21] for O2 /CO2 atmospheres. This chapter aims to summarize the previous findings and demonstrate several computational validation test cases applied to heterogeneous combustion/gasification under the influence of convection. Finally, the influence of the particle velocity on the particle oxidation behavior in the form of the particle Reynolds number and the ambient temperature Tin is studied. Different regimes are explored and discussed using a heterogeneous Damköhler number. 8.1.2 Setup and Model Formulation

In this work, a single spherical coal char particle is considered with a diameter ds , placed in a stationary position with the main gas flow passing around it. This setup corresponds to the case of a moving particle when the system of coordinates is located at the center of the particle. The diameter of the particle is varied between 200 μm and 2 mm. The wide range of particle sizes is chosen because of

207

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8 Pseudo-Steady-State Approach for Carbon Particle Combustion/Gasification

its practical importance for industrial applications: micrometer-sized coal particles are used in entrained-flow gasifiers, and millimeter-sized particles are utilized in fixed-bed gasifiers [23]. The inflow gas consists of 0.233 mass fraction of O2 , 0.001 mass fraction of H2 O, and 0.766 mass fraction of N2 . The inflow velocity u⃗ in is assumed to be uniform over the inlet cross section and is determined by means of the particle Reynolds number, as Rein =

�in u⃗ in ds �in

(8.1)

where �in and �in are the density and molecular viscosity, respectively, corresponding to the inflow gas temperature Tin and the inflow gas composition. The inflow gas temperature is varied between 1000 and 3000 K at intervals of 50 K in order to cover both kinetically controlled and diffusion-controlled regimes for all particle sizes chosen. Assumptions To proceed with the governing equations, the following basic

assumptions have been made: 1) The particle is spherical. 2) The particle consists of carbon only. 3) The volatilization of the particle (regression of the particle surface) is not included because of the steady-state character of the model. 4) Only surface radiation of the particle is considered. The influence of radiation in the gas phase has been investigated in recent works [21, 22]. 5) The gas flow is considered to be incompressible with pin = 105 Pa. 6) The buoyancy effect is neglected. We give below a more detailed explanation of the assumptions made above:

• The ambient gas pressure is limited to atmospheric pressure because of the lack of kinetic data for higher pressures (e.g., see the recent review [20]).

• The assumption concerning the Lewis number is basically used in modeling combustion (see [3]) as a robust approximation. At the same time, it should be noted that the use of this assumption makes it possible to validate the numerical model against the analytical solution (see Section 8.1.5). Both the modeling configuration and the size of the domain and grid used are illustrated in Figure 8.1 and Table 8.1. The choice of the domain size is explained in Section 8.1.5. This configuration and the neglecting of the buoyancy effect suggest that the problem can be considered as axisymmetric. That allows the use of the 2D cylindrical coordinates r and z to model the flow past the sphere. The chemistry is modeled with three heterogeneous reactions at the particle surface and three homogeneous reactions in the gas phase [3, 8].

8.1

Particle-Resolved CFD Simulations: Spherical Particles

Side Outlet

L3

Inlet

rs

Particle

Rotation axis

r L1

L2

z 40

r/d

30 20 10 0 −30 −20 −10

0

10

20

30

40

50

60

70

80

90

100

z/d

Figure 8.1 Schematic representation of the computational domain and the grid. (Published with permission from ACS Publications: [21]). Table 8.1 Case Rein = 0 Rein > 0

Domain size and grid resolution. L1

L2

L3

Number of nodes

75 ds 40 ds

75 ds 30 ds

75 ds 100 ds

23 205 59 668

Heterogeneous reactions: 2 C(s) + O2 −−−−→ 2 CO,

ΔR H = −9.2 MJ kg−1 C −1

(R1)

C(s) + CO2 −−−−→ 2 CO,

ΔR H = 14.4 MJ kg

C

(R2)

C(s) + H2 O −−−−→ CO + H2 ,

ΔR H = 10.9 MJ kg−1 C

(R3)

Homogeneous reactions: H2 O

CO + 0.5 O2 −−−−→ CO2 ,

ΔR H = −10.1 MJ kg−1 CO

(R4)

CO + H2 O −−−−→ CO2 + H2 ,

ΔR H = −1.47 MJ kg−1 CO

(R5)

CO2 + H2 −−−−→ CO + H2 O,

ΔR H = 1.47 MJ kg−1 CO

(R6)

The semi-global reaction rates of the chemical reactions are given in Tables 8.2 and 8.3. The rate expression for the reaction describing CO oxidation, Eq. (R4), was first proposed in [24]. It should be noted that global reaction rates are valid only in a relatively narrow range of conditions and should be used very cautiously.

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8 Pseudo-Steady-State Approach for Carbon Particle Combustion/Gasification

Table 8.2

Reaction rates for homogeneous reactions.

Equations

̂ Ri, r (mol m−3 s−1 )

(R4) (R5) (R6)

Ar (corr. units)

Ea,r (J mol−1 )

nr

kr, CO CCO CH0.5O CO0.25

1.25 × 1011

1.67 × 105

0

kr, CO CCO CH2 O kr, CO2 CCO2 CH2

2.74 × 106 1.00 × 108

8.36 × 104 1.205 × 105

0 0

2

Table 8.3

2

Source [3] [26] Equilibrium.

Reaction rates for heterogeneous reactions.

Equations

̂ Ri, r (mol m−2 s−1 )

Ar (corr. units)

Ea,r (J mol−1 )

nr

Source

(R1) (R2) (R3)

kr, O2 CO2 , s kr, CO2 CCO2 , s kr, H2 O CH2 O, s

1.500 × 105 4.605 11.25

1.494 × 105 1.751 × 105 1.751 × 105

0 1 1

[29] [30] [30]

8.1.3 Governing Equations

Taking into account the assumptions made above, the conservation equations for mass, impulse, species, and energy transport can be written in the following general form: ( ) ∇ ⋅ � u⃗ = 0 (8.2) ( ) ( ) (8.3) ∇ ⋅ � u⃗ ⊗ u⃗ = −∇p + ∇ ⋅ � ( ) ( ) ∇ ⋅ � u⃗ Yi = ∇ ⋅ � Di, m ∇Yi + Ri (8.4) N

R ∑ ( ) ΔR Hr, j Rr, j ∇ ⋅ � u⃗ H = ∇ ⋅ (� ∇T) −

(8.5)

r

where �=�

)] [( ∇⃗u + ∇⃗u T

(8.6)

is the stress tensor. i stands for the participating reactants O2 , CO2 , CO, H2 O, and H2 , and r stands for the homogeneous reactions (R4)–(R6). The mass fraction for N2 is calculated as N−1

Y N2 = 1 −

∑

Yi

(8.7)

i

In Eq. (8.3), the symbol ⊗ denotes the dyadic product of two vectors. All transport properties �, �, �, and Di, m are calculated as functions of the gas composition and temperature. For details see [18]. The net production rate of species Ri , where i stands for the ith species and r for the rth reaction (R4)–(R6), is computed as the sum of the reaction sources over

8.1

Particle-Resolved CFD Simulations: Spherical Particles

the number of homogeneous reactions in which the species are involved: ∑ ̂i, r Ri = Mi R

(8.8)

r

̂i, r is calculated as follows [25]: The molar rate of creation/destruction R [ )] ( N ) ( ∏ � ′j, r −� ′′j, r ′′ ′ ̂i, r = � − � R C kr i, r

i, r

j, r

(8.9)

j=1

for the homogeneous reactions, where � ′i, r is the stoichiometric coefficient for the reactant i in the homogeneous reaction r, and � ′′i, r is the stoichiometric coefficient for product i in reaction r. � ′j, r and � ′′j, r are the forward and backward rate exponents for each reactant and product species j in reaction r. The rate constant for reaction kr is computed using the extended Arrhenius expression ) ( −Ea,r (8.10) kr = Ar T nr exp Ru T where Ar is the pre-exponential factor, nr is the temperature exponent, Ea,r is the activation energy, and Ru is the universal gas constant. The values of Ar , nr , and Ea,r for homogeneous reactions are given in Table 8.2. 8.1.4 Boundary Conditions

Boundary conditions have to be specified at the inlet and at the outlet boundary as well as at the symmetry axis and the upper side of the computational domain (Figure 8.1). Additionally, interface equations have to be specified to account for the processes on the surface of the particle. At the symmetry axis as well as at the upper side, Neumann boundary conditions have to be specified in the form ∂p = 0, ∂r

∂⃗u = 0, ∂r

∂Yi = 0, ∂r

∂T =0 ∂r

(8.11)

At the inlet boundary, the values of gas velocity, mass fractions, and temperature should be given: u⃗ = u⃗ in ,

Yi = Yi, in ,

T = Tin

(8.12)

At the outlet boundary, the so-called outflow boundary conditions with an overall mass balance correction have to be specified (see [27] for details). As heterogeneous reactions affect the mass and energy balance at the interface between the particle surface and the gas phase, they have a significant influence on the boundary conditions for the gas species and the temperature. The convective and diffusive mass fluxes of the gas-phase species at the surface are balanced by the

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8 Pseudo-Steady-State Approach for Carbon Particle Combustion/Gasification

production/destruction rates of the gas-phase species by surface reactions (R1)– (R3) (see [28]): ∂Yi, s ̂i, s − ṁ ′′c Yi, s = Mi R �s Di, m ∂⃗ n ∑ ̂i, s Mi R ṁ ′′c = i

n⃗ ⋅ � ∇T ||gas =

∑

( ) ΔR Hr, j Rr, j, s + �s � Ts4 − Tin4

(8.13) (8.14) (8.15)

r

where Rr, j, s is the production rate of species j due to the surface reaction r, ṁ ′′C is the net mass flux between the surface and the gas in kilogram per square meter per second, the index “gas” refers to the gas side at the wall, and n⃗ is the vector normal to the wall. �s stands for the emissivity of the particle surface (�s ≈ 1.0 for solid carbon). The last term on the right-hand side of Eq. (8.15) describes surface radiation. The rate constant for the surface reaction kr is computed using the extended Arrhenius expression (8.10). The values of Ar , nr , and Ea,r for surface reactions are given in Table 8.3. Basically, on a chemically nonreacting solid surface, the fluid velocity on a solid wall is zero, which corresponds to the well-known “no-slip” boundary condition. However, if a heterogeneous chemical reaction occurs on the solid surface, then the velocity in the normal direction from the surface can be nonzero. This heterogeneous reaction-induced flow is called the Stefan flow, and characterizes the net mass flux between the surface and the gas. The Stefan velocity takes the following form: ′′

n⃗ ⋅ u⃗ =

ṁ c �

(8.16)

8.1.5 Numerics and Software Validation

The commercial software Fluent 14 [31] was adopted to solve the problem under consideration. The setup description of the model is given in Section 8.4. The governing equations (8.2)–(8.5) were solved following an implicit finitevolume technique. For pressure–velocity coupling, the semi-implicit method for pressure-linked equations (SIMPLE) was used (for details, see [32]). The convective terms in all equations were discretized using quadratic upstream interpolation for convective kinematics (QUICK) [33]. The scheme of the computational domain and the grid resolution near the surface are shown in Figure 8.1 including a zoomed view near the particle surface. It should be noted that the averaged linear size of a control volume (CV) in the normal direction near the particle surface was set to 10−6 m in order to resolve the flame sheet and the thermal and diffusive boundary layers properly. Because of the strong coupling between the species and energy conservation equations, the under-relaxation factors for T and Yi variables were set to 0.7. The

0.025

0.25

0.020

0.20

0.015 0.010 0.005

Analytics Numerics 1 Numerics 2

0.000 1000 1500 2000 2500 3000 (a)

Particle-Resolved CFD Simulations: Spherical Particles

Tin (K)

Figure 8.2 Surface-averaged carbon mass flux ṁ ′′ as a function of inlet gas temperC ature Tin for (a) ds = 2 mm and (b) ds = 200 μm. In both plots, “Numerics 1” corresponds to the numerical results without

· m′′C (kg m−2 s−1)

· m′′C (kg m−2 s−1)

8.1

0.15 0.10 0.05

Analytics Numerics 1 Numerics 2

0.00 1000 1500 2000 2500 3000 (b) Tin (K) Stefan flow and “Numerics 2” to those with Stefan flow. “Analytics” corresponds to the results of analytical calculations based on the two-film model. (Published with permission from Wiley: [22]).

iterations were stopped when the maximal normalized residual for all equations was < 10−7 . In order to reach convergence of the solution, ∼ 105 iterations were necessary. 8.1.5.1 Validation against Analytical Solution

The computer program was validated against an analytical solution for diffusive combustion (i.e., u⃗ in = 0) of a coal particle in dry air (YH2 O ≈ 0.001) based on the TFM. A short description of this model is given in Chapter 6. Fortran code of this model is given in the CD to this book. An extended description can be found in [3]. Because of the fact that the TFM is only valid for higher temperatures, the ambient temperature for the calculations was varied in a range between 1000 and 3000 K at intervals of 100 K . Figures 8.2 and 8.3 show how the surface-averaged carbon mass flux ṁ ′′C and the surface temperature Ts and the flame temperature Tmax , respectively, vary as a function of the inlet temperature Tin , which was predicted using CFD software and two-film analytic model. A comparison of the numerical and analytical results shows good agreement for the whole temperature range considered in the validation studies. In order to study the influence of the Stefan flow on the results for different ambient gas temperatures, two series of numerical calculations were performed. The first corresponds to the calculations with the Stefan flow taken into account and the second to calculations where the Stefan flow is neglected. As can be seen in Figure 8.2, considerable deviation (>1%) can be obtained only for the values of the surface-averaged carbon mass flux ṁ ′′C for the higher temperatures. For the maximum inlet temperature of 3000 K , the deviation reaches the value of 10% and therefore independent of the particle size. For an ideal case, the values of ṁ ′′C predicted numerically with the Stefan flow taken into account correspond exactly to the values of ṁ ′′C calculated using the TFM. At the same time, the impact of the Stefan flow on the surface and flame temperatures is negligible (see Figure 8.3).

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8 Pseudo-Steady-State Approach for Carbon Particle Combustion/Gasification

4500

3000

4000 3500 Tmax (K)

2500 Ts (K)

214

2000

1000 1000

1500

(a)

2000

2500 2000

Analytics Numerics 1 Numerics 2

1500

3000

1500 2500

1000 1000

3000

Tin (K)

Analytics Numerics 1 Numerics 2 1500

(b)

Figure 8.3 (a) Surface temperature Ts and (b) flame temperature Tmax as a function of inlet gas temperature Tin for ds = 2 mm. In both plots, “Numerics 1” corresponds to the numerical results without Stefan flow and

2000

2500

3000

Tin (K)

“Numerics 2” to those with Stefan flow. “Analytics” corresponds to the results of analytical calculations based on the two-film model. (Published with permission from Wiley: [22]).

dP = 5 mm TS

Graphite rod

Stagnation zone

Nozzle

X Y

(a)

Tin Yi uin

Flow direction ·

(b)

Figure 8.4 (a,b) Principal scheme of the experimental setup [34]. (Published with permission from IFP Energies nouvelles: [19].)

8.1.5.2 Validation against Experiments I: Laminar and Turbulent Regimes

To validate the software against experimental data, we repeat “numerically” the experiments by Makino et al. [34]. For this, we consider the 2D computational domain shown in Figure 8.4. In particular, a 2D rod of a diameter dp = 5 mm is placed (at coordinates 30dp and 40dp ) in the domain, which is 130dp long and 80dp high. The ambient airflow temperature Tin was set to 320 and 1280 K, respectively. The rod surface temperature Ts was varied between 1200 and 2500 K. In the experiments by Makino et al. [34], the carbon combustion rate was measured in the forward stagnation region (see Figure 8.4b). The velocity gradient a (stretch rate) in the forward stagnation point was varied between 820 and 40 000 s−1 , corresponding to Reynolds numbers 10–2000 as a function of Ts . The so-called PSS approach described in the previous section is utilized because of the fact that the

8.1

Particle-Resolved CFD Simulations: Spherical Particles

Table 8.4 Reaction rates for the heterogeneous reactions used for “repeating” the experiment of Makino et al. [34]. Here, [O2s ] is the molar concentration of O2 at the surface, and kc,O2 is the rate coefficient given by Eq. 8.10. Gl.

̂ Ri,s (kmol m−2 s−1 )

Ar

Er (J mol−1 )

nr

Source

R1 R1b R2 R3

kc,O2 [O2s ]0.5 kc,O2 [O2s ] kc,CO2 [CO2s ] kc,H2 O [H2 Os ]

3.007 ⋅ 105 593.83 4.016 ⋅ 108 1.205 ⋅ 109

1.4937 ⋅ 105 1.4965 ⋅ 105 2.477 ⋅ 105 2.477 ⋅ 105

0 1 0 0

[7] [29] [3] [3]

combustion time of the rod is large compared to the convective and diffusion timescales for the gas phase. The following basic assumptions have been made:

• The gas flow is treated as that of an ideal incompressible gas. • The porosity of the rod is not taken into account, thus the diffusion inside the • • • •

cylinder is neglected. The cylinder consists of carbon only. The radiation of the gas phase is not taken into account. The buoyancy effect is neglected. For high Re number flows (Re > 400), a Reynolds-averaged Navier–Stokes (RANS) model coupled with the finite rate/eddy dissipation model is utilized.

The chemistry is modeled using four semi-global heterogeneous reactions, where three of them are given by Eqs (R1)–(R3), and three semi-global homogeneous reactions given by Eqs (R4)–(R6). The additional heterogeneous reaction is C(s) + O2 −−−−→ CO2

(R1b)

The input parameters for the Arrhenius equation (Eq. (8.10)) for heterogeneous and homogeneous reactions are taken from different authors and are listed in Tables 8.4 and 8.2, respectively. For the high Re-number flows (Re < 400), the so-called transitional k-kl-� model was chosen. This RANS model was developed by Walters and Cokljat [35] for transitional flows. Applied to the modeling of nonisothermal flows at high Re past a cylindrical rod, the transitional k-kl-� model demonstrated the best performance among other RANS models relating adequate prediction of the surface-averaged Nu and surfacial values of Nu (for details see the work by Kestel et al. [36]). Finally, the k-kl-� model was chosen for a graphite cylinder reacting under turbulent flow conditions. The results of the simulations of the numerical study of a graphite cylinder reacting under laminar and turbulent hot air flow conditions are presented in Figures 8.5 and 8.6. In particular, Figure 8.5 depicts the comparison of the combustion rates at the stagnation point predicted numerically and experimentally by Makino et al. [34] for different velocity gradients a as a function of the surface

215

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8 Pseudo-Steady-State Approach for Carbon Particle Combustion/Gasification

0.035 0.1 −2 −1 m′′ s ) p (kg m

0.025 0.02 0.015 0.01

0.08 0.06 0.04

·

· (kg m−2 s−1) m′′ p

0.03

0.02

CFD Experiments

0.005

0 1000 1250 1500 1750 2000 2250 2500 Ts (K)

(a)

CFD Experiments

0 1000 1250 1500 1750 2000 2250 2500 Ts (K)

(b)

-> Re = [36 ∶ 167] and (b) a = 40 000 s−1 -> Re = [447 ∶ 2000]. Different values of Re at constant a are explained by different values of the viscosity and density, which are temperature dependent.

Figure 8.5 Carbon mass flux at the stagnation point as a function of the rod surface temperature predicted numerically and experimentally [34] at Tin = 320 K and different a corresponding to (a) a = 3300 s−1

0.05 0.10 0.15 0.20 0.25 0.30

1300 1557 1814 2071 2329 2586 2843 3100

1 Y/D

Y/D

1 0 −1

0 −1

0

1

(a)

2

3

4

5

0

3

4

5

320

YCO2 – mass fraction 740

0.001 0.002 0.003 0.004 0.005

1160 1580 2000

1

1 Y/D

Y/D

2 X/D

Temperature

0 −1

0 −1

0 (b)

1

X/D

1

2

3

4

5

X/D Figure 8.6 Contour plots of the temperature (upper line) and CO2 mass fraction (lower line) predicted numerically for different flow regimes: laminar (a) a = 3300 s−1

0

1

2

3

4

5

X/D and turbulent (b) a = 40 000 s−1 . Here, Tin = 320 K, Ts = 2000 K. (Published with permission from IFP Energies nouvelles: [19].)

8.1

Particle-Resolved CFD Simulations: Spherical Particles

temperature, with the airflow temperature Tin = 320 K. Here, different flow regimes were modeled including the laminar regime (Figure 8.6a), and the turbulent regime (Figure 8.6b). The grid used for the RANS calculations contained about 0.22 ⋅106 CV cells, and the grid for the laminar calculations consisted of about 25 ⋅103 CV cells. The analysis of the results reveals fairly good agreement between numerical and experimental data for all flow regimes at lower temperatures. In particular, as expected, all results show a distinct nonlinear dependency between Ts and the carbon consumption. The nonlinearity occurs for all flow regimes and characterizes two different combustion regimes: kinetically controlled governed by kinetics and diffusion-controlled characterized by approaching constant values for the combustion rates. The analysis of results shows that increase in Re leads to the shift of the kinetically controlled regime to higher values of the surface temperature. This effect is illustrated in Figure 8.6, which shows the contour plots of the temperature and CO2 mass fraction predicted numerically for laminar (Figure 8.6a) and turbulent flow (Figure 8.6b) regimes. The contour plots shown in Figure 8.6 correspond to the convection–diffusion-controlled regime. The so-called flame sheet around the rod can be detected in the laminar flow. The increase in the inflow Re (a number) leads to the flame extinction effect (no CO2 is produced) (see Figure 8.6b). Finally, it should be noted that the impact of turbulence on heterogeneous combustion should be studied more precisely using the so-called direct numerical simulations (DNS). 8.1.5.3 Validation against Experiments II: The impact of Particle Porosity

The next validation test case is based on the experimental data published by Bejarano and Levendis [37], who studied the combustion of single lignite and bituminous coal particles in different O2 /N2 and O2 /CO2 atmospheres with different oxygen concentrations and gas-flow temperatures. In this validation case, we compare particle surface temperatures predicted numerically against experimental data for lignite coal at ∼50% burnout [37]. Data were selected for an operating temperature of 1400 K in the drop-tube furnace and a particle size of 75–90 μm. Both O2 /N2 and O2 /CO2 atmospheres were considered. The oxygen content was varied between 20% and 60% (mole fraction), which largely covers the range of oxygen concentrations that we consider in this chapter. In the drop-tube furnace, the Reynolds number was estimated to be on the order of unity, and the effect of pyrolysis gases on the char burnout temperatures at ∼50% burnout time was assumed to be negligible. In contrast to the previous test cases, in this validation–simulations the gas radiation effect was taken into consideration using the P-1 radiation model [25, 38]. The complete model and boundary conditions can be found in the work by Richter et al. [21]. Additionally, particle porosity was modeled using a pseudoporous geometry approximated by defining a 2D axisymmetric problem. In this case, the effect of macroporosity was modeled using a geometry consisting of six gaps arranged at equal distances along the particle surface (see Figure 8.7b). These gaps

217

218

8 Pseudo-Steady-State Approach for Carbon Particle Combustion/Gasification

O2/CO2 atmosphere

3000

Ts (K)

2750

Solid particle Porous particle Measurements

2500 2250 2000 0.2

0.3

0.4

0.5

0.6

YO

2,∞

O2/N2 atmosphere

3000 2750 2500 2250

0.6 0.4

Solid particle Porous particle Measurements

2000 0.2 (a)

1450 1650 1850 2050 2250

0.8

r/d

Ts (K)

1

0.3

0.4

0.5

0.2 0

0.6

YO

2,∞

Figure 8.7 Validation against the data of Bejarano and Levendis [37] taking into account “porosity” effect. (a) Particle surface temperature as a function of ambient oxygen mass fraction (Tin = 1400 K, Re = 1). (b)

(b)

−1 −0.8 −0.6 −0.4 −0.2

0

0.2 0.4 0.6

8

1

x/d

Temperature distribution inside and around a single particle predicted numerically for Tin = 1400 K and YO2in = 0.4, Re = 1. (Published with permission from ACS Publications: [21].)

represent macropores that have access to the outside of the char. Heat transfer inside the solid particle was taken into account. The gaps lead to an increased surface area inside the particle. Figure 8.7b illustrates the temperature field in and around these gaps. Inside the gaps, no oxygen is present, so the Boudouard and gasification reactions (reactions R2 and R3, respectively) are the dominant heterogeneous reactions. For this reason, the temperature inside the gaps is decreased. Because of the extended endothermic reactions inside the gaps, the averaged surface temperature is decreased, as illustrated in Figure 8.7a. Compared to the results for solid particles, agreement with the measurements is improved. The deviation is between 1.5% and 3.8% for O2 /CO2 and between 0.5% and 1.4% for O2 /N2 . With these phenomena in mind, it can be summarized that the modeling of porous particles is far more challenging and remains the goal of future investigations. Some first steps in this direction are illustrated in Chapter 9.

8.1

Particle-Resolved CFD Simulations: Spherical Particles

8.1.6 Results: The Impact of Re on the Oxidation Regimes

The main subject of this section is the exploration of the influence of the relative particle velocity on the oxidation regimes for air atmosphere. To study the impact of the Reynolds number Rein and the ambient temperature Tin on the oxidation behaviour, several simulation runs were carried out for particle sizes ds = 200 μm and ds = 2 mm. In every run, the particle Reynolds number was fixed and the ambient temperature Tin was varied between 1000 and 3000 K in order to find the regime where the flame sheet appeared. The main goal was to show the existence of the three basic oxidation regimes for both particle sizes under consideration and to compare the conditions under which these regimes occurred. Before the analysis of the computational results, we present a short description of the basic processes occurring on the particle surface and close to the particle in the gas phase in the case of a quiescent ambient environment. According to the fundamental works [7, 9] devoted to modeling a nonmovable spherical particle in a quiescent dry-gas environment, the combustion of a carbon particle involves the diffusion of O2 through a stagnant layer and its reaction at the particle surface to produce CO according to the reaction (R1). The CO produced at the surface diffuses outward and is consumed in the flame sheet, producing CO2 in reaction (R4). At the same time, the resulting CO2 reacts with the carbon surface and produces additional CO in reaction (R2). Since CO oxidation occurs very fast, both CO and O2 are nearly zero at the flame sheet. There is no CO leakage through the flame sheet into the gas phase. The existence of the gas-phase flame also cuts off most of the supply of oxygen to the surface such that the reaction (R1) is now suppressed. The temperature T and CO2 concentration profiles peak at the flame sheet. This effect can be seen in Figure 8.8, which shows the spatial distribution of the species mass fractions as well as the dimensionless temperature T∕Tin predicted for ds = 2 mm and Tin = 1600 K. Because of the radiative heat loss, the surface temperature Ts is only slightly higher than the ambient temperature Tin . After this short introduction on the basic physics, we want to comment the dependence of the flame radius on the ambient conditions. Recently, Zhang et al. [4] introduced the so-called moving flame front (MFF) model, where the flame sheet location changes in a range between 1 and 50 rs for different ambient partial pressures of oxygen. This is a questionable assumption because, from the analytic TFM and from numerical simulations, it can be seen that during the combustion of a spherical carbon particle the flame radius remains within a range between 1.7 and 2 rs , the value decreasing while the O2 concentration increases. In particular, from Figure 8.8 it can be seen that the flame radius scaled with the particle radius approaches a value of 2. Before we discuss the numerical results, let us discuss the characterization of heterogeneous oxidation of a solid surface. Applied to the classical heat and mass transfer theory, the impact of the gas-phase convection and ambient temperature can be described accurately enough by means of nondimensional criteria, namely Reynolds, Sherwood, and Prandtl numbers, for different particle sizes. In order to

219

8 Pseudo-Steady-State Approach for Carbon Particle Combustion/Gasification

0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28

2

0.00 0.05 0.09 0.14 0.19 0.23 0.28 0.33

2

0.16

1.5

1

0.28

0.5

0.04

0 (a)

r/ds

r/ds

1.5

1 0.5 0

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 (b)

(c)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

1600 1814 2029 2243 2457 2671 2886 3100

2

0.13

1.5

2243

0.06

r/ds

1.5

1

0

0.33

z/ds

0.00 0.03 0.06 0.09 0.13 0.16 0.19 0.22

0.5

0.00

0.05

z/ds

2 r/ds

220

0.00

3100 1814

0

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 z/ds

1 0.5

(d)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 z/ds

Figure 8.8 Spatial distribution of the mass fraction of CO2 (a), the mass fraction of CO (b), the mass fraction of O2 (c), and the temperature (d) for ds = 2 mm, Rein = 0, Tin = 1600 K, and YH2 O = 10−3 . (Published with permission from Wiley: [22]).

be able to predict the oxidation regime for the given combination of ambient temperature, gas velocity, and particle size, the transport phenomena should be coupled with the reaction kinetics. For this purpose, the nondimensional Damköhler number will be utilized [39]. In the case of heterogeneous reactions, the surface Damköhler number Das is used, which is given by the ratio of the chemical reaction rate to the mass transfer rate as follows [39, 40]: �j,′ r −1 ∑ r kr Yj, r (8.17) Das = � with � as the mass transfer coefficient for the gas–solid interface. Here, r represents the heterogeneous reactions (R1)–(R3). Thus, the system is kinetically controlled if Das < 1 and, vice versa, Das > 1 presumes the existence of a diffusion-controlled regime. Following description given in [40], we have the following regimes:

• kinetically controlled regime: Da < 1 occurs when the chemical kinetic rate is much slower than the mass transfer (diffusion) rate. For fine particles, this regime is valid until higher temperatures (T < 2500 K) due to the very small diffusion resistance. • diffusion-controlled regime: Da > 1 occurs when the mass transfer rate is very slow compared to the kinetic rate, or the kinetics is so fast that oxygen reaching the external surface is immediately consumed by the char surface. This type of oxidation denotes the diffusion-controlled regime. This regime is basically valid for large particles even at relatively low temperatures.

8.1

Particle-Resolved CFD Simulations: Spherical Particles

• transitional regime: It occurs when the surfacial mass transfer and the kinetic rates are comparable, leading to Da ≈ 1. This regime is not trivial for modeling, because both effects must be taken into account. The mass transfer coefficient � in Eq. (8.17) is evaluated using the Sherwood number Sh as follows: Sh Dm �= (8.18) ds For the case Le = 1, the Sherwood number for a single burning coal particle can be calculated using the following correlation [3, 41]: 1

1

0.555 Rein2 Pr 3 Sh = 2 + [ ]1 .

(8.19)

2

(

1+1.232 4

)

Rein Pr 3

Here, Pr means the Prandtl number of the ambient gas phase. The ambient gas temperature Tin is taken into account by evaluating the gas-phase transport properties. Applied to the problem under consideration, taking into account, first, the order of reactions R1–R3, the following relations can be formulated for Damköhler numbers for the different heterogeneous reactions: Das, (R1) =

k(R1) �

,

Das, (R2) =

k(R2) �

,

Das, (R3) =

k(R3)

(8.20)

�

103

102

102

101

101

100 Das

Das

Figures 8.9 and 8.10 show the surface Damköhler number Das = Das, (R1) and Das = Das, (R2) + Das, (R3) plotted against the inlet gas temperature Tin for different Reynolds numbers. The shaded area represents the transition temperatures that

100

10−2

10−1 Rein = 10 Rein = 50 Rein = 100 Rein = 150

10−2 10−3 1000 (a)

10−1

1500

2000 Tin (K)

2500

Rein = 10 Rein = 50 Rein = 100 Rein = 150

10−3 10−4 1000

3000 (b)

1500

2000

2500

3000

Tin (K)

Figure 8.9 Surface Damköhler number Das = Das, (R1) as a function of the inlet gas temperature Tin for different Reynolds numbers Rein , ds = 2 mm (a), and ds = 200 μm (b). (Published with permission from Wiley: [22]).

221

102

101

101

100

100

10−1

10−1

10−2

Das

Das

8 Pseudo-Steady-State Approach for Carbon Particle Combustion/Gasification

10−2 10−3 10

−4

1500

(a)

2000 Tin (K)

2500

10−3 10−4

Rein = 10 Rein = 50 Rein = 100 Rein = 150

10−5 1000

Rein = 10 Rein = 50 Rein = 100 Rein = 150

10−5 10−6 1000

3000

1500

2000

2500

3000

Tin (K)

(b)

Figure 8.10 Surface Damköhler number Das = Das, (R2) + Das, (R3) as a function of the inlet gas temperature Tin for different Reynolds numbers Rein , ds = 2 mm (a), and ds = 200 μm (b). (Published with permission from Wiley: [22].)

correspond to Das ≈ 1. These figures illustrate that the increase in the particle Reynolds number leads to the shift of the kinetically controlled regime to lower temperatures. This fact is explained by the enhancement of the heat and mass transfer between the particle and the gas when Re increases. Besides, it can be clearly seen that a decrease in the particle diameter at constant temperature moves the system in the direction of the kinetically controlled regime, too. To explore the influence of Re on the surface-averaged carbon mass flux ṁ ′′C , we plot this quantity in Figure 8.11, which shows ṁ ′′C as a function of the

0.06 0.05

Rein = 0 Rein = 10 Rein = 50 Rein = 100 Rein = 150

· m′′C (kg m−2 s−1)

0.07

· m′′C (kg m−2 s−1)

222

0.04 0.03 0.02

(a)

0.4 0.3 0.2

0.01 0.00 1000

Rein = 0 0.7 Rein = 10 Rein = 50 0.6 Rein = 100 Rein = 150 0.5

0.1

1500

2000 2500 Tin (K)

0.0 1000

3000 (b)

1500

2000

2500

3000

Tin (K)

Figure 8.11 Surface-averaged carbon mass flux ṁ ′′ for ds = 2 mm (a) and ds = 200 μm (b) C as a function of inlet gas temperature Tin for the different Reynolds numbers Rein . (Published with permission from Wiley: [22].)

8.1

Particle-Resolved CFD Simulations: Spherical Particles

inlet temperature Tin for different Reynolds numbers. In particular, for a 2 mm particle (compare Figures 8.9a, 8.10a, and 8.11a), the different regimes can be clearly identified by analyzing the change of the Das values for different reactions:

• kinetically controlled regime for all reactions: Tin < 1200 K; • transitional regime for reaction (R1) and kinetically controlled regime for reactions (R2 + R3): Tin ≈ 1200 K;

• diffusion-controlled regime for reaction (R1) and kinetically controlled regime for reactions (R2 + R3): 1200 K < Tin ≲ 2000 K;

• diffusion-controlled regime for reaction (R1) and transitional regime for reactions (R2 + R3) (shaded in Figure 8.11a): 2000 K < Tin ≲ 2700 K;

• diffusion-controlled regime for all reactions: Tin > 2700 K(Re = 1), Tin > 3000 K (Re = 150). In contrast to the large particle, the transitional and diffusion regimes for a 200 μm particle cannot be distinguished as clearly and are shifted to higher ambient temperatures (see Figures 8.9b, 8.11b, and 8.10b):

• kinetically controlled regime for all reactions: Tin < 1300 K; • transitional regime for reaction (R1) and kinetically controlled regime for reactions (R2 + R3): Tin ≈ 1300 K;

• diffusion-controlled regime for reaction (R1) and kinetically controlled regime for reactions (R2 + R3): 1300 K < Tin ≲ 2700 K;

• diffusion-controlled regime for reaction (R1) and transitional regime for reactions (R2 + R3) (shaded in Figure 8.11b): Tin > 2700 K. This means that even at higher gas temperatures, of about 3000 K , the 200 �msized particles will still not reach a diffusion-controlled regime for all heterogeneous reactions. Finally, to illustrate the impact of the particle Reynolds number, Rein , on the spatial distribution of chemical species’ mass fraction, Yi , and the temperature, T, around the particle, we plot in Figures 8.12 and 8.13 zoomed views of T and YCO2 contour plots calculated for millimeter- and micrometer-sized particles for different ambient temperatures. For the analysis of these plots, we use Figure8.9, which serves as the identifier of kinetically controlled and diffusion-controlled regimes as functions of Tin and Rein (see discussions in the previous section). From Figure 8.9, it can be seen in the diffusion-controlled regime the so-called envelope flame (with maximum in T and YCO2 ) existing around a moving spherical particle, (see Figure 8.12b,d). The flame shape covers the particle and is elongated in the direction of the flow. This effect has been illustrated numerically and experimentally by many authors (e.g., see [15, 16]). A decrease in the ambient temperature at Rein = 100 leads to the transition from the envelope-flame regime to the so-called wake flame (see Figures 8.12a,c ). It can be seen that there is almost no CO oxidation in the front part of the particle surface. The flame, which can be detected at maximum CO2 or T, is established only in the backstream part of the flow (wake region). In previous studies [19], it was shown that the increase in the particle

223

8 Pseudo-Steady-State Approach for Carbon Particle Combustion/Gasification

2200 2272 2344 2416 2488 2560 2632 2704 2776 2848

1200 1286 1371 1457 1542 1628 1713 1799 1884 1970

1200

2200

1 r/ds

r/ds

1

0.5 0

0.5

1970 1713

−1

2848

1628

−0.5

0

(a)

0.001

0.048

0.5 z/ds

1

0.094

0.141

1.5

0

2

2848

−1

−0.5

0

(b)

0.187

0.234

0.280

0.5 z/ds

1

1.5

2

0.00 0.05 0.09 0.14 0.18 0.23 0.27 0.32

0.00

0.001

1 r/ds

r/ds

1

0.5

0.5 0.32

0.280

0

−1

−0.5

0

(c)

0.5 z/ds

1

1.5

0

2

−1

−0.5

0

(d)

0.5 z/ds

1

1.5

2

Figure 8.12 Zoomed view of the temperature contour plots (a,b) and YCO2 (c,d) predicted numerically for ds = 2 mm and Rein = 100. (a) and (c) Correspond to Tin = 1200 K and (b) and (d) refer to Tin = 2200. (Fig. 8.12a,b published with permission from IFP Energies nouvelles: [22].) 2400 2527 2654 2781 2908 3034 3161 3288 3415 3542

1000 1000 1001 1001 1001 1001 1002 1002 1002 1003

1000

2400

1

r/ds

r/ds

1

0.5

0.5

3161

1000

0

−1

−0.5

1002

0

(a)

0.00001

0.5 z/ds

0.00001

1

1.5

0

2

3542

−1

−0.5

0

(b)

0.00002

0.00003

0.5 z/ds

1

1.5

2

0.00 0.03 0.05 0.08 0.11 0.13 0.16 0.19 0.21 0.24 0.27

0.00

1

1

0.00001

r/ds

r/ds

224

0.5 0 (c)

0.5 0.00003

−1

−0.5

0

0.5 z/ds

1

1.5

2

0 (d)

0.27

−1

−0.5

0

0.5 z/ds

1

1.5

2

Figure 8.13 Zoomed view of the temperature contour plots (a,b) and YCO2 (c,d) predicted numerically for ds = 200 μm and Rein = 50. (a) and (c) Correspond to Tin = 1000 K, and (b) and (d) refer to Tin = 2400. (Published with permission from Wiley: [22].)

Reynolds number promotes the formation of the wake flame regime. Following findings by Raghavan et al. [15], the effect of the wake flame at high Re numbers is attributed to the flow separation in the recirculation zone. In particular, the extinction of the flame in the front part of the particle is attributed to the lower flow residence time in comparison to the reaction time. However, in the wake

8.2

Particle-Resolved CFD Simulations: Nonspherical Particles

region, the recirculation flow increases the residence time for the reactants. In a recent work by Schulze et al. [19], it was shown numerically that, in the case of low particle Re numbers flows, where the recirculation does not exist, a similar wake flame regime can be observed. This effect was explained by the Stefan flow, which modifies the boundary layer thickness around the particle for low Re number flow regimes. Next, the analysis of Figures 8.13 and 8.9b reveals that the decrease in the particle radius leads to the shift of the kinetically controlled regime to higher ambient temperatures compared to larger particles. Correspondingly, the wake flame regime prevails up to Tin ≳ 2700 K at moderate values of Rein .

8.2 Particle-Resolved CFD Simulations: Nonspherical Particles 8.2.1 Introduction

In practice, coal/biomass particles are irregular in shape and have unique volume and density [42, 43]. There has been significant progress in modeling and understanding the processes occurring during thermal conversion of spherical or nearly spherical particles. Recently, also several experimental and numerical studies have been published in the literature that discover the impact of the particle shape on the carbon burnout and temperature distribution on the particle surface. In particular, Sampath et al. [42] showed experimentally that the particle shape and density have significant effect on the temperature histories of coal particles. They confirmed that the assumption of a spherical shape results in underestimation of the particle surface area. For example, if a particle has the shape of a parallelepiped, cylinder, or ellipsoid of equal volume, then its surface area is larger than that of a sphere of the same volume [42]. Recent works are mainly focused on biomass particles, 1) as there the impact of sphericity is more significant than on coal/char particles. In particular, Lu et al. [44] carried out experimental and theoretical investigations of the impact of biomass particle shape and size on the particle drying, heating, and reaction rate. Spherical, cylindrical, and flake-like wood particles were investigated. It was shown that the assumption of spherical or isothermal conditions for chemically reacting particles leads to large errors at most biomass particle sizes of practical interest. The results by Lu et al. [44] revealed that nonspherical particles larger than 200–300 μm can hardly reproduce correctly the conversion times using a spherical model approach. For example, for large millimeter-sized particles the conversion time of the sphere was up to 2.7 times higher than for the cylindrical/plate-like particles. In most of the works, 0D or 1D models were used to simulate the particle–gas interaction. For instance, Peters and Bruch [45] developed a transient 1D model for heating up, drying, and pyrolysis, which took into account shape-specific heat and mass 1) Biomass particles commonly have more irregular shapes and larger sizes than pulverized coal [44].

225

226

8 Pseudo-Steady-State Approach for Carbon Particle Combustion/Gasification

transfer coefficients for modeling the particle–gas interaction. The comparison of the 1D model results with the experimental results on large (4–12 mm) wood particles showed good agreement. However, 1D-based models are not capable of resolving the structure of the fluid flow surrounding the particle and, thus, they do not reflect correctly the interfacial heat and mass transfer during the particle oxidation. Recently, Yang et al. [46] presented a CFD-based two-dimensional model for a cylindrical particle of biomass surrounded by a passing gas stream. The flow boundary layer was modeled using a CFD code to explore the structure of the fluid flow surrounding the particle. The particle size was in the range 10 μm to 20 mm. Different subprocesses such as moisture evaporation, devolatilization, tar cracking, gas-phase reactions, and char gasification were examined. The authors were analyzing single cylindrical biomass particles with a 2D axisymmetric approach including transient particle shrinking as well as fluid flow at all stages of conversion (evaporation, devolatilization, and char burnout). The results revealed that, for particles larger than 150–200 μm, thermal gradients inside the particles play a significant role. Other authors have modeled whole burners using Euler–Lagrangian particle tracking with modified 0D submodels for nonspherical biomass particles. Recently, Gubba et al. [43] simulated a combustion test facility using a 1D approach for modeling the radial thermal gradients inside differently shaped biomass particles. The analysis of the simulations showed that, while using the 1D model, the flame shape of the coal/biomass cofiring burner was in excellent agreement with the experimental results. However, calculations made with a 0D model, which used the spherical particle assumption, showed large disagreement regarding the flame shape. In 2002, Gera et al. [47] modeled a utility boiler fired with cylindrical switch grass particles using 0D models. The particle–fluid interaction was modeled using a heat transfer coefficient and a diffusion rate coefficient. The sphericity of the particles was taken into account for the mass transport using an enhancement factor based on the Sherwood number, which was introduced by Grow [48]. The simulations for the millimeter-sized particles revealed that the aspect ratio of the particles played a major role on the consumption time of the particles. The mentioned studies clearly demonstrate that the particle shape has to be considered when dealing with nonspherical carbonaceous reacting particles. So far, the revised studies have shown that accounting the impact of particle shape on the burnout history and particle temperature is a critical issue in understanding coal combustion and gasification. However, most of the studies were just focused on two or three different shapes, which are especially important for biomass applications, while coal is randomly shaped. Furthermore, all of them except the study of Yang et al. were using 0D or 1D models, which were not taking into account the surrounding fluid flow. There are only a few papers in the literature that resolve the particle–fluid interaction directly. Motivated by this fact, in this work we illustrate the influence of the particle shape by means of six different geometries, and compare the results with those of a spherical particle.

8.2

Particle-Resolved CFD Simulations: Nonspherical Particles

227

8.2.2 Shapes of Particles

As sample geometries, an ellipsoidal, a cylindrical, a conical, a rhomboidal, and two different spherical particles with surface enlargements were chosen. The ellipsoidal form was selected because it resembles the drop-like shape that develops during combustion and gasification due to the relative speed of the particles (see also Chapter 7). The cylindrical particle represents a chip-like biomass particle. However the ratio of height and diameter is much larger for our geometry than it would be in reality. The conical and rhomboidal particles correspond to rough particle fractures that can occur at the milling process or as a result of fracturing during pyrolysis. Finally, the particles with surface enlargements depict a particle with surface roughness. In the following, we will call that shape sponge. All considered particle shapes are axisymmetric. Furthermore, the volumes of the different particles are equal to that of the sphere, while their surface areas are different. The diameter of the sphere used was 2 mm. Figures 8.14 and 8.15 (upper row) give an impression of the shapes considered, and table 8.5 lists the important geometrical features of the different particles. Here, D is the expansion in direction of the rotational axis of the particles, while H denotes the maximum expansion in radial direction. A is the external surface area of the particles. For all particle shapes, a grid study was conducted to guarantee a gridindependent solution. The grids that were used are shown in Figures 8.14 and 8.15 (lower row). They consist of between 21 000 (ellipsoid) and 92 000 (cylinder) CVs.

Sphere

Ellipsoid

Cylinder

Figure 8.14 Different particle shapes considered in this study. (upper row) with close-ups of the used meshs (lower row)

Sponge

228

8 Pseudo-Steady-State Approach for Carbon Particle Combustion/Gasification

Cone

Rhombus

Star

Figure 8.15 Different particle shapes considered. (upper row) with close-ups of the used meshs (lower row). Table 8.5

Some important geometric parameters for the particular shapes.

Parameter

Sphere

Ellipsoid

Cylinder

Sponge

D∕dsphere H∕dsphere A∕Asphere Cell volumes

1 1 1 27 750

1.59 0.79 1.08 21 410

1.10 0.55 1.21 91 848

1.03 0.52 1.41 44 630

8.2.3 Results

To illustrate the effects of shape on the char conversion process, Figures 8.16 and 8.17 show the YCO2 mass fraction and the temperature distribution, respectively, of different particle shapes at different ambient inflow temperatures 1200, 1400 and 2000 K corresponding to the three regimes described for a spherical particle. A comparative analysis of figures reveals that, qualitatively, the temperature distribution is comparable for all particle shapes investigated in this work. However, a more detailed view reveals differences especially in the position of the areas of the maximum temperature and CO2 mass fraction. It can be seen that the temperature distribution for the spherical particle is more homogeneous than for the

8.2

Particle-Resolved CFD Simulations: Nonspherical Particles

0.00 0.01 0.01 0.01 0.02 0.02 0.02 0.03 0.03 0.04 0.04

1 R/D 0.5

0

0

−1

−0.5

(a)

0

0.5 Z/D

1

1.5

2

1

−0.5

0

0.5 Z/D

1

1.5

0

2

0

−0.5

0

0.5 Z/D

1

1.5

2

−1

−0.5

0

0.5 Z/D

1

1.5

0

2

−0.5

0

0.5 Z/D

1

1.5

−1

−0.5

0

0.5 Z/D

1

1.5

0

2

1.5

1.5

1.5

0

R/D

2

R/D

2

(d)

1

0

0.00 0.00 0.01 0.01 0.01 0.02 0.02 0.02 0.02 0.03 0.03

−0.5

0

0.5 Z/D

1

1.5

−1

−0.5

0

0.5 Z/D

1

1.5

0

2

0.00 0.02 0.05 0.07 0.09 0.12 0.14 0.16 0.18 0.21 0.23

0.00 0.00 0.01 0.01 0.01 0.01 0.02 0.02 0.02 0.02 0.03

1.5

1.5

0

R/D

1.5

R/D

2

(f)

1

0

0

−1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 Z/D

0

0.5 Z/D

1

1.5

2

2

−1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 Z/D

−1

−0.5

0

0.5 Z/D

1

1.5

2

−1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 Z/D

R/D 0.5

0

0

1.5

1

R/D

R/D 0.5

−0.5

1

0.00 0.03 0.06 0.09 0.13 0.16 0.19 0.22 0.25 0.28 0.31

1

−1

0.5 Z/D

1

0.00 0.03 0.06 0.08 0.11 0.14 0.16 0.19 0.22 0.24 0.27

0.00 0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.04 0.05 0.05

1

0

0.5

0.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 Z/D

−0.5

0.00 0.03 0.06 0.09 0.12 0.15 0.17 0.20 0.23 0.26 0.29

2

0.5

−1

0.5

2

1

2

R/D 0

2

1.5

0.00 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 0.27 0.30

0.5

−1

1

1

R/D

R/D 0

(e)

0

−1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 Z/D

1

0.5

0.5 Z/D

1

0.00 0.03 0.05 0.08 0.11 0.13 0.16 0.18 0.21 0.23 0.26

1

0

0.5

0.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 Z/D

−0.5

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

0.00 0.03 0.05 0.08 0.11 0.13 0.16 0.18 0.21 0.23 0.26

0.00 0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.04 0.05 0.05

0.5

−1

0.5

2

1

2

R/D 0

2

1.5

1

R/D

R/D

−1

1

0.00 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 0.27 0.30

0.5

(c)

0.5 Z/D

0.5

1

1

0

0.00 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 0.27 0.30

0.00 0.03 0.05 0.08 0.11 0.13 0.16 0.18 0.21 0.23 0.26

0.5

−0.5

R/D 0.5

0

−1

−1

1

R/D

R/D 0.5

0.00 0.00 0.01 0.01 0.01 0.02 0.02 0.02 0.02 0.03 0.03

R/D

−1

1

(b)

R/D

0.5

0.00 0.03 0.06 0.08 0.11 0.14 0.16 0.19 0.22 0.24 0.27

0.00 0.01 0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.04 0.04

(g)

1

R/D

R/D 0.5

0

0.00 0.03 0.06 0.09 0.13 0.16 0.19 0.22 0.25 0.28 0.31

0.00 0.03 0.06 0.08 0.11 0.14 0.16 0.19 0.22 0.24 0.27

1

229

0.5

−1

−0.5

0

0.5 Z/D

1

1.5

2

0

−1

−0.5

0

0.5 Z/D

Figure 8.16 YCO2 contour plots predicted numerically for different particle shapes at inflow temperatures of T∞ = 1200 K (left column), T∞ = 1400 K (center column), and T∞ = 2000 K (right column).

1

1.5

2

230

1200

8 Pseudo-Steady-State Approach for Carbon Particle Combustion/Gasification

1217

1234

1250

1267

1284

1300

1317

1333

1400

1350

1578

1622

1666

1711

1755

−1

1200

−0.5

1219

1237

0.5 Z/D

0 1256

1274

1292

1 1311

1.5 1329

1348

1366

−1

−0.5

1465

1530

0.5 Z/D

0 1595

1660

1725

1 1790

1.5 1855

1920

−0.5

1214

1228

0 1242

0.5 Z/D 1256

1270

1 1283

1.5 1297

1311

−1

−0.5

1458

1516

0 1574

0.5 Z/D 1631

1689

1 1747

1.5 1805

1862

0

2 1920

1200

−0.5

1218

1236

0.5 Z/D

0

1254

1272

1290

1

1308

1.5

1326

1344

1362

−1

1400

−0.5

1454

1509

0.5 Z/D

0 1563

1617

1671

1 1725

1.5 1779

1833

0

2

1.5

1.5

0.5 0

−1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 Z/D 1200

1212

1225

1237

1249

1261

1273

1285

1297

1309

1465

1530

1594

1659

1724

1788

1853

1917

1200

−0.5

1211

1223

0.5 Z/D

0

1234

1245

1256

1

1267

1.5

1278

1289

0

2

1300

−1

1400

−0.5

1504

1607

0.5 Z/D

0 1710

1814

1917

1 2020

1.5 2123

2227

0

2

1.5

1.5

0.5 0

−1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 Z/D 1200

1216

1233

1249

1265

1281

1297

1313

1329

1345

1453

1507

1560

1614

1667

1721

1774

1828

0.5

0

0

0

0.5 Z/D

1

1.5

2

−0.5

2054

2108

0 2162

0.5 Z/D 2216

2270

1 2323

1.5 2377

2431

2 2485

−0.5

2052

2103

0.5 Z/D

0 2155

2206

2257

1 2309

1.5 2360

2412

2 2463

2056

2112

2168

2224

2280

2336

2392

2448

2504

−0.5

2056

2111

0.5 Z/D

0

2167

2222

2277

1

2333

1.5

2388

2444

2

2499

−1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 Z/D 2000

1881

2062

2125

2187

2249

2312

2374

2436

2499

2561

1

R/D

R/D 0.5

−0.5

2 2443

1

0

−1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 Z/D

1

−1

2393

0.5

1400

1

1.5 2344

R/D

0

(f)

R/D

1.5

R/D

2

1

−1

2000

2330

2

0.5

1 2295

0.5

2

1

2246

1

0.5

−1

2197

−1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 Z/D 2000

1982

R/D

R/D 0

(e)

0.5 Z/D

0 2148

1

0

−1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 Z/D

1

0.5

2099

0.5

1400

1

−0.5

2050

R/D

0

(d)

R/D

1.5

R/D

2

1

−1

2000

1887

2

0.5

2357

0.5

2

1

2317

R/D 0

2

−1

2000

0.5

−1

2278

1

R/D

R/D 0

2238

1

1

1

2198

0.5

1400

1325

0.5

2159

R/D 0

2

−1

2000

1985

R/D 1200

(c)

0

2

0.5

−1

2119

0.5

1

R/D 0

2079

1

1400

1

2040

R/D 0

2

0.5

(b)

2000

1800

R/D

R/D 0

(a)

R/D

1533

0.5

0.5

R/D

1489

1

1

(g)

1444

0.5

−1

−0.5

0

0.5 Z/D

1

1.5

2

0

−1

−0.5

0

0.5 Z/D

1

1.5

2

Figure 8.17 Temperature distribution predicted numerically for different particle shapes at inflow temperatures of T∞ = 1200 K (left column), T∞ = 1400 K (center column), and T∞ = 2000 K (right column).

8.2

Particle-Resolved CFD Simulations: Nonspherical Particles

other shapes, which is somehow an expected result. The result of simulations showed that, independent of the shape of particles, at high ambient temperatures (T∞ ≥ 2000; diffusion-controlled regime) the flame sheet exists around a moving particle for almost all shapes considered in this work. At the same time, a more irregular shape leads to the establishment of large temperature differences between upstream and downstream parts of the particle. In particular, in the front part of the particle in respect to the flow direction the surface temperature is higher in comparison to the temperature in the downstream (rear) part of the particle. This effect is attributed to two phenomena:

• The complex shape of a particle is responsible for (in comparison to the sphere) more CO2 in the flow separation zone behind a particle due to the suitable conditions (low velocity value) for CO oxidation; • The endothermic Boudouard reaction (C + CO2 ) results in a decrease in the surface temperature due to more CO2 in the downstream part of the particles with irregular shapes in comparison to near-spherical particles. Increase in the flow-resistance gradient leads to an enhancement of this effect. However, it should be noted that, in reality, if a particle rotates, for example, because of turbulence, this effect might be negligible. It can be seen that the increase in aerodynamic resistance of the particle leads to the enhancement of so-called wake flame regime, where the flame exists only in the downstream part of the particle. For details, see the previous section. At lower temperatures (T∞ ≤ 1200 K), the oxidation zone is located on the surface of particle independent of the shape. 8.2.3.1 Integral Characteristics

Next we discuss the impact of particle shapes on the integral characteristics, for instance, particle-averaged carbon mass flux and carbon consumption rate. As discussed in the previous section, in the kinetically controlled regime, the carbon conversion is limited by the velocity of the heterogeneous reactions. Hence, the local carbon mass flux is a function of the surface temperature, while the carbon consumption rate is a function of the surface temperature and the surface area. The temperature results from the energy balance at the particle surface. Regarding the different particle shapes, the heat transfer coefficient 𝛼, which plays a role in the heat transfer to the particle, is in general shape dependent. That means that the surface temperature and thus the carbon mass flux and the consumption rate vary with the shape. For a lower value of 𝛼, the convective heat transfer is smaller. Therefore, the surface temperatures and the carbon consumption rate increase. This effect is illustrated in Figure 8.18. On (a) , we can see the local 𝛼 over the nondimensionalized rectification length of the particle, L∗ = l∕L, where L is the rectified curve length between the forward and backward stagnation point of the rz plane of the particle, and l is the run length on the curve. On (b) , we can see the ′′ impact of 𝛼 on the local carbon mass flux ṁ c . It can be seen that the deviations of ′′ ṁ ′′c are small compared to the particle-averaged carbon mass flux ṁ p .

231

(a)

10 9 8 7 6 5 4 3 2 1 0

8 Pseudo-Steady-State Approach for Carbon Particle Combustion/Gasification

1.01 Cylinder Sponge

1.005

Average

1 · / m′′ m′′ c p

α /α part.-avg.

232

Average

0.995 0.99 Cylinder Sponge

0.985 0

20

40 60 L∗ (%)

80

0.98

100 (b)

Figure 8.18 (a) Local heat transfer coefficient 𝛼 predicted numerically for cylindrical and sponge shapes at Re = 100 and (b) the relative carbon mass flux

0

20

40 60 L∗ (%)

80

100

′′

calculated using the particle-averaged ṁ p for u∞ = 10 m s−1 and T∞ = 1000 K over the non-dimensionalized rectification length of the particle L∗ = l∕L.

Finally, it is obvious that the impact of the shape on 𝛼 is negligible for the kinetically controlled regime, and the influence of the particle shape on the carbon consumption rate is only a function of the surface. Therefore, it is possible to express the overall carbon mass flux of a nonspherical particle as ṁ p,shaped = ṁ p,sphere

Ashaped Asphere

(8.21)

for Da < 1. As discussed in Section 8.1.6, the carbon conversion in the diffusion-controlled regime is limited by the species transport to the surface, which is characterized by the mass transfer coefficient. The relation between the Reynolds number and �, which was discussed in Section 8.1.6, is also valid for the nonspherical particles. However, in addition, these effects are now superimposed by shape-dependent effects that are connected with the attributes of the particle form. In general, we can divide between the exposed and covered areas. In areas that are more exposed to the flow, such as the edges on the cylinder geometry or the surface extensions of the sponge shape, the boundary layer thickness decreases. This causes an enhancement in the species transport and hence an increase in the carbon mass flux. At recesses the flow stagnates. Here, the mass transport and ṁ ′′c are low compared to the particle-averaged values (see Figure 8.19a). With increasing Re, the boundary layer decreases and thus the overall carbon mass flux increases. In the case of nonspherical particles, the differences between the exposed and covered particle areas increase with an increasing Reynolds number. This finally leads to an increase in the amplitude of ṁ ′′c between the exposed and covered areas. The impact of this effect on ṁ ′′c is well depicted in Figure 8.19 b for the sponge shape. The analysis of Figure 8.20a reveals that the particle-averaged carbon mass flux of the nonspherical particles is lower than that of the sphere.

8.2

Particle-Resolved CFD Simulations: Nonspherical Particles

233

5 6

Sphere Ellipse Cylinder Sponge

4 3

V = 0.1 m s−1 V = 1 m s−1 V = 10 m s−1

4 · / m′′ m′′ c p

· / m′′ m′′ c p

5

Average

2

3 Average

2 1

1 0 0

50 L∗ (%)

(a)

0

100

0

20

40 60 L∗ (%)

(b)

80

100

Figure 8.19 Nondimensional carbon flux predicted numerically (a) for different shaped particles at u∞ = 10 m s−1 and (b) for the sponge shape at different u∞ values. T∞ = 2000 K, dp = 2 mm.

5×10−7

0.03

m· p (kg s−1)

· (kg m−2 s−1) m′′ p

0.04

0.02

0 1000 (a)

1500

2000

2500

Sphere Ellipse Cylinder Sponge

1×10−7 0 1000

3000

T∞ (K)

3×10−7 2×10−7

Sphere Ellipse Cylinder Sponge

0.01

4×10−7

(b)

1500

2000

2500

T∞ (K)

Figure 8.20 (a) Surface-averaged carbon mass flux and (b) carbon consumption rate predicted numerically for different shapes at u∞ = 10 m s−1 and T∞ = 2000 K, dp = 2 mm.

This can be explained due to the lower mass transfer coefficient � of the nonspherical shapes. To get a relation ṁ p,r,shape ∕ṁ p,r,sphere for the diffusion-controlled regime, the influence of � has to be taken into account. Therefore, the Sherwood number can be used. This leads to an extended expression of equation (8.21): ṁ p,shaped = ṁ p,sphere

Ashaped Shshaped Asphere Shsphere

.

(8.22)

Equations (8.21) and (8.22) are just valid for calculating the carbon mass flux of one single heterogeneous reaction r. In cases where several homogeneous reactions participate in the carbon conversion, the correlations are still valid if all reactions are in the same regime. However, if there is just one reaction that dom∑ inates the process, the equations are also valid for ṁ p = ṁ p,r .

3000

234

8 Pseudo-Steady-State Approach for Carbon Particle Combustion/Gasification

0.04

0.03 m· p (kg s−1)

· (kg m−2 s−1) m′′ p

5e−07

0.02 Sphere Cylinder Sponge Sphere corr. to cylinder Sphere corr. to sponge

0.01

0 1000 (a)

1500

2000 T∞ (K)

2500

4e−07 3e−07 2e−07

Sphere Cylinder Sponge Sphere corr. to cylinder Sphere corr. to sponge

1e−07 0 1000

3000 (b)

1500

2000 T∞ (K)

2500

3000

Figure 8.21 (a) Surface-averaged carbon mass flux and (b) carbon consumption rate predicted numerically for different shapes at u∞ = 10 m s and T∞ = 2000 K, dp = 2 mm.

In Figure 8.21, the surface averaged carbon mass flux (a) and the carbon consumption rate (b) of the cylinder and the sponge are compared with the carbon mass flux/consumption rate generated via Eq. (8.21) for the kinetic regime and via Eq. (8.22) for the diffusion-controlled regime, respectively, for different temperatures at an inlet velocity of u∞ = 10 m s−1 and dry air conditions. The transportcontrolled regime was assumed for all three heterogeneous reactions (R1)—(R3) at temperatures where a flame occurs (T∞ ≳ 1200 K). The transitional regime was neglected. For calculating the Sherwood number of the sphere equation (8.19) was utilized. In general, there are much more Nusselt correlations than Sherwood correlations available in the literature (e.g., [49]) for nonspherical shapes. Therefore, for the nonspherical particles, the Nusselt number correlations were used instead of Sherwood correlations. This is acceptable when we assume that the Lewis number is equal to unity. The plots show that Eqs (8.21) and (8.22) work exactly for a wide range of ambient temperatures. In the temperature range where the ignition occurs and Das ≈ 1, there are some deviations of up to 10% for the cylinder and 15% for the sponge shape. This is obvious because the assumptions made for deriving Eqs (8.21) and (8.22) are for Das < 1 or Das > 1, respectively. Furthermore, the ignition temperature of the sphere is different from that of the nonspherical geometries. Therefore, in certain cases, the nonspherical particle is already ignited while the spherical one is not. Thus Eqs (8.21) and (8.22) should be used with care for this temperature range. For the temperature range between 2200 and 2600 K, also some deviations occur which are about 1% for the cylinder and between 1% and 3% for the sponge. This is caused by the fact that the carbon consumption rate is not dominated any more by the fully transport-controlled heterogeneous oxidation reaction, but the Bouduard reaction and the water-gas shift reaction, which are still kinetically controlled, gain influence.

8.4

Setup of Heterogeneous Reactions in ANSYS FLUENT

8.3 Conclusions

In this chapter, we discussed the influence of Re on the carbon consumption of spherical and nonspherical particles. We showed that depending on the heterogeneous kinetics and the Reynolds number three different regimes can be detected: a kinetically controlled regime, where the reaction kinetics mainly control the carbon consumption; a diffusion-controlled regime, where transport processes dominate; and a transitional regime where both mechanisms compete with each other. The temperature range of the regimes depends on the Reynolds number, the reactivity of the coal, the particle diameter, and the shape of the particle. In case of nonspherical particles, correlations for the kinetically and the diffusion-controlled regimes were presented, which allowed us to express the carbon consumption of a nonspherical particle via the carbon consumption of a sphere. The correlations worked quite well for a wide range of temperatures.

8.4 Setup of Heterogeneous Reactions in ANSYS FLUENT

In this Section, we give a short tutorial for the use of heterogeneous reactions model in commercial CFD software ANSYS-Fluent® [31]. The tutorial is addressed to people who have basic knowledge about how to use Fluent®. Thus the tutorial is limited to the implementation of the reaction system as well as the boundary conditions. For further details, the reader is referred to Fluent® user guide [27] and theory guide [25]. 8.4.1 Step 1: Species Transport Settings

Generally, in order to be able to simulate chemically reacting gases and surfaces, it is necessary to activate option “Species” and “Energy” in the menu “Define” –> “Models”. In particular, the following steps have to be carried out:

• Open CFD-Fluent graphic user interface (GUI). • Load and scale your mesh. • Choose in the “Problem Setup” panel the option “Models” and then double-click in the “Models” panel the option “Species.”

• A new “Species” window pops up. • Now select the following options in this order (some options become just available, when choosing other options): —Species transport (enables Eq. (8.4)); —“Reactions” → “Volumetric” (homogeneous reactions); —“Reactions” → “Wall Surface” (heterogeneous reactions); —“Wall Surface Reaction Options” → Heat of Surface Reactions ;

235

236

8 Pseudo-Steady-State Approach for Carbon Particle Combustion/Gasification

Figure 8.22 Overview of the settings on the “Species Model” panel.

—“Wall Surface Reaction Options” → Mass Deposition Source (enables Stefan flow). • In addition, the following may be necessary: • “Inlet Diffusion” (necessary for low inflow velocities); • “Diffusion Energy Source“ (enthalpy transport due to species diffusion); • “Full Multicomponent Diffusion” (the Maxwell–Stefan equations will be used to obtain the diffusive mass flux); • “Thermal Diffusion” (includes Soret diffusion). An overview over the settings at the “Species Model” panel is shown in Figure 8.22. 8.4.2 Step 2: Define Species and Mixtures

When you have done the settings in the “Species Model” panel, FLUENT generates automatically a new mixture in the “Materials” panel called the mixture-template. The mixture contains already three “fluid” species, namely H2 O, O2 , and N2 , which corresponds to “dry air.” For adding the missing species CO, CO2 , and H2

• Open the mixture-template in the “Materials” panel; • Click on the “FLUENT Database...” button → and the “FLUENT Database Materials” panel will pop up;

• Select “fluid” as material type;

8.4

Setup of Heterogeneous Reactions in ANSYS FLUENT

Figure 8.23 Setup of the “Species” panel.

• Choose CO, CO2 , H2 , and “Carbon Solid (C)” (this will be the carbon for the heterogeneous reactions) from the “FLUENT Fluid Materials” list and copy them. Now it is possible to add the new materials to the mixture. Therefore, open the mixture template and then “species” panel via the “edit” button on the “Mixture Species field.” In the “Species” panel, add the solid species “Carbon Solid (C)” to the “Selected Solid Species” field and the gaseous species CO, CO2 , as well as H2 to the “Selected Species” field (see Figure 8.23). In this field, it is important that N2 as inert species is listed at the last place, since the last species will close the mass balance in FLUENT®. The material properties for the particular species as well as the mixture are constant and correspond to the standard conditions by default. As the properties are temperature dependent and the temperature for the considered cases differs strongly from the standard conditions, it is useful to change the property settings to kinetic theory. To do so, go to the “mixture-template” and change the thermal conductivity and the viscosity from “constant” to “ideal- gas-mixinglaw” and the mass diffusivity to “kinetic-theory.” Then open the “Edit Material” panel of the particular species and change there the thermal conductivity and the viscosity from “constant” to “kinetic-theory” as well. 8.4.3 Step 3: Define Reactions

After adding all species to the mixture, the particular reactions can be defined. Therefore, open the “Reactions” panel in the mixture panel (see also Figure 8.24).

237

238

8 Pseudo-Steady-State Approach for Carbon Particle Combustion/Gasification

Figure 8.24 The “Reactions” panel.

At the “Reactions” panel

• Select the total number of reactions; • Choose the type of reaction: “Volumetric” (homogeneous reaction) or “Wall surface” (heterogeneous reaction);

• Select the participating reactant and product species and its stoichiometric coefficients and rate exponents;

• Enter the Arrhenius coefficients (cf. Eq. 8.10) A, EA , and n. In the case of a heterogeneous reaction, “Carbon Solid (C)” will appear in the species drop-down list. You have to choose it as one of the reaction partners. The stoichiometric coefficient and the rate exponent are always unity for the carbon. 8.4.4 Step 4: Boundary Settings

When the species transport model is enabled, the new field “species” is available at the inlet and the wall boundary panels. At the inlet species panel you have to

8.4

Setup of Heterogeneous Reactions in ANSYS FLUENT

enter the gas composition that enters the domain. At the wall boundary species panel of the wall, which represents the particle, choose the reaction option. Now the heterogeneous reactions at this surface are enabled.

Nomenclature Latin

A C d D Da Ea H k Le ṁ ′′c ṁ ′′p ṁ p M n⃗ N p Pr r R Ru ̂ R Re Sh T u⃗ Y

surface (m2 ) molar concentration (mol m−3 ) diameter (m) diffusion coefficient (m2 s−1 ) Damköhler number (–) activation energy (J mol−1 ) enthalpy (J kg−1 ) reaction rate (corresponding unit) Lewis number (–) local carbon mass flux (kg m−2 s−1 ) particle averaged carbon mass flux (kg m−2 S−1 ) carbon consumption rate (kg S−1 ) molecular weight (kg mol−1 ) normal vector (–) number of species (–) pressure (Pa) prandtl number (–) radius (m) mass production rate (kg m−3 s−1 ) universal gas constant (J mol−1 K−1 ) molar production rate (mol m−3 s−1 ) Reynolds number (–) Sherwood number (–) temperature (K) velocity (m s−1 ) mass fraction (–)

Greek

𝛼 � ΔR H

heat transfer coefficient (W m−2 K−1 ) mass transfer coefficient (m s−1 ) enthalpy of reaction (J kg−1 )

239

240

8 Pseudo-Steady-State Approach for Carbon Particle Combustion/Gasification

�s �′ � ′′ � � �′ � ′′ � � �

emissivity of the particle surface (–) rate exponent of reactant (–) rate exponent of product (–) thermal conductivity (W m−1 K−1 ) dynamic viscosity (Pa s) stoichiometric coefficient for reactant (–) stoichiometric coefficient for product (–) density (kg m−3 ) stefan-Boltzmann constant (W m−2 K−4 ) stress tensor (Pa)

Indices

i j r p s (s) ∞∕in m gas sphere shaped

referring to species i referring to species j referring to reaction r referring to the particle referring to the particle surface referring to solid phase (in reactions) referring to ambient conditions referring to mixture property referring to gas phase referring to spherical particle referring to shaped particle.

References 1. Nusselt, W. (1924) Der Verbren-

2.

3. 4.

5.

nungsvorgang in der Kohlenstaubfeuerung. Zeitschrift des Vereins Deutscher Ingenieure, 68, 124. Burke, S.P. and Schumann, T.E.W. (1931) Kinetics of a type of heterogeneous reaction: the mechanism of combustion of pulverized fuel. Industrial and Engineering Chemistry, 23 (4), 406–413. Turns, S.R. (2006) An Introduction to Combustion, 2nd edn, McGraw-Hill. Zhang, M., Yu, J., and Xu, X. (2005) A new flame sheet model to reflect the influence of the oxidation of co on the combustion of a carbon particle. Combustion and Flame, 143 (3), 150–158. Chen, L., Yong, S.Z., and Ghoniem, A.F. (2012) Oxy-fuel combustion of

pulverized coal: characterization, fundamentals, stabilization and CFD modeling. Progress in Energy and Combustion Science, 38, 156–214. 6. Kumar, M. and Ghoniem, A.F. (2012) Multiphysics simulations of entrained flow gasification. part II: Constructing and validating the overall model. Energy & Fuels, 26, 464–479. 7. Caram, H.S. and Amundson, N.R. (1977) Diffusion and reaction in a stagnant boundary layer about a carbon particle. Industrial and Engineering Chemistry Fundamentals, 16, 171–181. 8. Mon, E. and Amundson, N.R. (1978) Diffusion and reaction in a stagnant boundary layer about a carbon particle. 2nd extension. Industrial Engineering

References

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

Chemistry Fundamentals, 17 (4), 313–321. Sundaresan, S. and Amundson, N.R. (1980) Diffusion and reaction in a stagnant boundary layer about a carbon particle. 5. pseudo-steady-state structure and parameter sensitivity. Industrial and Engineering Chemistry Fundamentals, 19, 344–351. Smith, D.F. and Gudmundsen, A. (1931) Mechanism of combustion of individual particles of solid fuels. Industrial and Engineering Chemistry, 23, 277–285. Stauch, R. and Maas, U. (2009) Transient detailed numerical simulation of the combustion of carbon particles. International Journal of Heat and Mass Transfer, 52 (19–20), 4584–4591. Matsui, K., Kôyama, A., and Uehara, K. (1975) Fluid-mechanical effects on the combustion rate of solid carbon. Combustion and Flame, 25, 57–66. Chomiak, J. and Sarofim, A.F. (1984) Combustion rate of carbon: Fifty years after. International Communications in Heat and Mass Transfer, 11, 3–14. Ha, M.Y. and Choi, B.R. (1994) A numerical study on the combustion of a single carbon particle entrained in a steady flow. Combustion and Flame, 97, 1–16. Raghavan, V., Babu, V., Sundararajan, T., and Natarajan, R. (2005) Flame shapes and burning rates of spherical fluid particles in a mixed convective environment. International Journal of Heat and Mass Transfer, 48, 5354–5370. Higuera, F.J. (2008) Combustion of coal particle in a stream of dry gas. Combustion and Flame, 152, 230–244. Yi, F., Fan, J., Li, D., Lu, S., and Luo, K. (2011) Three-dimensional timedependent numerical simulation of a quiescent carbon combustion in air. Fuel, 90, 1522–1528. Kestel, M., Nikrityuk, P., Hennig, O., and Hasse, C. (2012) Numerical study of the partial oxidation of a coal particle in steam and dry air atmospheres. IMA Journal of Applied Mathematics, 77, 32–46. Schulze, S., Kestel, M., Safronov, D., and Nikrityuk, P.A. (2013) From detailed description of chemical reacting coal

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particles to subgrid models for CFD. Oil & Gas Science and Technology, 68, 1007–1026. Nikrityuk, P.A., Gräbner, M., Kestel, M., and Meyer, B. (2013) Numerical study of the influence of heterogeneous kinetics on the carbon consumption by oxidation of a single coal particle. Fuel, 114, 88–98. Richter, A., Nikrityuk, P.A., and Kestel, M. (2013) Numerical investigation of a chemically reacting carbon particle moving in a hot o2/co2 atmosphere. Industrial and Engineering Chemistry Research, 52, 5815–5824. Safronov, D., Kestel, M., Nikrityuk, P.A., and Meyer, B. (2014) Particle resolved simulations of carbon oxidation in a laminar flow. Canadian Journal of Chemical Engineering, accepted for publication. Higman, C. and van der Burgt, M. (2008) Gasification, 2nd edn, Elsevier GPP, Gulf Professional Publishing, Amsterdam U.A. Dryer, F.L. and Glassman, I. (1973) High temperature oxidation of CO and CH4 . Proceedings of the Combustion Institute, 14, 987–1003. ANSYS Inc. (2011) ANSYS-FLUENT Theory Guide, Release 13.0. Jones, W.P. and Lindstedt, R.P. (1988) Global reaction schemes for hydrocarbon combustion. Combustion and Flame, 73, 233–249. ANSYS Inc. (2011) ANSYS-FLUENT User Guide, Release 13.0. Kee, R.J., Coltrin, M.E., and Glarborg, P. (2003) Chemically Reacting Flow: Theory & Practice. John Wiley & Sons, Inc., Hoboken, NJ. Anil, W. (2005) Introduction to Computational Fluid Dynamics, Cambridge University Press, New York. Libby, P.A. and Blake, T.R. (1981) Burning carbon particles in the presence of water vapor. Combustion and Flame, 41 (0), 123–147. ANSYS, Inc. (2013) ANSYS-FLUENT V 14.0 – Commercially available CFD software package based on the Finite Volume method, Southpointe, 275 Technology Drive, Canonsburg, PA 15317, U.S.A., www.ansys.com.

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and temperature histories: Impact of particle irregularity on temperature predictions and measurements. 26th Symposium on Combustion/The Combustion Institute, pp. 3179–3188. Gubba, S.R., Ma, L., Pourkashanian, M., and Williams, A. (2011) Influence of particle shape and internal thermal gradients of biomass particles on pulverised coal/biomass co-fired flames. Fuel Processing Technology, 92, 2185–2195. Lu, H., Ip, E., Scott, J., Foster, P., Vickers, M., and Baxter, L.L. (2010) Effect of particle shape and size on devolatilization of biomass particle. Fuel, 89, 1156–1168. Peters, B. and Bruch, C. (2003) Drying and pyrolysis of wood particles: Experiments and simulation. Journal of Analytical and Applied Pyrolysis, 70 (2), 233–250. Yang, Y.B., Sharifi, V.N., Swithenbank, J., Ma, L., Darvell, L.I., Jones, J.M., Pourkashanian, M., and Williams, A. (2008) Combustion of a single particle of biomass. Energy & Fuels, 22, 306–316. Gera, D., Mathur, M.P., Freeman, M.C., and Robinson, A. (2002) Effect of large aspect ratio of biomass particles on carbon burnout in a utility boiler. Energy & Fuels, 16, 1523–1532. Grow, D.T. (1990) Mass and heat transfer to an ellipsoidal particle. Combustion and Flame, 80, 209–213. Richter, A. and Nikrityuk, P.A. (2012) Drag forces and heat transfer coefficients for spherical, cuboidal and ellipsoidal particles in cross flow at sub-critical reynolds numbers. International Journal of Heat and Mass Transfer, 55, 1343–1354. Tu, C.M., Davis, H. and Hottel, H.C. (1934) Combustion rate of carbon –combustion of spheres in flowing gas streams. Ind. Eng. Chem., 26 (7), 749–757.

243

9 Pore-Resolved Simulation of Char Particle Combustion/Gasification Andreas Richter, Matthias Kestel, and Petr A. Nikrityuk

9.1 Introduction

As discussed in the previous chapter, computer simulation models have become well-established tools for the understanding of high-temperature conversion processes at the particulate level (e.g., see [1–7]). Nearly all computational fluid dynamics (CFD) investigations of single particles given in the literature consider char conversion at the particle surface only. However, it is a well-known fact that coal particles become porous [8–10] after devolatilization, and also that char reactions occur inside the particle. Modeling chemically reacting porous particles is a challenging problem, which has great importance in energy process engineering and chemical engineering. Previous and recent studies used mainly a so-called macroscopic representation of porosity or particle-averaged models to describe carbon conversion because of the intrinsic-based character of heterogeneous reactions inside the porous particle (e.g., see reviews in [8, 11, 12]). In such ′′′ models, the specific surface ratio S is a “virtual” input parameter. Pioneering works on the chemistry and fluid mechanics relating to the combustion and gasification of porous carbon char have been analyzed in extensive reviews by Walker [13] and Mulcahy and Smith [14]. In particular, one of the first fundamental works was published by Hottel and coworkers in the 1930s. They studied experimentally the reaction of graphitic – “Graphon” – spheres with air-oxygen as a function of temperature and flow velocity of the gasifying agent using a thermogravimetric method. Their work was the basis for a well-known theory for gas–solid reactions in porous solids. In addition to other findings, they described for the first time the transition from the kinetically controlled regime to the diffusion-controlled regime during carbon combustion [15]. Now it is well understood and described that gasification can be kinetically or diffusion-controlled depending on the ambient conditions (gas composition and the temperature) and pore structure of the char including the specific surface Gasification Processes: Modeling and Simulation, First Edition. Edited by Petr A. Nikrityuk and Bernd Meyer. © 2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA.

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value. First, theoretical descriptions of the correlation between the internal surface of char and its reactivity including the role of diffusion1) were reviewed by Laurendeau [8]. Many pore structure models have been proposed (for reviewing works until the end of 1970 see e.g. Szekely et al. [11]). A theory for the tree-like pore structure of carbon char has been presented by Simons in series of papers [16–18]. In particular, a pore distribution function specifying the size distribution of the tree trunks has been derived statistically and verified empirically. The pore structure was completely defined using measured values of the porosity, internal surface area, and the particle radius. It was shown that the oxidation rate of carbon char is very sensitive to the structure of the pore tree [16–18]. Szekely et al. [11] proposed a very simple grain model for modeling a porous particle. In particular, this model assumes that a porous particle consists of fine grains of particles with or without some binding agents. The overall shape was basically approximated by a long cylinder or a sphere. It should be noted that, in general, the assumption of grains packed into a sphere is not accurate in the sense of a real porous structure. However, this approach allows performing well-controlled experiments or numerical simulations where all characteristics are well known. From the computational point of view, the modeling of char particles resolving all pores is nearly impossible. This is due to the fact that the complex geometry inside porous char particles is not a priori known, and it is also very expensive to apply such complex structures in CFD-based models (see e.g., the scanning electron microscopy (SEM) measurements of Brix et al. [19]). During the conversion process, the particle can change its inner structure, for example, as a result of a continuous change of the pore size, which finally increases the modeling effort. As an alternative, it is feasible to study fundamental phenomena by considering simplified geometries. The key characteristics for such particle shapes are the porosity � and the ratio of the total surface area (some authors only take the inner surface into account). Closely connected to the surface ratio is the volume-specific surface area S′′′ (m2 m−3 ). In the literature, a wide range of S′′′ values are given for coal char particles at different conversion stages. The measurement technique, coal type, and atmospheric composition and temperature significantly influence the measured values of S′′′ , leading to possible values between 105 and 3 ⋅ 108 m2 mm−3 (see the reviews by Laurendeau [8] and van Heek [20]). The change of the pore system with the progress of reactions can be taken into account by means of the theory of Bhatia and Perlmutter [21, 22], who developed a statistics-based model for the development of the internal pore structure during gasification in the kinetically controlled regime. However, the critical issue is that we still do not know quite well the characteristic parameters in the transitional pore-diffusion regime, where the Bhatia and Perlmutter theory is not valid. Moreover, until now the roles of homogeneous reactions inside a porous particle and of the Stefan flow are not well understood. In this view, in order to understand the impact of a porous particle’s velocity on the oxidation rates, in this chapter we use different idealized pore shapes taking into account homogeneous and heterogeneous reactions inside the 1) Diffusion inside the char can be defined by Knudsen diffusion or by continuum diffusion.

9.2

Model Assumptions and Chemistry

particle including the resulting Stefan flow inside the pores. In this chapter, the specific surface ratio is given by the geometry of pores.

9.2 Model Assumptions and Chemistry

In this chapter, we study the char conversion effects of different pseudo-porous structures, and compare all the results with those for a nonpermeable (solid) particle. The application we focus on is entrained-flow reactors. Even though the Reynolds numbers referring to gas flows are large in such systems, the particle Reynolds numbers are generally small because of the small particle sizes [23]. In particular, for 90 μm particles, the typical particle Reynolds numbers at atmospheric pressure are, for example, in the range 0.1–45. Larger particle Reynolds numbers were reported only for fluidized-bed and fixed-bed applications, and for high-pressure systems. The focus on moderate particle Reynolds numbers allows us to consider two-dimensional axisymmetric cases, because the flow field is expected to be laminar and axisymmetric. Figure 9.1 illustrates the different particle shapes that are considered here. It should be noted that the figure shows only a slice in the xy-plane, and the geometry is a body of rotation around the x-axis. The first porous geometries (shapes b and c) are comparable to a nonporous particle but feature long gaps that show at the center of the particle. The gaps are similar to fractures that can be observed sometimes during char conversion. Table 9.1 provides the characteristics of the different particle shapes, such as the total surface area ratio, volume-specific surface area, porosity, and smallest and largest length scales of the substructures. For shapes b and c, the total surface ratio is between 1.67 and 2.4, so the surface enhancement compared to a solid particle is relatively low. The resultant specific surface area is of order 105 , and the porosity is between 0.03 and 0.06, which is very low compared to, for example, the value 0.6 published in [24]. As a second approach, the porous particle was modeled as a cluster of 27 rings (shape d) (see Figure 9.1c). In comparison with shapes b and c, the surface area is only slightly increased but the porosity exceeds 0.55, which is closer to the published data. As a third approach, the porous particle was modeled as a set of 10 layers, which are equidistantly placed and are unconnected (shape e). Although the particle porosity is comparable to that for shape d, the surface area is increased by a factor of 5, which leads to a specific surface area of 106 m2 m−3 . The model considered in this chapter follows the experimental studies on coals and chars by Bejarano and Levendis [25]. In their work, the authors studied the combustion of single lignite and bituminous coal particles in a drop-tube furnace, and considered different O2 /N2 and O2 /CO2 atmospheres with different oxygen concentrations and gas-flow temperatures. Here, we focus on an operating temperature of 1400 K and a particle size of 75–90 μm. Only one O2 /N2 atmosphere with 20% oxygen content is considered here. In the drop-tube furnace, finally the particle falls at its terminal velocity, which is approximately 0.1 m s−1 and

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9 Pore-Resolved Simulation of Char Particle Combustion/Gasification

(a)

(b)

(c)

(d)

(e) Figure 9.1 Considered particle shapes. (a) Solid spherical Particle. (b–e) Different approaches to modeling porous particle structures: (b) short gaps, (c) long gaps, (d) rings, (e) layers. Surface areas that take part in radiation processes are marked with thick lines. Table 9.1

Characteristics of different particles. Particle diameter Dp equals 90 μm. ′′′

m2 m−3

Shape

Description

S∕Ssolid

S

a b c d e

Nonporous Short gaps Long gaps 27 Rings 10 Layers

1.00 1.67 2.40 3.13 7.68

6.67 ⋅ 104 1.11 ⋅ 105 1.60 ⋅ 105 2.08 ⋅ 105 5.12 ⋅ 105

�

Minimum pore size

Pore length

0.0 0.034 0.059 0.55 0.54

— 1∕60 Dp 1∕60 Dp 1∕28 Dp 1∕100 Dp

— 1∕10 Dp 1∕4 Dp — 1∕40 Dp

corresponds to a Reynolds number 0.04. For simplicity, the effect of pyrolysis gases on the char burnout temperatures at ∼50% burnout time was assumed to be negligible. The different particles are placed in a nearly dry, uniform oxygen/nitrogen mixture (YH2 O = 0.001). The consumption time of the particle is always large compared to the convective and diffusion timescales for the gas phase, thus the pseudo-steady-state (PSS) approach can be used [15, 26, 27].

9.2

Model Assumptions and Chemistry

Buoyancy effects of the particle are not considered. The particle consists of carbon only. The chemistry is modeled using semi-global heterogeneous and homogeneous reactions, written as follows: 1 CO + O2 + H2 O −−−−→ CO2 + H2 O 2 CO + H2 O −−−−→ CO2 + H2

(R2)

CO2 + H2 −−−−→ CO + H2 O

(R3)

(R1)

C + CO2 −−−−→ 2CO

(R4)

2C + O2 −−−−→ 2CO

(R5)

C + O2 −−−−→ CO2

(R6)

C + H2 O −−−−→ CO + H2

(R7)

The corresponding pre-exponential factor A and the activation energy E for the heterogeneous and homogeneous reactions are given in Table 9.2, together with the corresponding literature. In contrast to the previous chapter, two heterogeneous reactions involving oxygen are applied, leading to the simultaneous production of carbon monoxide and carbon dioxide. The Arrhenius expression for reaction (R6) is valid only for T∞ ≤ 1650 K. Above the critical temperature, the reaction rate coefficient is replaced by k = 2.632 ⋅ 10−5 T 2 − 0.03353T

(9.1)

see [28]. Since ANSYS Fluent® does not allow for the native definition of reaction rate coefficients, the reaction rate given by Eq. (9.1) of the heterogeneous reaction (R6) was implemented using the so-called user-defined function (UDF). The reactions include water as a catalytic species. A detailed study of the influence of water was provided by Kestel et al. [3]. The authors showed that, even for low water content, the catalytic effect of water vapor cannot be neglected. The model assumptions described above lead to a system of 2D, axisymmetric, steady-state Navier–Stokes equations coupled with species and energy conservation equations. Gas–gas and solid–gas radiation were taken into account by Table 9.2

Reaction mechanisms for homogeneous and heterogeneous reactions.

Equations

A

E, J kmol−1

n

References

R1 R2 R3 R4 R5 R6 R7

2.24 ⋅ 1012 m kmol−0.75 s−1 2.75 ⋅ 109 m kmol−1 s−1 9.98 ⋅ 1010 4.605 3.007 ⋅ 105 593.83 m s−1 K−1 11.25 m s−1 K−1

1.6736 ⋅ 108 8.368 ⋅ 107 1.205 ⋅ 108 1.751 ⋅ 108 1.4937 ⋅ 108 1.4965 ⋅ 108 1.751 ⋅ 108

0 0 0 1 0 1 1

Turns [28] Jones and Lindstedt [29] Richter et al. [6] Libby and Blake [30] Caram and Amundson [31] Date [32] Libby and Blake [30]

247

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9 Pore-Resolved Simulation of Char Particle Combustion/Gasification

applying a P1 radiation model. Details about the governing equations, material properties, boundary treatment, and the calculation of chemical source terms are provided in Chapter 8. Details on the radiation model setup used in this work can be found in [6]. Some remarks should be made concerning the heat transfer inside the solids and the radiation. In the previous chapter, for the validation against data from Bejarano and Levendis [25], the heat transfer inside the nonporous spherical particle was not considered. At the particle wall, radiative heat transfer was applied via the P1 model, and an additional Stefan–Boltzmann radiation term was set to incorporate the radiation to the particle far field. In this chapter, the calculations for the pseudo-porous particles incorporate heat transfer inside the particles, and all results are compared against those for a nonporous particle with heat transfer in the solid body. The consideration of heat transfer in solid bodies prevents the application of a Stefan–Boltzmann radiation term in ANSYS Fluent®. For that reason, the results for the solid particle deviate from those discussed in the previous section. Special attention was paid to the definition of the radiative heat exchange at the solid bodies. Because of the significance of radiative heat transfer and the uncertainties in the setting up of the radiation boundary conditions, the approximation of radiation is a critical point in particle modeling, and has to be checked carefully. In spite of these uncertainties, the fundamental principles concerning intraparticle phenomena can be captured using the assumptions made above. Here, the following assumptions were made: In the case of shape a, the whole particle radiates. For shapes b and c (particles with gaps), only the outer surface takes part in radiation, and not the gaps. In the case of shape d, all faces that belong to the outer two rows and face outward take part in the radiation processes. Because of the fact that for shape d the radiative surface area is larger compared to the solid particle, the emission coefficient was adjusted to match the radiative surface of the solid particle. For shape e, the assumption was made that only the outer surface radiates. To highlight the final settings, all radiative zones are marked in Figure 9.1. It can be concluded that for all particle shapes the radiative surface is approximately equal. 9.2.1 Numerical scheme, discretization, and software validation

In this work, ANSYS-FLUENT® V 14.0 [33] was used to solve the set of governing equations. The pressure-based coupled algorithm was applied; this denotes the solution of the momentum equations and the pressure-based continuity equation as a coupled system of equations. The remaining equations (species conservation, incident radiation, energy conservation) are solved in a segregated way. The spatial discretization of convective terms is based on the quadratic upstream interpolation for convective kinematics (QUICK) scheme [34]. Figure 9.2 shows the computational domain and numerical setup. To avoid blockage effects, the domain extends 40 Dp in a radial direction, 30 Dp in an upstream direction, and 100 Dp in a downstream direction, with Dp the particle

9.3 Small Porous Particle: 90 �m

Outlet

40d

Symmetry

u∞ T∞ Porous particle Rotational axis 30d

100d

Figure 9.2 Computational domain and numerical setup.

diameter. Details of the numerical grid are shown in Figure 9.3. For the solid particle, the first 20 cell rows around the particle surface have a length of 0.01 Dp only, and the particle is discretized with 100 cells along the half circumference. This fine resolution guarantees the resolution of all near-wall effects. Additionally, grid independence was ensured by additional calculations with coarser and finer grids (not shown here). Shapes b and c follow this discretization, but the resolution of the gaps leads to much finer cells inside and around the gaps, which increases the number of cells by a factor of 2–3. Shape d features a much finer resolution around the rings in order to ensure that effects that occur at length scales related to those substructures are resolved. For that reason, the final grid has ∼8 times more cells than the nonporous particle. A special case is geometry e. The different layers are interrupted and form small gaps. Because of the Stefan flow from the particle center, a jet flow at the gaps may be possible. For that reason, the gaps are discretized, for example, at the outer layer with 13 cells. In the case of jet flow, recirculation occurs. Since it is not a priori known whether a jet flow occurs and how large the recirculation domain is, the flow field in the vicinity of the particle is meshed equidistantly. That is why the number of cells is 22 times larger than that for the solid particle. The heat transfer inside the solid was considered, too. For that reason, the particles themselves are also meshed. Depending on the specific shape, the mesh of the solid bodies consists of triangle or quadrilateral cells, which are about the same size. All meshes related to the fluid consist of quadrilateral elements only. The different mesh sizes are tabulated in Table 9.3. The numerical model used in this chapter has been extensively applied in previous works. The validation of the software we use is described in detail in Chapter 8.

9.3 Small Porous Particle: 90 �m

In this section, the influence of the inner particle structure on the char conversion process of a 90 μm particle falling at its terminal velocity in a hot atmosphere at

249

250

9 Pore-Resolved Simulation of Char Particle Combustion/Gasification

0.6

0.6

0.4

0.4

0.4

0.2

0

y /d

y/d

y/d

0.6

0.2

0.2

−0.6

−0.4

−0.2

0

0

−0.6

−0.4

Solid particle

−0.6

−0.4

0.15

0.15 y /d

0.2

0.1

0.05

−0.2

0

x /d

Short gaps

(b)

0.2

y /d

0

0

x /d

x /d

(a)

−0.2

(c)

Long gaps

0.1

0.05

0 −0.55

−0.5

−0.45

−0.4

−0.35

0

−0.55

−0.5

x /d

(d)

−0.45

−0.4

−0.35

x /d

Rings

(e)

Layers

Figure 9.3 Numerical grids for different particles. Table 9.3

Grid size for different particle shapes.

Shape

Description

Number of control volumes

a b c d e

Nonporous Short gaps Long gaps 27 rings 10 layers

2.23 ⋅ 104 4.03 ⋅ 104 6.38 ⋅ 104 1.57 ⋅ 105 4.47 ⋅ 105

about 1400 K is discussed. The atmosphere consists of 20% oxygen, balanced with nitrogen. Integral characteristics referring to the surface-averaged carbon consumption ṁ C and the surface-averaged temperature Ts were calculated using the following equations: N

ṁ C =

p 1 ∑ 1 ṁ d S , Np i=1 Sp,i ∮Sp,i C,i p,i

(9.2)

9.3 Small Porous Particle: 90 �m

N

Tsurf =

p 1 ∑ 1 T dS . Np i=1 Sp,i ∮Sp,i s,i p,ii

(9.3)

Volume-averaged quantities inside the porous structures are calculated using the following relations: Yi,inside =

|u|inside =

1 Y d V, V ∫V i 1 V ∫V

√

u2x + u2y + u2z d V .

(9.4)

(9.5)

The values of the integrals are given in Table 9.4. Figures 9.4 and 9.5 illustrate the impact of the porous structure on the temperature field inside and around the reacting char particle. As shown in these figures, the highest temperatures for porous and nonporous particles are located directly at the outer surface of the particle, where exothermic reactions, namely partial oxidation reactions, take place. For a porous particle, additional reactions take place inside the particle and influence the temperature distribution along the particle. As shown in Figure 9.6b, the oxygen content is zero inside the pores, except for a very small region close to the inflow sections of the pores. For that reason, only endothermic reactions are present inside the particle, namely the Boudouard reaction (reaction (R4)) and the gasification reaction (reaction (R7)). For example, because of the Boudouard reaction, the CO2 content decreases inside the particle, while additional CO is produced (compare Figure 9.6). The endothermic reaction rates are relatively low at 1400 K, so only a moderate temperature drop of ∼10 K in the inward direction can be detected for shapes b and c. The situation changes for geometry d, which consists of several rings. Because of the large pore size, oxygen penetrates the outer part of the particle and reacts with carbon at the two outer rows. The internal surface is increased, too. The DamköhlerII numbers for the endothermic reactions at 1400 K are much smaller than unity. For that reason, the heterogeneous reactions are kinetically controlled, and hence they directly scale with the reacting surface. For shape d, it explains the increased reaction rates and therefore the larger temperature drop of ≈ 30 K from particle surface to center point, compared to shapes b and c. For the last geometry, which consists of 10 layers, this effect is amplified because of the 3 times larger reacting surface. The temperature gradient along the particle radius exceeds 50 K. It becomes obvious that in this case intraparticle reactions no longer scale with the internal surface, but are limited as a result of the diffusive mass transport to the solid surface. Even though the maximum CO2 drop occurs for the layers, which indicates the largest reaction rates of the Boudouard reaction, the CO2 content does not become zero (see also Table 9.4). The CO2 distributions inside the gaps or between the layers are highlighted in Figure 9.5. Additionally, Table 9.4 lists the outer surface temperature Tout and the particle-averaged temperature Tsurf .

251

252

9 Pore-Resolved Simulation of Char Particle Combustion/Gasification 1.25 1700 1800 1900 2000 2100 2200 2300

1 00 20 50

00

21

21

20

2170

50

00 21

0.5

50 20

r/Dp

0.75

0.25 0 −1

−0.75 −0.5 −0.25

0

0.25

0.75

0.5

1

x/Dp

(a)

Solid particle

1.25

1.25 1700 1800 1900 2000 2100 2200 2300

00

50 20 00 00

21 22

0.25

0.5

0.75

0 −1

1

2240

0

00 21 0 22

215

0

−0.75 −0.5 −0.25

x/Dp

(b)

(c)

1.25 1700 1800 1900 2000 2100 2200 2300 205

0

1

2280

Rings

0 220

80

90

0.25

2150

21

0

21

0 218 2170

−0.75 −0.5 −0.25

0.5

2200

50

50

21

1700 1800 1900 2000 2100 2200 2300

2270

22

r/Dp

r/Dp

0.75

0.75

2250 2240

2260

2230

0.25

0.5

0.75

0 −1

1

x/Dp

(d)

0.5

Long gaps

1

2100

0.25 0 −1

0.25

1.25

0.75 0.5

0 x/Dp

Short gaps

1

50

00

0.25

−0.75 −0.5 −0.25

21 0 225 2250

0

0 −1

0.5

21

0.25

2230

2240

0.75

00

r/Dp

20

21

0.5

00

50

20

5 20

0.75

1700 1800 1900 2000 2100 2200 2300

1

20

215

00

r/Dp

1

−0.75 −0.5 −0.25

0

0.25

0.5

0.75

1

x/Dp

(e)

Layers

Figure 9.4 Temperature field for 90 μm particles with different porous structures. The gas velocity is approximately zero. T∞ = 1400 K.

The ambient gas-flow velocity of 0.1 m s−1 corresponds to the terminal velocity of a 90 μm particle falling in a drop-tube reactor at similar atmospheric conditions. Because of the low gas-flow velocity, and therefore the low Reynolds number (0.04), the momentum and thermal boundary layers are large. The ratio of the thermal boundary layer and the particle diameter can be estimated as �t ∕Dp = 2.84Re−0.6 ≈ 20 [35], which is close to the numerically estimated value 22. For reacting particles, Stefan flow is present, which is the carbon mass flow from

9.3 Small Porous Particle: 90 �m

0.2

0.2 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16

2190 2200 2210 2220 2230 2240 2250 2260 2270

0.15 r/Dp

r/Dp

0.15 0.1

0.1 0.05

0.05 0 −0.6

−0.55

−0.5

−0.45

−0.4

−0.35

−0.3

0 −0.6

−0.25

−0.55

−0.5

x/Dp

(a)

0.2

2235

2247.5

2260

(b)

2272.5

−0.35

−0.3

−0.25

0

0.025 0.05 0.075

0.1

0.125 0.15

0.15 r/Dp

r/Dp

−0.4

Long gaps, YCO2

0.2

2285

0.15 0.1 0.05

0.1 0.05

−0.55

−0.5

−0.45 −0.4

−0.35

−0.3

0 −0.6

−0.25

x/Dp

(c)

−0.45

x/Dp

Long gaps, T

0 −0.6

253

Layers, T

−0.55

−0.5

−0.45 −0.4

−0.35

x/Dp

(d)

Layers, YCO2

Figure 9.5 Zoomed views of the temperature field inside 90 μm porous particles with different internal structures.

the particle surface due to heterogeneous reactions. The Stefan flow widens the boundary layer, which is the reason for the larger boundary layer thickness compared to the semiempirical correlation. For a nonpermeable particle, the Stefan flow is of order 1, and therefore 10 times larger than for ambient gas flow. This is illustrated in Figure 9.7a. For the different porous structures, the additional mass flow from intraparticle reactions are conducted to the outside of the particle. Depending on the pore size of the different shapes, the flow field gets accelerated and forms small jets penetrating the boundary layer. This effect can be seen in Figures 9.7b,c for particles with radial gaps, and is a maximum for particles that consist of several layers. The maximum velocity is between 2.8 and 3.1 m s−1 for the geometries with gaps, and achieves 7.9 m s−1 for the layers. The situation is different for geometry d. The gaps between the different rings are relatively large, which prevents the formation of jets. The maximum velocity is comparable with that for a nonpermeable particle. The question is how the Stefan flow affects the carbon conversion process. As discussed previously, inside the pores only endothermic heterogeneous reactions are present, namely the Boudouard and gasification reactions. The surface temperatures are ∼2100 K for T∞ = 1400 K and YO2 = 0.2. At these conditions, the

−0.3

−0.25

254

9 Pore-Resolved Simulation of Char Particle Combustion/Gasification

0.20 0.15 Yo2

T (K)

2200

Solid Short gaps Long gaps Rings Layers

2000 Solid Short gaps Long gaps Rings Layers

1800 −2

−1

0

0.10 0.05

1

0.00

2

−2

−1

x/Dp T

(a)

1

0.2

Solid Short gaps Long gaps Rings Layers

0.25

0.15 Yco2

0.2 0.15

0.1 Solid Short gaps Long gaps Rings Layers

0.1 0.05 0.05 0

−2

−1

0

1

0

2

−2

−1

x/Dp

(c)

2

Yo2

(b)

0.3

YCO

0 x/Dp

0

1

2

x/Dp

(d)

YCO

YCO2

Figure 9.6 Temperature and species mass fractions along the symmetry axis. Y∞,O2 = 0.2, |u|∞ = 0.1 ms−1 (Re ≈ 0.04), Dp = 90 μm, T∞ = 1400 K.

′′

Table 9.4 Carbon consumption rate ṁ C , carbon mass flux ṁ C , average temperature at outer surfaces (Tout ), temperature averaged over all particle walls (Tsurf ), and volumeaveraged velocity magnitude (|⃗u|inside ), and YCO,inside mass fraction inside the porous particles. Particle size is 90 μm, T∞ = 1400 K. Shape

ṁ C �g s−1

ṁ ′′ C kg s−1 m−2

Tout (K)

Tsurf (K)

|⃗ u|inside m s−1

YCO,inside

Nonporous Short gaps Long gaps Rings Layers

5.20 5.48 5.83 5.21 6.12

0.204 0.129 0.096 0.065 0.031

2177 2227 2253 2199 2281

2177 2234 2254 2194 2262

— 0.757 0.767 0.566 0.358

— 0.160 0.132 0.153 0.083

9.3 Small Porous Particle: 90 �m

255

1

r/Dp

0.2

0.1

0 −0.6

−0.5

−0.4

−0.2

−0.3

x/Dp

Solid particle

(a)

1

1

r/Dp

0.2

r/Dp

0.2

0.1

0 −0.6

0.1

−0.5

−0.4

−0.3

0 −0.6

−0.2

−0.5

x/Dp

(b)

−0.4

−0.3

(c)

Short gaps

Long gaps

1

1 0.2

r/Dp

r/Dp

0.2

0.1

0.1

0 −0.6

−0.5

−0.4

−0.3

0 −0.6

−0.2

x/Dp

(d)

−0.2

x/Dp

Rings

−0.5

−0.4

−0.3

x/Dp

(e)

Layers

Figure 9.7 Stefan flow at the particle surface. Y∞,O2 = 0.2, |u|∞ = 0.1 ms−1 (Re ≈ 0.04), Dp = 90 μm, T∞ = 1400 K. For rings, only every fifth vector is plotted, for layers every twentieth.

DamköhlerII numbers for the endothermic reactions are much smaller than unity, so the intraparticle reactions are mainly controlled by the reaction kinetics, and less by the diffusive species transport. At the outer particle surface, exothermic reactions take place because of the presence of oxygen. The DamköhlerII numbers of the exothermic reactions are larger than unity, so the modified mass transport

−0.2

256

9 Pore-Resolved Simulation of Char Particle Combustion/Gasification

due to the Stefan flow changes the reaction rates. The jet flow, on the other hand, influences locally the temperature and species concentration around the particle. The effect is not dominant, but can be identified in Figure 9.4, for example, for shapes b and d. The overall carbon consumption rate, together with the carbon mass flux, is listed in Table 9.4. As shown in this table, the carbon consumption is maximum for shape e, which features the maximum internal surface. It is worth noting that the carbon consumption in this setup depends on the exothermic reactions at the particle’s outside and on the intraparticle endothermic reactions. For that reason, a linear dependence between the internal surface and carbon consumption cannot be identified, and the carbon mass flux is not constant for the different shapes. This becomes obvious for shape d, which consists of several rings. The internal surface is 3 times larger compared to a solid particle, but because of the ring shape the outer surface is smaller. In that case, the intraparticle reactions are increased, but the exothermic reactions at the particle outside are reduced. Finally, the carbon consumption is similar to that for a nonporous particle. 9.3.1 Influence of Gas Temperature

In this section, we study the influence of the ambient gas temperature on the species and temperature distribution in and around porous particles. The model setup is similar to the one used in the previous section, that is, the particle diameter is 90 μm and the oxygen content in the far field is 0.2. Calculations were carried out for T∞ = 2500 K, and the results are compared against data taken from the previous section. To focus on the basic phenomena related to the gas temperature, only rings and layers are considered and compared against nonpermeable particles. At higher gas temperatures, the endothermic and exothermic reaction rates are significantly increased. Considering a nonpermeable particle, the oxygen at the particle surface is zero. For that reason, only endothermic reactions, mainly the Boudouard reaction, take place at the particle surface. Because of the Boudouard reaction, CO2 is consumed. The CO produced diffuses away, and reacts with oxygen to form CO2 in a small sheet around the particle, the so-called flame sheet. The carbon dioxide created in that way diffuses back to the particle surface and is a new reactant for the Boudouard reaction. The formation of the flame sheet, together with the maximum CO content close to the particle and the CO2 peak at the flame center, is illustrated in Figure 9.8. From this figure, it becomes obvious that porous particles, at least the small ones, feature comparable characteristics. All of the shapes studied exhibit a characteristic flame sheet at 2500 K, which is detached from the particle surface. Inside the particles, the endothermic reactions are controlled by the reaction kinetics and, at the same time, by the diffusive species transport. The endothermic reactions are maximized for the layers, which holds true at 1400 K and 2500 K. For that

9.4

Table 9.5 Shape

Nonporous Rings Layers

Large Porous Particle: 2 mm

Characteristic values inside the porous structures for T∞ = 2500 K. Dp = 90 μm. ′′

ṁ C (�g s−1 )

(kg (s−1 m−2 ) )

Tout (K)

Tsurf (K)

|⃗ u|inside (m s−1 )

YCO2 ,inside

9.01 7.58 9.34

0.354 0.095 0.048

2874 2919 3063

2874 2864 3009

— 1.271 0.714

— 0.116 0.027

ṁ

C

reason, this geometry features the lowest CO2 and the largest CO content inside the particle. The situation becomes more difficult if the overall carbon consumption is compared for different particle shapes, see Table 9.5. Results for 1400 K show that all porous particles feature an increased carbon consumption rate. This does not hold true for higher gas temperatures. At these temperatures, almost all oxygen is consumed within the flame sheet, and oxygen cannot reach the particle surface. The temperature distribution depends therefore on the amount of CO produced by the Boudouard reaction inside and on the outside of the particle. The endothermic reactions are maximized for shape d, which features the maximum inner surface. This is illustrated by the minimum CO2 content and the maximum CO content inside the particle. The temperature drop inside the particle, caused by the endothermic reactions, and the temperature rise outside of the particle due to the increased amount of CO are the reasons for the larger temperature gradients seen in Figure 9.8a. The corresponding carbon consumption rates are listed in Table 9.5. The nonpermeable particle shows the largest carbon flux, since all heterogeneous reactions are concentrated at the particle’s outside. With increasing surface, the carbon flux decreases, but because of the diffusion-limited reactions, the relation between carbon flux and inner surface is not linear. Interestingly, the overall carbon consumption for the rings is below that for a nonpermeable sphere, whereas the layers show the maximum carbon consumption.

9.4 Large Porous Particle: 2 mm 9.4.1 Small Reynolds Numbers

In chemical reactors, the particle size is generally not limited to small values. In fluidized-bed reactors, particles can achieve millimeter scales, but also in some entrained-flow reactors particle larger than 1 mm can be found. For that reason, the impact of the particle geometry on the conversion process of a 2 mm particle is considered in this section. Consequently, larger particles should be modeled as a geometrical extension of the particles discussed previously, which means,

257

258

9 Pore-Resolved Simulation of Char Particle Combustion/Gasification

Solid, 1400 K Rings, 1400 K Layers, 1400 K Solid, 2500 K Rings, 2500 K Layers, 2500 K

0.15

YO2

T − T∞ (K)

800 600

0.1 Solid, 1400 K Rings, 1400 K Layers, 1400 K Solid, 2500 K Rings, 2500 K Layers, 2500 K

0.05

400

0

200

−2

−1

0

1

2

−2

−1

x/Dp (a)

(b)

T 0.35

Solid, 1400 K Rings, 1400 K Layers, 1400 K Solid, 2500 K Rings, 2500 K Layers, 2500 K

0.3

0.2 0.15

(c)

0.2 0.15 0.1 0.5

0 x/Dp YCO

1

Solid, 1400 K Rings, 1400 K Layers, 1400 K Solid, 2500 K Rings, 2500 K Layers, 2500 K

0.25

0.5 −1

2

Y O2

0.1

−2

1

0.3

YCO2

YCO

0.25

0

0 x/Dp

0

2

−2 (d)

−1

0 x/Dp

1

2

YCO2

Figure 9.8 Temperature and species mass fractions along symmetry axis for different gas temperatures. Dp = 90 μm.

9.4

Large Porous Particle: 2 mm

for example, adding additional layers or rings. But, this strategy would result in a rapidly increasing modeling and calculation effort, which prevents a detailed investigation. As an alternative, the larger particles are modeled by a scale-up of the previously defined particles. The disadvantage of that approach is that the surface density S′′′ is changed by a factor of Dp,old ∕Dp,new . The change from 90 μm to 2 mm particles leads to a value of S′′′ that is only 5% of the old one, so it is 3000 m2 m−3 for the nonpermeable particle and 23 040 m2 m−3 for layers. This, in general, prevents a direct comparison between particles of different sizes, but it is still possible to study the influence of Re and T∞ on the carbon conversion rate for such particles. In order to compare the results for different particle sizes, the gas-flow velocity was kept constant at 0.1 m s−1 , which results in an increased Reynolds number for the 2 mm particles. At 1400 K, the flame sheet is detached from the particle surface, which is illustrated in Figure 9.9. The impact of the enhanced inner surface becomes obvious by the reduced temperature and CO2 distribution inside the particle. Even for larger particles at the same gas velocity, the Reynolds numbers are below 1, and hence the Sherwood number is between 2 and 2.5. For that reason, the diffusive mass transport varies for different particle sizes with approximately the ratio of the different particle diameters. The exothermic reactions, which means reactions involving oxygen, occur only at the particle’s outside and dominate the overall carbon consumption. Even at 1400 K, the exothermic processes are mainly controlled by diffusive mass transport. In Table 9.6, the carbon consumption rates for different particle shapes at 1400 K and 2500 K ambient gas temperatures are listed. A comparison with Table 9.4 reveals that for nonpermeable particles there is an increase in the carbon conversion rate by a factor between 17.4 (T∞ = 1400 K) and 29.9 (2500 K). This value is comparable to the particle diameter ratio, which is 22. For porous particles, the situation is not straightforward. While for 90 μm particles at 1400 K, the increased inner surface results in carbon conversion rates that are increased up to 18%, 2 mm particles behave more like a nonpermeable particle at 1400 K, with only a minor influence of the particle porosity on the carbon conversion rates and surface temperatures (maximum 3%). At 2500 K, the regime changes. Because of the increased gas temperature, also the endothermic reactions are partially controlled by mass diffusivity. In that situation, the particle behavior is like that for a nonpermeable sphere with a reduced outer surface. The increased reaction rates inside the particle due to the enhanced inner surface are not capable of compensating the effect. In summary, at 2500 K the carbon conversion rates of porous particles are reduced by up to 42%. 9.4.2 Large Reynolds Numbers

The terminal velocity of a 2 mm particle in a 20% oxygen/80% nitrogen atmosphere at 1400 K is ∼11.5 m s−1 . This value is two orders of magnitude above the gas velocity considered in the previous section, and results in a Reynolds number of

259

9 Pore-Resolved Simulation of Char Particle Combustion/Gasification

1400 1200

T − T∞

1000 800 600

Solid, 1400 K Rings, 1400 K Layers, 1400 K Solid, 2500 K Rings, 2500 K Layers, 2500 K

400 200 0 −1.5

−1

−0.5

0

0.5

1

1.5

x/ Dp (a)

T 0.3 0.25 0.2

YCO2

260

0.15

Solid, 1400 K Rings, 1400 K Layers, 1400 K Solid, 2500 K Rings, 2500 K Layers, 2500 K

0.1 0.05 0 −1.5

−1

−0.5

0

0.5

1

1.5

x /Dp (b)

YCO2

Figure 9.9 Temperature and species mass fractions along the symmetry axis for different gas temperatures. Dp = 2 mm.

115. For such Reynolds numbers, the flow field around a nonpermeable sphere is still in steady state and axisymmetric, but compared to the conditions in the previous section the convective heat and mass transfer are significantly enhanced. We study this situation in the following section. Figure 9.10 shows the velocity field for different particle shapes at 1400 K and 2500 K. For the rings, the flow field deeply penetrates the porous structure at smaller temperatures, and some fractions of the gas flow pass the whole particle. If the gas temperature is increased, the Reynolds number drops by a factor of approximately 2.6 because of the decreased density and increased viscosity. The boundary layer thickness increases, and even for the rings the outer gas flow cannot penetrate the porous structure. The particle behaves in that case like a nonporous structure (not shown here), but the outer surface is reduced because of the discontinuous particle structure.

9.4

Large Porous Particle: 2 mm

261

Table 9.6 Characteristics for 2 mm porous particles at different ambient gas temperatures. |⃗u|∞ = 0.1 m s−1 . ′′

T∞ (K)

ṁ C (�g s−1 )

ṁ C (kg (s−1 m2 ))

Tout (K)

Tsurf (K)

|⃗ u|inside (m s−1 )

YCO2 ,inside

Nonporous Rings Layers

1400 1400 1400

90.7 87.3 93.2

0.00722 0.00222 0.00097

1536 1529 1553

1536 1514 1533

— 0.042 0.042

— 0.201 0.204

Nonporous Rings Layers

2500 2500 2500

269 190 198

0.0214 0.00483 0.00205

2518 2510 2530

2518 2484 2491

— 0.070 0.068

— 0.251 0.248

Shape

0.15

1

1

r/Dp

r/Dp

0.15

0.1

0.05

0.05

0 −0.6

0.1

−0.5

0 −0.6

−0.3

−0.4

−0.5

x/Dp

Rings, T∞ = 1400 K

(a)

−0.4 x/Dp

Layers, T∞ = 1400 K

(b) 0.15

1

1

r/Dp

r/Dp

0.15

0.1

0.05

0.05

0 −0.6

−0.5

−0.4

0 −0.6

−0.3

Rings, T∞ = 2500 K

−0.5

−0.4 x/Dp

x/Dp

(c)

0.1

(d)

Layers, T∞ = 2500 K

Figure 9.10 Velocity field around particles at larger Reynolds numbers. Dp = 2 mm, |u|∞ = 11.5 m s−1 . For rings, only every fifth vector is plotted, for layers every twentieth.

For the layers, the situation is different. At both temperatures, the flow field is similar to that for a nonpermeable sphere. At the small gaps, the Stefan flow from inside the particle dominates the mass flow, so the outer flow field cannot penetrate the porous structure at the open gaps. This impact of the outer gas flow is observable in the temperature field. From Figure 9.11, it can be seen that the flame sheet at T∞ = 1400 K is shifted in the

262

9 Pore-Resolved Simulation of Char Particle Combustion/Gasification

downstream direction, and in the region close to the forward stagnation point the flame sheet is partially located inside the porous particle. Because of the internal endothermic reactions, the temperature inside the particle drops by about 250 K. At 2500 K, the flame sheet is mainly located at the outside of the particle. The impact of the endothermic reactions on the inner temperature is large, with a temperature drop of up to 750 K. Figure 9.12 gives the temperature and CO2 distribution along the symmetry axis of the particle. At T∞ = 1400 K, the CO2 mass fraction is minimized if the layers are considered. At 2500 K, the CO2 inside all considered porous particles is fully consumed, which indicates diffusion-controlled endothermic processes inside the particle. Both figures show that at 2500 K gas temperature the flame sheet around the porous particles is located closer to the particle surface, partially inside the pores, compared to the nonpermeable particle. The overall carbon conversion rates reflect the findings discussed above. At 1400 K, the conversion rates are comparable for all geometries. The reduced outer surface of the porous particles, which results in reduced exothermic reactions, is compensated by the increased endothermic reactions inside the particle. At 2500 K, the inner processes are limited by the diffusive CO2 transport. The impact 0.75

0.75 1400 1500 1600 1700 1800

0.5 1775

0.25

r/Dp

1400

r/Dp

0.5

1600

1800

1700

−0.5

−0.25

0

1600 1800

0.25

0

0.5

1750

−0.5

−0.25

x/Dp

(a)

0

0.25

0.5

x/Dp

(b)

Rings, T∞ = 1400 K 0.75

1825

1850

0.25 1550

0

1400 1500 1600 1700 1800

1400

Layers, T∞ = 1400 K 0.75

2500

2500 2450 2650 2850 3050 3250

2400 2600 2800 3000 3200 3400

0.5 r/Dp

r/Dp

0.5

3400

3100

3400

0.25

3200

0.25 2450 2450

0

−0.5

−0.25

3050

0 0

0.25

0.5

Rings, T∞ = 2500 K

−0.25

0

2500

0.25

0.5

x/Dp

x/Dp

(c)

−0.5

2500

(d)

Layers, T∞ = 1400 K

Figure 9.11 Temperature field for 2 mm particles at different gas temperatures. |⃗u|∞ = 11.5 m s−1 .

9.4

Large Porous Particle: 2 mm

263

1400 1200

T − T∞

1000 800 600

Solid, 1400 K Rings, 1400 K

400

Layers, 1400 K Solid, 2500 K

200

Rings, 2500 K Layers, 2500 K

0 −1.5

−1

−0.5

0

0.5

1

1.5

x/ Dp (a)

T − T∞

0.3 0.25

YCO2

0.2 0.15

Solid, 1400 K Rings, 1400 K

0.1

Layers, 1400 K Solid, 2500 K

0.05

Rings, 2500 K Layers, 2500 K

0 −1.5

−1

−0.5

0

0.5

1

1.5

x/ Dp (b)

CO2

Figure 9.12 Temperature and species mass fractions along symmetry axis. Y∞,O2 = 0.2, |u|∞ = 11.5 m s−1 , Dp = 2 mm.

264

9 Pore-Resolved Simulation of Char Particle Combustion/Gasification

Table 9.7

Characteristic values for 2 mm particles. |u|∞ = 11.5 m s−1 .

Shape

T∞ (K)

ṁ C (�g s−1 )

ṁ ′′ C (kg (s−1 m2 ))

Tout (K)

Tsurf (K)

|u|inside (m s−1 )

YCO2 ,inside

Nonporous Rings Layers

1400 1400 1400

315 287 318

0.0258 0.00731 0.00330

1691 1670 1719

1691 1638 1695

— 8.41 8.25

— 0.0680 0.0646

Nonporous Rings Layers

2500 2500 2500

568 370 389

0.0452 0.00940 0.00403

2545 2542 2580

2545 2496 2523

— 7.76 7.62

— 0.104 0.104

of the reduced reactions at the outside of the particles cannot be compensated by the internal reactions anymore. For that reason, the overall carbon consumption rate is reduced by up to 35%. The carbon mass flux indeed is relatively small compared to smaller particles. These findings are similar to those for 2 mm particles at 0.1 m s−1 gas-flow velocity, with carbon mass flux increased by a factor between 2 and 3.5. Finally, in conclusion, it should be noted that the use of the PSS approach for the modeling of carbon oxidation in O2 -based atmosphere should be carefully validated against simulations, taking into account the pore growth effect.

9.5 3D Simulations under Gasification Conditions

As discussed in Section 9.1, the real porosity is defined by a random pore structure, which is originally three-dimensional. From the modeling effort and because of the massive computational costs, it is impossible to resolve and simulate realistic three-dimensional porous structures. As a first step toward the simulation of a fully resolved porous particle, we performed numerical calculations of char particle conversion under gasification conditions. We focus on the gasification regime, since in that regime the largest influence of the intraparticular effects is expected. In this section, we present selected results of CFD-based simulations of a 3D porous particle moving in a hot gas, which has the composition YCO2 = 0.3, YCO = 0.2, YH2 O = 0.3, and relaxed to the chemical equilibrium. The gas composition is a typical composition that can be found in autothermic entrained-flow gasifiers, or in endothermic reactors for biomass conversion (see e.g., Carbo-V technology). The internal structure of the porous particle was obtained numerically using a three-dimensional cut from a 3D column of spherical particles sedimented into a cylindrical cavity [36]. The mesh generation of such packed-bed structures is complex in general, since the well-known contact point problem leads to a variety of geometrical singularities. To overcome this problem, we use additional bridges between pores (see (a)). This strategy allows numerical meshes that feature a significantly increased mesh quality. The validity of this approach was

9.5

3D Simulations under Gasification Conditions

discussed in detail by Dixon et al. [37]. Figure 9.13 illustrates the porous particle. ̇ 6, The total number of control volumes in the whole domain is about 5.110 6 with about 3.5 ⋅ 10 control volumes inside the particle. The total surface ratio between porous and nonporous spheres is 3.13, giving a specific surface ratio of ′′′ S ≈ 105 m2 m−3 for a 200 μm particle. Because of the stiff chemical system and the unstructured mesh, and of the large number of control volumes, it required approximately 5 ⋅ 105 iterations to achieve a fully converged solution. This results in a computational time of nearly 2 months, involving 64 Intel™ Xeon™computing cores. This computational effort prohibits a systematic numerical analysis of char gasification under different conditions, but it allows first insights into the complex intraparticle char conversion. In this work, we concentrated on the char gasification of a 200 μm char particle at one dedicated atmosphere, one Reynolds number (0.01), and two ambient gas temperatures, namely 1250 K and 1800 K. The temperature distribution inside the particle at T∞ = 1250 K is illustrated in Figure 9.14a. It can be seen that toward the particle center the temperature drops because of the endothermic character of the heterogeneous reactions, namely the Boudouard reaction C + CO2 (R4) and the gasification reaction C + H2 O (R7). Exothermic reactions are not active because of the lack of oxygen. It should be noted that even though a temperature gradient is present inside the particle, the magnitude of the gradient with 3000 K m−1 is very small and can be neglected in simplified models (e.g., for submodel development). The velocity field for T∞ = 1250 K is depicted in Figure 9.14b. The main carbon mass flow is produced inside the porous structure, leading to a continuous mass flow from the inside of the particle to the ambient gas (Stefan flow). The situation changes with increasing gas temperature. At 1800 K, a distinct temperature gradient of about 2.93 ⋅ 105 K m−1 can be seen in Figure 9.15a, which is ≈100 times larger than the previous results for T∞ = 1250 K. This temperature gradient evokes a different carbon conversion rate depending on the radial position inside

(a)

(b)

Figure 9.13 3D view of a porous particle (Dp = 200 μm) used in this work. The total number of control volumes is about 5.1 ⋅ 106 . The total surface ratio between porous and non′′′ porous spheres is 3.12, giving a specific surface ratio of S of ≈ 105 m2 /m−3 .

265

266

9 Pore-Resolved Simulation of Char Particle Combustion/Gasification

Z x

Y

Temperature (K) 1249.54 1249.5 1249.45 1249.41 1249.37 1249.33 1249.29 1249.24 1249.2

Z x Y

(a)

(b)

Figure 9.14 Temperature distribution inside the porous particle (a) and streamlines of the gas flow predicted numerically for Tin = 1250 K, Dp = 200 μm (b). The ambient gas composition corresponds to YCO2 = 0.3, YCO = 0.2, YH2 O = 0.3. The streamlines are colored with the gas temperature. The Reynolds number is 0.01.

the porous particle. Modeling of such processes is more complex, since the temperature gradient along with the temperature-dependent reaction rates has to be taken into account. The overall carbon conversion rate is increased significantly, by about a factor of 120 compared to the conversion at 1250 K ambient gas temperature, which leads to an increased Stefan flow from the particle to the gas flow, and therefore the diffusive boundary layer is enhanced clearly. This phenomenon is illustrated in Figure 9.15b. The mass flow from the inside of the particle can exit the structure only at distinct locations. This leads to the formation of a variety of small jets consisting of CO and H2 O. This effect, together with the increased

z x

Y

Temperature (K) 1775.95 1772.29 1768.62 1764.96 1761.3 1757.64 1753.98 1750.31 1746.65

z x Y

Figure 9.15 (a) Temperature distribution and (b) streamlines of the gas flow for Tin = 1800 K.

9.6

Conclusions

boundary layer due to the Stefan flow, might have a significant influence on the terminal velocity of small-sized coal particles used in drop-tube experiments.

9.6 Conclusions

The analysis of the numerical simulations reveals that the Stefan flow plays a significant role in species transport inside the pores and between the particle and the surrounding gas, as well as in the heat transfer over the whole domain. For instance, at an ambient temperature of 1400 K, the velocity in a pore can reach values of about 8 m s−1 for a 200 μm particle. Additionally, the numerical simulations confirmed that in a kinetically controlled regime the small pores enhance the total carbon conversion rate in comparison to the case without pores. In oxygen-rich conditions, the increase of the pore surface (by adding small pores into the particle) leads to an increase in the carbon conversion rate and in the surface-averaged temperature at moderate gas temperatures of about 1400 K. In that case, all heterogeneous reactions are in a kinetically controlled or in a transitional regime. On the contrary, at high ambient temperatures (T∞ = 2500 K), the carbon conversion rate and the surface-averaged temperature reflect a balance between the endothermic and exothermic heterogeneous reactions, which depends on the mass transport toward and inside the different idealized pore structures. For 2 mm particles under oxidation conditions, the characteristics are different: the heterogeneous reactions are mainly controlled by the outer surface area. For that reason, the carbon conversion rate drops depending on the particle porosity. The analysis of porous particles undergoing gasification reveals clearly that, with increasing ambient gas temperature, the radial temperature distribution shows a significant temperature gradient, which is caused by the endothermic reactions that occur primary inside the particle.

Nomenclature Latin

A DP Da E k ′′ ṁ C N Re

Frequency factor, corresponding unit Particle diameter (m) Damköhler number (–) Activation energy (J kmol−1 ) Reaction rate, corresponding unit Carbon mass flux (kg s−1 m2 ) Number (–) Reynolds number (–)

267

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9 Pore-Resolved Simulation of Char Particle Combustion/Gasification

Carbon consumption rate (kg s−1 ) Surface (m2 ) Specific surface (m2 m3 ) Temperature (K) Velocity vector ((ux , uy , uz )T , m s−1 ) Volume (m3 ) Coordinate system ((x, y, z)T , m) Mass fraction (–)

ṁ C S S′′′ T u⃗ V x⃗ Y

Greek

�t �

Thickness of thermal boundary layer (–) Porosity (–)

Indices

∞ inside out p solid surf

Ambient gas condition Inside the porous particle Outer surface Part Solid fraction of the particle Surface

References 1. Lee, J., Tomboulides, A.G., Orszag, S.A.,

Yetter, R.A., and Dryer, F.L. (1996) A transient two-dimensional chemically reactive flow model: fuel particle combustion in a nonquiescent environment. 26th Symposium (International) on Combustion/The Combustion Institute, pp. 3059–3065. 2. Higuera, F.J. (2008) Combustion of coal particle in a stream of dry gas. Combustion and Flame, 152, 230–244. 3. Kestel, M., Nikrityuk, P., Hennig, O., and Hasse, C. (2012) Numerical study of the partial oxidation of a coal particle in steam and dry air atmospheres. IMA Journal of Applied Mathematics, 77, 32–46. 4. Schulze, S., Kestel, M., Safronov, D., and Nikrityuk, P.A. (2013) From detailed

description of chemical reacting coal particles to subgrid models for CFD: model development and validation. Oil & Gas Science and Technology, 68, 1007–1026. 5. Nikrityuk, P.A., Gräbner, M., Kestel, M., and Meyer, B. (2013) Numerical study of the influence of heterogeneous kinetics on the carbon consumption by oxidation of a single coal particle. Fuel, 114, 88–98. 6. Richter, A., Nikrityuk, P.A., and Kestel, M. (2013) Numerical investigation of a chemically reacting carbon particle moving in a hot O2 /CO2 atmosphere. Industrial and Engineering Chemistry Research, 52, 5815–5824. 7. Safronov, D., Kestel, M., Nikrityuk, P.A., and Meyer, B. (2014) Particle resolved

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20. van Heek, K.H. (2000) Progress of coal

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science in the 20th century. Fuel, 79, 1–26. Bhatia, S.K. and Perlmutter, D.D. (1980) A random pore model for fluid-solid reactions: I isothermal, kinetic control. AIChE Journal, 26 (3), 379–385. Bhatia, S.K. and Perlmutter, D.D. (1981) A random pore model for fluid-solid reactions: II. Diffusion and transport effects. AIChE Journal, 27 (2), 247–254. Smoot, L.D. and Smith, P.J. (1985) Coal Combustion and Gasification, The Plenum Chemical Engineering Series, Springer. Xu, Q., Pang, S., and Levi, T. (2011) Reaction kinetics and producer gas compositions of steam gasification of coal and biomass blend chars, part 2: mathematical modelling and model validation. Chemical Engineering Science, 66 (10), 2232–2240. Bejarano, P.A. and Levendis, Y.A. (2008) Single-coal-particle combustion in O2 /n2 and O2 /CO2 environments. Combustion and Flame, 153, 270–287. Sundaresan, S. and Amundson, N.R. (1980) Diffusion and reaction in a stagnant boundary layer about a carbon particle. 5. pseudo-steady-state structure and parameter sensitivity. Industrial and Engineering Chemistry Fundamentals, 19:344–351. Turns, S.R. (2000) An Introduction to Combustion: Concepts and Applications, McGraw-Hill. Turns, S.R. (2006) An Introduction to Combustion, 2nd edn, McGraw-Hill. Jones, W.P. and Lindstedt, R.P. (1988) Global reaction schemes for hydrocarbon combustion. Combustion and Flame, 73, 233–249. Libby, P.A. and Blake, T.R. (1981) Burning carbon particles in the presence of water vapor. Combustion and Flame, 41, 123–147. Caram, H.S. and Amundson, N.R. (1977) Diffusion and reaction in a stagnant boundary layer about a carbon particle. Industrial and Engineering Chemistry Fundamentals, 16, 171–181. Date, A. (2005) Introduction to Computational Fluid Dynamics, Cambridge University Press, New York.

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reynolds numbers. International Journal of Heat and Mass Transfer, 55, V 14.0 – Commercially available CFD 1343–1354. software package based on the Finite Volume method. Southpointe, 275 Tech- 36. Wittig, K., Schmidt, R., Schulze, S., and nology Drive, Canonsburg, PA 15317, Nikrityuk, P.A. (2013) 3d numerical U.S.A., www.ansys.com. simulation of a porous particle heating. Proceeding of the 21st Annual Con34. Leonard, B.P. (1979) A stable and accuference of the CFD Society of Canada, rate convective modeling procedure Sherbrooke, Quebec, Canada, May 69, based on quadratic upstream interpo2013. lation. Computer Methods in Applied Mechanics and Engineering, 19, 59–98. 37. Dixon, A.G., Nijemeisland, M., and Stitt, 35. Richter, A. and Nikrityuk, P.A. (2012) E.H. (2013) Systematic mesh development for 3d cfd simulation of fixed beds: Drag forces and heat transfer coefficients for spherical, cuboidal and ellipsoidal contact points study. Computers and Chemical Engineering, 48, 135–153. particles in cross flow at sub-critical

271

10 Subgrid Models for Particle Devolatilization-Combustion-Gasification Sebastian Schulze, Robin Schmidt, and Petr A. Nikrityuk

..as a global view, coal is, and will remain, the most important source of energy and feedstock K.H. van Heek [69]

10.1 Subgrid Model for the Devolatilization/Combustion of a Moving Coal Particle 10.1.1 State of the Art Pyrolysis Models One major process that determines the ignition behavior of coal is pyrolysis [1, 2]. Pyrolysis, or devolatilization, is the thermal decomposition of coal [3]. The pyrolysis process is complex, involving the nature of the coal, the heating rate, the maximum particle temperature, pressure, and particle size [4]. During pyrolysis, light gases, for example, CO2 , CO, H2 , H2 O, CH4 , C2 H4 , C2 H6 , and tar as a heavy fraction, are released from the coal. In the past decades, numerous mathematical models have been published describing the pyrolysis of a coal particle. These models differ significantly both in terms of their computational effort and in the physics included. In the following, the most important model approaches are mentioned; for a more detailed overview, see [5–7]. The simplest models describe the complex pyrolysis process using one global reaction rate. It is assumed that the pyrolysis rate is proportional to the amount of volatile species left in the particle. Using a single pyrolysis rate results in a mathematically simple model but one that applies only in cases with very similar ambient conditions [5, 6]. An increase in flexibility is obtained when multiple reactions are taken into account. Some schemes for multiple-reaction models are reported in [5, 6]. In principle, multistep models can predict the pyrolysis of coal and biomass (see the model of Sommariva et al. [8]). They describe coal pyrolysis with 20 reactions Gasification Processes: Modeling and Simulation, First Edition. Edited by Petr A. Nikrityuk and Bernd Meyer. © 2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA.

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representing the pyrolysis kinetics of three base coals. The coal under investigation was expressed as a linear combination of the base coals (for further details see [8]). It must be noted that the idea of multistep kinetics is old, having been originally developed by Pitt in 1962 [9]. Pitt [9] posited that coal pyrolysis can be described as a set of independent first-order reactions. Later, the idea was extended by Anthony and Howard [5], who stated that the activation energy for the decomposition of differed chemical bonds can be expressed using a Gaussian distribution. Models based on this assumption are called distributed activation energy models (DAEMs); for further details and examples, see [5, 10]. Finally, the most detailed type of model is the so-called network model. This assumes that pyrolysis is a depolymerization process, breaking the large molecular structures making up the coal into smaller fragments. The fragments can undergo a series of conversion processes such as repolymerization or further cracking. The models allow the prediction of the pyrolysis kinetics, light gas composition, and tar release. Three well-known network models are the CPD [11], FLASHCHAIN [12], and FG-DVC [13] models. Network pyrolysis models require large amounts of computational time and thus they can rarely be used directly in large-scale simulations. One way to reduce the numerical effort is to approximate the pyrolysis process obtained using a detailed network model with simple one- or two-rate expressions (e.g., see the work by Vascellari et al. [14]). However, because of the coupling between the pyrolysis process, the particle heating rate, and the maximal particle temperature, the calculation procedure must be repeated until convergence is reached. Heterogeneous Char Conversion Models In a gasifier, pyrolysis is followed by the

conversion of the remaining char. A review of the mainstream subgrid models for char combustion has been carried out by Edge et al. [15]. One of the first subgrid models was developed by Tu et al. [16] in 1934. That model included the diffusion resistance of oxygen and the chemical kinetics at the particle surface. The thickness of the boundary, and thus the diffusion resistance, was calculated as a function of the Re number (for details, see [16]). Nearly 40 year later, in 1971, Baum and Street [17] developed their widely used model. As in the model by Tu et al. [16], diffusion resistance and the chemical kinetics were included in the model. The overall resistance was calculated as a series of both resistances similarly to an electric circuit. However, unlike the model by Tu et al. [16], the Baum and Street model assumes that there is a stagnant ambient gas phase. The major drawback of both models is that only the reaction C + O2 is considered, while gasification reactions of the char with carbon dioxide or water are neglected. The coal conversion is normally divided into three regimes depending on the temperature. At low temperatures, the conversion rate is controlled by the reaction kinetics. With increasing temperature, the chemical kinetics accelerate and a mixed regime is reached where both the reaction kinetics and the diffusion rate determine the conversion rates. Finally, at very high temperatures, the conversion rate is limited by the diffusion rate to the particle [15]. In the kinetically controlled

10.1

Subgrid Model for the Devolatilization/Combustion of a Moving Coal Particle

regime in particular, the overall heterogeneous reaction rate depends on the inner surface of the particle. Thus, some conversion models consider the intrinsic surface of the particle. On the basis of the handling of the reacting surface of the particles, char conversion models can be divided in two basic types: surface-based models and intrinsic models. In surface-based models, the heterogeneous reactions are assumed to occur solely at the outer surface, as in the model by Tu et al. [16]. On the other hand, intrinsic conversion models take the inner surface of the particle into account to calculate the reaction rate. Meanwhile, different approaches are used to express the development of the inner surface; for example, the original Baum and Street model [17] used a so-called surface factor. The surface factor as a function of the char burnout was calculated by an empirical correlation [17]. Today, the random-pore model is widely applied to predict the progress of the intrinsic surface area during char burnout; for further information see the works by Bhatia and Perlmutter [18] and by Gupta and Bhatia [19]. Finally, one of the most detailed models for char conversion is the carbon burnout kinetic (CBK) model, originally published in 1998 by Hurt et al. [20]. Today, a number of different CBK models are available in the literature (for an overview, see [15, 21, 22]). The CBK/G model was basically developed for coal gasification. That model uses a Langmuir–Hinshelwood mechanism with eight reactions to describe the heterogeneous char conversion, and includes the annealing mechanism of the original CBK model. The concentrations of the gas species at the particle surface are calculated as a balance of the reaction rates and the diffusion through the boundary layer. To account for the intrinsic reactions, the concentration profile inside the particle is approximated using effectiveness factors based on the Thiele modulus. The development of the total particle surface during conversion is calculated using the random-pore model. For a more detailed description of the CBK/G model, see [21, 22]. All models cited in this paragraph assume that only heterogeneous reaction take place during the char conversion. The models do not consider the combustion of volatiles or CO, which are formed during char conversion, in the vicinity of the particle. Char Conversion Models with Homogeneous Reaction In general, particle conver-

sion models can be divided into three categories, namely one-film, two-film, and continuous models [23–25]. Figure 10.1 shows some characteristics of these models. The one-film models consider only the heterogeneous reactions at the particle; any homogeneous reactions in the boundary layer are neglected. Following this definition, the models discussed in the previous paragraph can be classified as one-film models. Two-film models, on the other hand, assume that CO or volatiles oxidize rapidly in a flame sheet within the boundary layer. The most realistic models are the continuous models, where the homogeneous reactions are allowed to take place

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Subgrid Models for Particle Devolatilization-Combustion-Gasification

Flame

CO2

Surface: Yj, T

Surface: Yj, T

δ O2 T

CO

CO2

O2 T

r

r (a)

One film model

(b)

H-zone CO2

Surface: Yj, T

Surface: Yj, T

274

O2 T

CO

Two film model

H-zone

CO

O2 CO2

r (c)

Original H-zone model

T

r (d)

Extended H-zone model

Figure 10.1 (a–d) Comparison of the temperature and concentration gradients in different subgrid models: adapted from [24, 33]. The radius of the boundary layer is �.

anywhere in the boundary layer without simplifications [23–25]. Thus, continuous models are 1D, particle-resolved simulations. The continuous models provide a fundamental understanding of the ignition and flame structure around the particle; for example, see the work of Sotirchos and Amundson [26] who investigated the steady-state conversion of a porous char particle. The combustion of volatiles was taken into account in detailed numerical models, such as the finite-volume method of Sadhukhan et al. [10] or the numerical model by Du and Annamalai [27]. However, at this point it should be noted that continuous models require too much computational power to be used as subgrid models in large-scale computational fluid dynamics (CFD) simulations. Thus, in the following, some analytical and semianalytical models are discussed (e.g., see [25, 28–30]). The various models use different methods to capture the variable energy feedback of the homogeneous oxidation reactions. A simple method was presented by Chern and Hayhurst [28], who used a factor to describe the amount of CO that is oxidized near the particle and contributes to the particle heat balance. They observed that the amount of CO oxidation depends on the material combusted in the fluidized bed; for Daw Mill char, the factor was approximately zero and for Rietspruit char it was nearly unity; for details see [28]. Li and You [29] developed a one-film and a two-film model for carbon burning and the burning of volatiles in a stagnant gas atmosphere. The volatiles were represented by methane. Their results showed that the flame position was different for volatile combustion and the combustion of CO from the carbon oxidation. Another model based on the two-film approach is the moving flame front (MFF) model developed by Zhang et al. [25]. In this model, the conversion rate of the particle is calculated as an explicit algebraic function of the heterogeneous kinetics and the diffusion rates [25, 31]. The model assumes that

10.1

Subgrid Model for the Devolatilization/Combustion of a Moving Coal Particle

there is a quiescent gas atmosphere around the particle and an infinitely fast chemical reaction in the homogeneous phase. The only heterogeneous reaction the original model [25] considered was C + O2 . Later, the MFF model was extended to include the gasification reaction with CO2 [31] and a finite reaction rate was considered for the homogeneous reactions [32]. At this point, it must be noted that the MFF model was successfully validated against experimental results in the sense of the particle surface temperature as a function of the ambient oxygen concentration [25]. The flame radius predicted by the MFF model varied with the oxygen concentration. In particular, it was reported by Zhang et al. [25] that the flame radius increased from 1 Rp to 50 Rp as the oxygen concentration increased. This trend is nonintuitive and was not reproduced in particle-resolved CFD-based simulations (see Safronov et al. [33, 34]). A different approach for the description of char conversion was chosen by Schulze et al. [33]. They developed a semiempirical model by introducing a so-called homogeneous reaction layer (H-zone) next to the particle (e.g., see Figure 10.1c). In the virtual H-zone single-film (VHZ-SF) model, it was assumed that the temperature of that layer was equal to the particle temperature and one energy equation was solved. The thickness of the homogeneous reaction layer, and thus the energy production due to the homogeneous reactions, was a function of the Re number, the particle diameter, and the ambient air composition [33]. The validation of the model against detailed particle-resolved simulations showed good agreement. However, the main disadvantage of the model is that the thickness of the homogeneous reaction layer is an input parameter that depends on the ambient conditions and the particle velocity. It must be noted that the conversion model can have a significant influence on the results of large-scale reactor simulations. The effect of the homogeneous reaction near the particle on the performance of an entrained-flow gasifier was investigated by Kumar and Ghoniem [35]. They compared the Baum and Street model with the MFF model and observed that in the combustion zone the yield of the key species was noticeably different in the two models. Vascellari et al. [36] investigated the influence of the particle conversion model in the case of MILD coal combustion. They stated that using the VHZ-SF model improves the prediction of the CO concentration compared to the Baum and Street model. The survey of the literature shows that the energy feedback of the homogeneous reaction in the vicinity of a coal particle is very important to predict the correct particle temperature and carbon consumption during the coal devolatilization and char conversion. However, to reduce the numerical effort in large-scale reactor simulations, experimentally determined rate constants are normally used for the heterogeneous reactions and for the release of the volatiles. Furthermore, it is assumed that the volatiles and CO from the heterogeneous reactions do not react in the vicinity of the coal, and the direct energy feedback is neglected (e.g., see [28, 37]). When analyzing the existing models for char conversion, it must be noted that no model accounts for all the relevant heat and mass transfer processes in the boundary layer and inside the particle. From that point of view, it is the subject

275

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Subgrid Models for Particle Devolatilization-Combustion-Gasification

of this chapter to develop and validate a subgrid model for coal ignition, combustion, and gasification. For that reason, a homogeneous reaction layer similar to that in [33] is used. In particular, the model is extended by adding a pyrolysis model to predict both homogeneous and heterogeneous ignition. To improve the model prediction, the characteristic temperature of the homogeneous reaction layer and its thickness are calculated dynamically. In the following section, the model is explained in detail and the validation against numerical and experimental results is discussed. 10.1.2 Model Formulation

The kinetics of pyrolysis is determined by the coal type, the heating rate, and the maximum particle temperature [4]. Additionally, the particle heating is coupled with the pyrolysis kinetics because the volatiles can ignite outside the particle and increase both the particle heating rate and the maximum particle temperature. The focus of the subgrid model presented in this chapter is on energy feedback modeling because the kinetics of the pyrolysis is best determined by experiments or detailed models such as the CPD or FLASHCHAIN models. Before we proceed with the model description, the following assumptions are made:

• • • •

• • • • • • •

The particle temperature is homogeneous. The pyrolysis kinetics can be described by a single Arrhenius rate expression. The homogeneous gas-phase reaction takes place inside the boundary layer. The transport resistance between the homogeneous reaction layer and the particle surface is neglected for the volatiles and for the products of the heterogeneous reactions. Liquid tar formation is neglected. The heterogeneous reactions are calculated solely on the particle surface, and reactions inside the pores are neglected. Ash is instantaneously removed from the particle surface. Gas radiation is neglected. The Lewis number is unity. The gas phase is ideal. Buoyancy is neglected.

The scheme of the model is shown in Figure 10.2. The model considers three different regions: the homogeneous reaction layer, the particle surface for the heterogeneous reactions, and the inner particle where the pyrolysis takes place. The homogeneous reactions, which deliver heat to the particle, take place in the boundary layer of the particle. The part of the boundary layer where the homogeneous reactions are assumed to occur is called the homogeneous reaction layer or H-zone. The thickness of the H-zone, rh , is equal to the boundary layer thickness

10.1

Subgrid Model for the Devolatilization/Combustion of a Moving Coal Particle

Homogenous reactions

T ∞ C∞ Tinit

Heterogeneous reactions

rinit

Th Ch CS

rP TP Pyrolysis

O2, H2O

δ rT/C

CO, CO2, CH4, H2

CO2, CO, CH4, H2

Figure 10.2 Scheme of the pyrolysis and particle conversion model [38].

� given by Turns [24] �=

Nu ⋅r Nu − 2 p

(10.1)

However, for small Re numbers, the boundary layer thickness is very large. Thus, to avoid numerical problems, the H-zone thickness is limited [38, 39]: ) ( Nu rh = MIN , 5 ⋅ rP (10.2) Nu − 2 It should be noted that the thickness of the thermal and the species boundary layer is the same because of the assumption that the Lewis number is unity. According to Eq. (10.2), rh increases rapidly if Nu approaches 2. To avoid numerical instabilities, the thickness of the homogeneous reaction layer, rh , is limited to 5. The maximum value of rh is set at 5 to ensure that the homogeneous reactions take place near the particle surface even for small Re numbers. The volume of the homogeneous reaction layer, Vh , is calculated as follows: [ ] Vh = MIN 4∕3 π (rh3 − rP3 ), 26 ⋅ 4∕3 π rP3 (10.3) From a numerical point of view, Vh is limited to 26 Vp . The limits of both rh and Vh are artificial limits that have been tested in several validation cases [38]. Basically, the model solves two energy equations: one for the particle temperature TP , and one for the temperature of the homogeneous layer, Th . However, to predict the homogeneous ignition of the volatiles, a third temperature, namely the initiation temperature Tinit , is necessary. The energy conservation of the particle

277

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Subgrid Models for Particle Devolatilization-Combustion-Gasification

is calculated as follows [38, 39]: VP (� cp )P,eff

∑ d TP 4 0 = �� � SP (T∞ Ṙ S,j ΔR HS,j − TP4 ) + �in (Th − TP ) − dt ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏟ j heat conduction ⏟⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏟ radiation heterogeneous reactions

−

∑

] [ Ṙ p,i Mi Δp H 0 + cp,g (T∞ − TP )

(10.4)

i

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ heat of pyrolysis and sensible heating of volatiles 0 where Ṙ S is the heterogeneous reaction rate, Ṙ p is the pyrolysis rate, and ΔR HS,i , 0 Δp H are the heat of reaction and the pyrolysis heat, respectively. The heat transfer coefficient between the particle and the homogeneous reaction layer, �in , is calculated as follows: 4π �g (10.5) �in = 1 1 − rP rT

where rT is the radial position of the H-zone temperature, which is calculated as (10.6)

rT = fT rh + (1 − fT ) rP

where fT is a blending factor (fT = 0.1 as a default value). Using the steady-state assumption for the temperature of the H-zone, Th , the energy balance takes the following form [38, 39]: ∑ 0 �g (T∞ − Th ) − �in (Th − Tp ) − =0 (10.7) Ṙ h,j ΔR Hh,j j

⏟⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏟

⏟⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏟

⏟⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏟

conduction to gas phase

conduction to particle

homogeneous reactions

Solving the equation for Th leads to ∑ 0 − i Ṙ h,i ΔR Hh,i + �g T∞ + �i Tp Th = �g + �i

(10.8)

where Ṙ h is the reaction rate in the homogeneous layer. The heat transfer coefficient between the H-Zone and the ambient gas, �g , is calculated as follows: 1 (10.9) �g = 2 rP 1 − Nu �g SP �in This equation is derived using the overall heat transfer coefficient, which can be expressed by a serial setup of the heat transfer resistances in the boundary layer 1∕�g and 1∕�i : 1 1 1 = + � �g �i

(10.10)

10.1

Subgrid Model for the Devolatilization/Combustion of a Moving Coal Particle

where � is the overall heat transfer coefficient of a nonreacting particle, given by 2 rp �= (10.11) Nu �g SP with Sp as the particle surface. The Nu number is determined by the Ranz–Marschall equation [40]: Nu = 2 + 0.6 Re1∕2 Pr1∕3

(10.12)

with the particle Reynolds number Re = �g cp,g

uslip �g Dp �g

and the Prandtl number Pr =

. �g As a result of the small fT value of 0.1, the temperatures Th and TP are more or less equal. Thus, the ignition of the volatiles is delayed because the auto-ignition temperature is reached only for a significantly heated particle. This problem is solved by introducing the temperature Tinit in the position rinit : (10.13)

rinit = finit rh,con + (1 − finit ) rP

where finit (finit > fT ) is the blending factor and rh,con is the thickness of the H-zone with included Stefan flow. The enlargement of the boundary layer by the Stefan flow is taken into account to guarantee a stable combustion of the volatiles after ignition. rh,con is calculated as follows [38, 39]: ) ( Nucon 2r � 5 ⋅ rP Nucon = P con rh,con = min (10.14) Nucon − 2 �g and [41] �con =

ṁ ′′g cp,g

(10.15)

exp(ṁ ′′g cp,g ∕�) − 1

where m′′′ g is the surface-specific mass flow from the particle. The model solves the species balance equations for the H-zone and simplified balance equations for the particle surface. In order to reduce the numerical effort, the species source terms of the pyrolysis and the surface reactions are included in the species balance of the H-zone and are not set at the particle surface. The balance equation of species i take the following form [38, 39]: ∑ ∑ �i, j Ṙ h, j + Ṙ p, i − ṅ con, i �i, j Ṙ S, j + (10.16) �g (C∞, i − Ch, i ) = j

j

⏟⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏟

⏟⏞⏞⏞⏟⏞⏞⏞⏟

⏟⏞⏞⏞⏟⏞⏞⏞⏟

⏟⏟⏟

⏟⏟⏟

mass transfer with the gas phase

heterogeneous reactions

homogeneous reactions

pyrolysis

Stefan flow

with the stoichiometric coefficient �i, j of species i in reaction j, the convective flux due to the Stefan flow ṅ con,i , and the mass transfer coefficient to the gas phase �g given by 4π Dg 1 �g = �in = (10.17) 1 2 rp 1 1 − − rP rC Sh Dg SP �in

279

280

10

Subgrid Models for Particle Devolatilization-Combustion-Gasification

where �in is the mass transfer coefficient between the particle surface and the H-zone. The position of the representative concentration in the H-zone, rC , is calculated using a blending factor fC (fC = fT ): (10.18)

rC = fC rp + (1 − fC ) rh

The convective flux ṅ con, i is calculated from the volume change due to reactions and pyrolysis using the ideal gas assumption: ) ( ∑ ∑ ∑ Ru �g ̇ ̇ ̇ Ym,i (10.19) ṅ con, i = RS,j ΔR ns,j + Rh,j ΔR nh,j + Rp,i T m pg j j i ⏟⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏟ ⏟⏟⏟ heterogeneous reactions

pyrolysis

homogeneous reactions

where ΔR nj is the change of the molar number in the reaction j, and Ym is the mean mass fraction between the H-zone and the ambient gas phase. Finally, a simplified species balance is solved at the particle surface to calculate the surface concentration Cs,i of the species i which are involved in heterogeneous reactions: ∑ �in (CS,j − Ch,j ) = − (10.20) �i,j Ṙ S,j j

The properties of the gas phase are calculated using the temperature Th and the mean mass fractions Ym = 0.5 (Y∞ + Yh ). The relations to calculate �i, g and �i, g are taken from [42], and cp, i, g is calculated in line with [43]. 10.1.2.1 Semiglobal Chemical Reactions

The models uses a reaction scheme with three semiglobal heterogeneous and six semiglobal homogeneous reactions: 2C + O2 −−−−→ 2CO,

Ṙ 1 = k1 ⋅ CO2 ⋅ SP

(R1)

C + CO2 −−−−→ 2CO,

Ṙ 2 = k2 ⋅ CCO2 ⋅ SP

(R2)

C + H2 O −−−−→ CO + H2

Ṙ 3 = k3 ⋅ CH2 O ⋅ SP

CO + 0.5O2 −−−−→ CO2 ,

Ṙ 4 = k4 ⋅

0.25 CCO CO 2

(R3) CH0.5O 2

⋅ Vh

(R4)

−−−−−−− ⇀ CO + H2 O ↽ − CO2 + H2 , Ṙ 5 = k5,f ⋅ CCO CH2 O − k5,b ⋅ CCO2 CH2 ⋅ Vh (R5) CH4 + 0.5O2 −−−−→ CO + 2H2 , CH4 + H2 O −−−−→ CO + 3H2 , H2 + 0.5O2 −−−−→ H2 O,

0.5 1.25 Ṙ 6 = k6 ⋅ CCH CO ⋅ Vh 4

Ṙ 7 = k7 ⋅ CCH4 CH2 O ⋅ Vh

Ṙ 8 = k8 ⋅

C6 H6 + 3O2 −−−−→ 6CO + 3H2 ,

2

CH0.5 2

2.25 CO 2

CH−1O 2

⋅ Vh

1.85 Ṙ 9 = k9 ⋅ CC−0.1 H CO ⋅ Vh 6

6

2

(R6) (R7) (R8) (R9)

Table 10.1 summarizes the kinetic parameters for the extended Arrhenius equation ] [ −EA (10.21) kR = AR T nR exp (Ru T)

10.1

Table 10.1

Subgrid Model for the Devolatilization/Combustion of a Moving Coal Particle

Kinetic parameters for the homogeneous and heterogeneous reactions.

Reaction No.

AR

R1 R2 R3 R4 R5,f R5,b R6 R7 R8 R9

3.007 × 105 ms−1 4.605 × 100 m (s K)−1 1.125 × 101 m (s K)−1 1.260 × 1010 (m3 ∕mol)0.75 s−1 2.750 × 106 (m3 ∕mol) s−1 3.533 × 1011 (m3 ∕mol) s−1 2.473 × 109 (m3 ∕mol)0.75 s−1 3.000 × 105 (m3 ∕mol) s−1 1.690 × 1014 (m3 ∕mol)0.75 K s−1 7.5895 × 106 (m3 ∕mol)0.75 K s−1

a

nR

E A in J mol−1

References

0 1 1 0 0 −1 0 0 −1 0

1.4937 × 105 1.751 × 105 1.751 × 105 1.674 × 105 8.368 × 104 1.2734 × 105 1.255 × 105 1.255 × 105 1.674 × 105 1.255 × 105

[24] [44] [44] [24] [45] a

[45] [45] [45] [46]

Kinetic parameters are obtained from the chemical equilibrium constant.

The pyrolysis rate is calculated for each species using a single-rate Arrhenius expression: ] [ d mbv, i −EA = AR exp mbv, i (10.22) dt (Ru T) with the mass of volatiles bound to the coal matrix mbv . 10.1.3 CFD-based Model

A detailed description of the CFD-based model can be found in [33, 47, 48]. In this work, we consider a spherical porous coal particle in a symmetric gas flow. Gas radiation and buoyancy are neglected. The gas flow on the porous particle is modeled by adding a Darcy term (� Ku⃗ ) to the momentum conservation equation. The governing equations for mass and momentum conservation can be found in [33, 47, 48]. The species and energy conservation equations have the following form: ) ∂ ( � Yi + ∇ ⋅ (�⃗uYi ) = ∇ ⋅ (�Di ∇Yi ) + Mi Ṙ ′′′ (10.23) i ∂t ∑ ∑ ∂ ̇ ′′′ − Δp H 0 Ṙ ′′′ ΔR Hj0 Ṙ ′′′ (� h) + ∇ ⋅ (�⃗uh) = ∇ ⋅ (�∇T) − p, i + Qrad (10.24) h, j ∂t i j ⏟⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏟ homogeneous reaction

�=

Ru T

p ∑ i

Yi ∕Mi

,

pyrolysis

(10.25)

, the homogeneous reaction rate Ṙ ′′′ , the pyrolysis with the radiative heat flux Q̇ ′′′ rad h, j ̇ ′′′ = ∑ �h, i,j Ṙ ′′′ + ∑ �p, i,j Ṙ ′′′ . , and the overall species production rate R rate Ṙ ′′′ j j p, i i p, j h, j

281

10

Subgrid Models for Particle Devolatilization-Combustion-Gasification

L1

T∞

Porous particle dp /2

R

T0

Outlet 

uin Inlet 

282

L2

Symmetry axis

Z Figure 10.3 Scheme of the CFD setup [39].

The general scheme of the setup including the domain boundary conditions is shown in Figure 10.3. No boundary conditions were used at the particle surface, and the conservation equations were solved in the whole domain. The permeability K of the porous particle was set at 10−12 m2 and infinity in the gas phase. The particle temperature was homogeneous, which was enforced by setting a high thermal conductivity in the solid phase (�P = 100 W (m K)−1 ). The domain size was 130 dp in the Z-direction and 40 dp in the R-direction. The particle’s center of mass was located at the symmetry axis (R = 0) and Lp = 3∕13 L1 . The entire domain was meshed with 27 750 CVs. Depending on Rein and T∞ , the time step Δt was between 1 × 10−6 and 1 × 10−3 s. The CFD modeling was carried out using the commercial software ANSYS-FLUENT [49]. The source terms, which describe the pyrolysis in the energy and species conservation equations, were implemented into the software using user-defined functions (UDF). Details on the numerics can be found in [47]. 10.1.4 Model Validation

The pyrolysis and homogeneous ignition of a single coal particle is investigated. Concentrating on homogeneous ignition, heterogeneous reactions are neglected. The coal particle with a diameter of 2 or 0.2 mm is placed in a hot oxidizing air stream: Y∞,O2 = 0.233, Y∞,H2 O = 0.001, and Y∞,N2 = 0.766. The gas temperature T∞ is set at 1000 or 1400 K, and the gas velocity is set such that the inlet Re numbers are 1, 10, and 100 (only for the case dP = 2 mm). The volatiles are assumed to consist of CH4 (3.77%), CO (52.40%), CO2 (27.68%), and N2 (16.15%). A singlerate Arrhenius expression is used to calculate the pyrolysis rate of each species. The kinetic parameters are AR = 72054.9 s−1 and EA = 54.285 kJ mol−1 [14]. The overall amount of volatile matter is 37.1%. The volatiles composition is calculated on the basis of the assumption that all hydrogen, oxygen, and nitrogen is released during the pyrolysis. Using the lower heating values of the coal and the pyrolysis gas composition, the heat of pyrolysis Δp H 0 is calculated as 1.3175 MJ kg−1 . The initial cold coal particle (TP = 300 K) is assumed to be dry with the properties �p = 849.4 kg m−3 , cp,P = 1000 J (kg K)−1 , and �� = 1. During the conversion, the

10.1

Subgrid Model for the Devolatilization/Combustion of a Moving Coal Particle

283

mass of the particle is constant and equal to the initial mass. In the subgrid model, the position of the initiation temperature is calculated using finit = 0.3. To illustrate the dynamics of the conversion process, Figure 10.4 depicts a series of temperature contour plots in the case dP = 2 mm, T∞ = 1400, and Rein = 10. The time history of the particle temperature Tp and the remaining amount of bound volatiles are shown in Figures 10.5 and 10.6 for dP = 2 mm and 0.2 mm, respectively. The figures show that the ignition time decreases as the ambient temperature and the Re number increase. An analysis of the figures reveals that the conversion in general can be divided into four phases. At the beginning, the particle heating is slow, governed by radiation and heat conduction through the boundary layer. The first phase ends when the pyrolysis gases ignite (e.g., see Figure 10.4a). Because of the large heat production in the vicinity of the particle, the particle temperature increases rapidly. However, after the ignition phase the heating rate decreases. This effect is a result of the increasing pyrolysis reaction which is endothermic. At the same time, the oxidation reactions in the vicinity of the coal particle are limited by the oxygen diffusion, which is additionally reduced by the Stefan flow. As a result, in the CFD simulations the flame radius increases (see Figures 10.4b–d). When the pyrolysis ends, the particle is again rapidly heated as a result of the combustion of the final volatiles leaving the particle. A comparison 500

631

762

893 1024 1156 1287 1418 1549 1680

534

r/d

r/d

2 0

–2

0

2

4

6

585

–2

0

(b) 500

893 1201 1510 1818 2126 2434 2743 3051 3359

2

4 z/d t = 0.18 s

6

8

823 1157 1485 1814 2142 2471 2799 3128 3456

4 r/d

4 r/d

2 0

8

z/d t = 0.16 s

(a)

2

–2

0

2

4

6

t = 0.22 s

2 0

8

z/d (c)

998 1230 1462 1694 1925 2157 2389 2621

4

4

0

766

–2

0

2

4

6

z/d (d)

t = 0.3 s

Figure 10.4 (a-d) Time series of temperature contour plots of a pyrolyzing particle at T∞ = 1400 K, dP = 2 mm, and Rein = 10 [39].

8

284

10

Subgrid Models for Particle Devolatilization-Combustion-Gasification

between the temperature time histories of the small particles shows that the ignition temperature increases as the particle size decreases, which is in qualitative agreement with experimental results (e.g., see [50]). Furthermore, for small particles the energy transfer from the flame significantly increases and the particle temperature exceeds the ambient temperature at the end of pyrolysis. After the detailed discussion of the CFD results, the results of the subgrid model are discussed. Figures 10.5 and 10.6 show that the subgrid model is able to predict the ignition times of the particles in most cases. It can be seen that the four phases observed in the CFD simulations are reproduced for differently sized particles, Re numbers, and different ambient temperatures without tuning of the model parameters. The good agreement between the ignition times of the subgrid model and direct numerical simulation (DNS) is achieved by setting finit at 0.3. Here, it must be noted that the subgrid model solves no equation for the flame position. The combustion of volatiles and thus the heat production as well as the energy feedback from the H-zone to the particle surface are limited by oxygen diffusion. At the ignition of the volatiles, the oxygen concentration in the H-zone tends to zero and unburned volatiles diffuse to the ambient phase. Figure 10.7 shows the 1000

1.0

900

Re = 1 Re = 10

0.8 mbv /m0bv

T (K)

800 700 600

0.6 0.4

500 300 0.0

0.5

1.0

1.5

2.0

0.0 0.0

2.5

t (s) 1400

1.0

1200

0.8

1000 800 600 400

Re = 1 Re = 10 Re = 100

t (s)

1.0

1.5

2.0

2.5

Re = 1 Re = 10 Re = 100

0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 (c)

0.5

t (s)

(b)

mbv /m0bv

T (K)

(a)

0.2

Re = 1 Re = 10

400

(d)

t (s)

Figure 10.5 Time history of the particle temperature predicted by the subgrid and the CFD model for a 2 mm particle at different ambient temperatures T∞ = 1000 K (a, b), T∞ = 1400 K (c,d), and different inlet Re numbers. The symbols represent the CFD result, and lines represent the results of the subgrid model.

10.1

Subgrid Model for the Devolatilization/Combustion of a Moving Coal Particle

1.0

1600 1400

mbv /m0bv

T (K)

1000 800 600

t (s)

(b)

1800

t (s) 1.0

1600

Re = 1 Re = 10

0.8

mbv /m0bv

1400 T (K)

0.4

0.0 0.000 0.020 0.040 0.060 0.080 0.010 0.120

0.000 0.020 0.040 0.060 0.080 0.100 0.120 (a)

0.6

0.2

Re = 1 Re = 10

400

1200 1000 800 600

0.01

(c)

0.02

0.03

0.6 0.4 0.2

Re = 1 Re = 10

400 0.00

Re = 1 Re = 10

0.8

1200

0.04

0.0 0.00

0.05 (d)

t (s)

0.01

0.02

0.03

t (s)

Figure 10.6 Time history of the particle temperature predicted by the subgrid and the CFD model for a 0.2 mm particle at different ambient temperatures T∞ = 1000 K (a, b), T∞ = 1400 K (c, d), and different inlet Re numbers. The symbols represent the CFD result, and lines represent the results of the subgrid model.

0.8 O2 CO2 CO N2 H2O CH4

0.7 0.6

Y

0.5 0.4 0.3 0.2 0.1 0.0 0.0

0.2

0.4

0.6 0.8 t (s)

285

1.0

1.2

1.4

Figure 10.7 Time history of the species mass fractions in the H-Zone in the case dP = 2 mm, T∞ = 1400 K, and Rein = 10.

0.04

0.05

286

10

495

676

Subgrid Models for Particle Devolatilization-Combustion-Gasification

858 1039 1220 1402 1583 1764 1946 2127

752

2 0

(a)

850

900

950 1000 1050 1100 1150

4 r/d

r/d

4

800

–2

0

2

4 z/d

6

2 0

8 (b)

–2

0

2

4

6

8

z/d

Figure 10.8 Contour plot of the temperature around the pyrolyzing particle in the moment of ignition T∞ = 1000 K (a) dP = 2 mm, Rein = 100, t = 0.29 s and (b) dP = 0.2 mm, Rein = 10, t = 0.03 s [39].

time history of the species mass fractions in the H-zone for a 2 mm particle at Tin = 1400 K and Rein = 10. However, in the cases dP = 2 mm, Re = 100, and dP = 0.2 mm, Re = 10 both at T∞ = 1000 K, large deviations are observed between the ignition time predicted by CFD and by the subgrid model. In this case, the ignition occurs away from the particle surface (see the temperature contour plots in Figure 10.8). Next, the subgrid model is validated using experimental data [51], where coal particles with an average diameter of 100 μm were fed into a hot coflow. The particle feed rate was sufficiently low so that they ignited independently of each other. Ignition of the volatiles was measured using CH∗ chemiluminescence. The proximate and ultimate analyses of the coal are given in Table 10.2. In the subgrid model, the initial apparent density of the coal is considered at 1100 kg/m3 , heat capacity at 1500 J (kg K)−1 [52], and �� set at unity. The particle diameter is fixed at 100 μm and the particle density decreases with the release of the volatiles. A single-rate kinetic expression for the coal pyrolysis in the experiment was published by Vascellari et al. [14]. The parameters are AR = 72054.9 s−1 and EA = 54.285 kJ mol−1 [14]. At high ambient temperatures, the amount of volatiles is 54.75% (daf ) [14]. In this work, the volatiles are composed of CH4 (28.98%), C6 H6 (36.04%), CO (32.18%), and N2 (2.8%). Benzene, with its low hydrogen/carbon ratio, must be included in order to fulfill the element balance for the high volatile content. The heat of pyrolysis is calculated as 704.2 kJ kg−1 , assuming that the heating value of the residual char is equal to that of graphite. Molina and Shaddix [51] investigated the coal ignition in four different gas atmospheres; the different gas compositions are given in Table 10.3. In the subgrid model, the ambient gas composition is assumed to be constant, and the ambient temperature is varied according to the temperature profile measured in the reactor (see [51]). The ignition times predicted by the subgrid model and the experimental data [51] for the different gas atmospheres are summarized in Table 10.4. It can be seen that the subgrid model qualitatively reproduces the trends observed during the experiments. In detail, the ignition time decreases with increasing oxygen content, and CO2 delays the ignition. The deviation in the ignition time is in the expected

10.1

Subgrid Model for the Devolatilization/Combustion of a Moving Coal Particle

Table 10.2 Proximate and ultimate analysis of the Pittsburgh seam high-volatile bituminous coal [14] Proximate analysis (m%) Volatile matter Fixed carbon Moisture Ash Higher heating value

Table 10.3

O2 H2 O CO2 N2

Ultimate analysis (m%)

35.89 56.46 0.47 6.95

C H O N

75.23 5.16 9.83 1.43

30.94 MJ kg1

S

2.0

Ambient gas composition in mol% [51].

N2,21

N2,30

CO2,21

CO2,30

21.00 12.27 1.65 65.08

30.00 12.34 1.65 56.00

21.00 13.69 65.31 0.00

30.00 12.98 57.02 0.00

Table 10.4 Comparison between the predicted ignition time and the experimental results of Molina and Shaddix [51]. The confidence interval is given in brackets. Atmosphere N2,21 N2,30 CO2,21 CO2,30

Subgrid model

Molina and Shaddix [51]

Error (%)

24.5 ms 20.5 ms 31.5 ms 25.8 ms

28.0 (26.9–28.8) ms 26.2 (25.3–26.7) ms 30.5 (29.6–31.4) ms 27.6 (26.9–28.3) ms

12.5 21.7 3.2 6.5

range, at most 21.7%. However, it should be noted that in this work we used singlerate pyrolysis kinetics. From this point of view, the model can be enhanced by coupling with a network pyrolysis model, for example, the CPD model [11]. Finally, the validity of the model for char conversion in different atmospheres is shown. In this case, the nonporous particle consists of pure carbon, and the results of a detailed particle-resolved CFD model have been reported by Schulze et al. [33]. The particle diameter is 2 mm, and the chemical reactions are modeled using reactions (R1)–(R5). The ambient temperature is varied between 1000 and 2800 K, and the Re number is in the range "zero" (1 × 10−2 ) to 100. Two different gas atmospheres are studied: the combustion (Y∞ O2 = 0.233, YH2 O = 0.001, Y N2 = 0.766), and the reduced (Y∞ O2 = 0.11, YH2 O = 0.074, YN2 = 0.816) atmosphere. The general setup of the validation case is similar to the case presented in Figure 10.2, and the detailed description of the particle-resolved CFD model is given in [33].

287

288

10

Subgrid Models for Particle Devolatilization-Combustion-Gasification

The conversion process is characterized using the temperature difference between the particle surface and the ambient gas phase ΔTS : (10.26)

ΔTS = TP − T∞ and the surface-specific carbon consumption ṁ ′′C is ṁ ′′C = MC ⋅ (2Ṙ 1 + Ṙ 2 + Ṙ 3 ) ∕ S

(10.27)

with the molar mass of carbon MC . Figure 10.9 shows the comparison between the CFD [33] and the subgrid models. The figure shows that the developed subgrid model can reproduce the conversion of the char particle for different gas compositions and Re numbers without tuning the model parameters (fT = fC = 0.1). In detail, the largest deviations between both models are observed around the ignition temperature (1200 K ≤ T∞ ≤ 1400 K) and the start of the gasification reactions (R2) and (R3), which are visible as break points in the plots of the carbon consumption rate between 1900 and 2500 K. For a more detailed discussion, see [38]. Combustion atmosphere (Y∞ O2 = 0.233, YH2O = 0.001, YN2 = 0.0766): 600 Re = 0 Re = 10 Re = 100

0.05 0.04

Re = 0 Re = 10 Re = 100

500

ΔTs (K)

m′C (kg (m2 s)–1)

0.06

0.03 0.02

400 300 200

0.01

100

0.00 1000 (a)

1500

2000

0 1000

2500 (b)

T∞ (K)

1500

2000

2500

T∞ (K)

Reduced atmosphere (Y∞ O2 = 0.11, YH2O = 0.074, YN2 = 0.816): 300 Re = 0 Re = 10 Re = 100

0.04 0.03 0.02 0.01

120 100 50

0.00 1000 (c)

Re = 0 Re = 10 Re = 100

200

ΔTs (K)

m′C (kg (m2 s)–1)

0.05

1500

2000

T∞ (K)

0 1000

2500 (d)

1500

2000

2500

T∞ (K)

Figure 10.9 Temperature difference ΔTS (a,c) and specific carbon consumption ṁ ′′ (b,d) for C the steady-state conversion of a 2 mm char particle for different gas atmospheres and Re numbers [33, 38]. The symbols represent the CFD result, and lines represent the results of the subgrid model.

10.1

Subgrid Model for the Devolatilization/Combustion of a Moving Coal Particle

0.030 CFD Subgrid fC = 0.05 Subgrid fC = 0.1 Subgrid fC = 0.2

ΔTs (K)

200 150 100 50

(a)

1500 T∞ (K)

2000

2 –1 m′′ C (kg (m s) )

250

0 1000

289

0.025 0.020 0.015 0.010 0.005 0.000 1000

2500 (b)

CFD Subgrid fC = 0.05 Subgrid fC = 0.1 Subgrid fC = 0.2

1500

2000 T∞ (K)

Figure 10.10 Temperature difference ΔTs (a) and the specific carbon consumption rate ṁ ′′ C (b) in the combustion atmosphere, without pyrolysis, for a 2 mm particle at different ambient temperatures T∞ and blending factors fC (fT = 0.1, Rein = 10).

The results reported so far are obtained using the blending factors fT and fC equal to 0.1. To recapitulate, the blending factors describe the position of the characteristic temperature and gas composition inside the homogeneous reaction layer. Thus they are essential in the description of the heat and mass exchange between the particle and the ambient phase. To demonstrate the influence of the blending factors on the model prediction, the blending values are varied between 0.05 and 0.2. The different parameters are compared using a 2 mm particle in the gasification atmosphere and an inlet Re number of 10. The results for different fC values are shown in Figure 10.10. The figure shows the influence of the factor fT . Figure 10.10b demonstrates that varying fC has only a minor influence on the carbon consumption rate. It can be seen that, when fC = 0.2, the carbon conversion increases slightly in the ambient temperature interval of 1600–2100 K. On the other hand, an analysis of Figure 10.10a reveals that fC influences the particle temperature throughout the T∞ interval. This is explained by the fact that increasing fC values enhance the oxygen diffusion to the homogeneous reaction layer and thus the exothermic CO oxidation. The second parameter fT has a significant influence on the energy feedback from the homogeneous reactions to the particle. To be specific, a decreasing fT value, which increases the energy feedback, leads to higher particle temperatures and larger ΔTs values (see Figure 10.11a). Despite the influence of fT on the particle temperature, the influence of the parameter on the carbon mass flux is very weak, which can be seen in Figure 10.11b. For the sake of completeness, it must be noted that in this validation case the results are independent of finit , which was verified using finit = 0.1 and 0.3. Conclusions A novel semiempirical model for coal particle devolatilization and

oxidation in a stream of hot gas has been developed and validated against CFDbased particle-resolved simulations. The distinguishing feature of this new model is its ability to predict the ignition of a dried raw coal particle at the particulate level. This is achieved by coupling the effects of the heterogeneous and homogeneous reactions near the particle surface. The model can be easily coupled with an

2500

290

10

Subgrid Models for Particle Devolatilization-Combustion-Gasification

0.030 CFD Subgrid fT = 0.05 Subgrid fT = 0.1 Subgrid fT = 0.2

ΔTs (K)

200 150 100 50 0 1000 (a)

1500 T∞ (K)

2000

2 –1 m′′ C (kg (m s) )

250

0.025 0.020 0.015 0.010 0.005 0.000 1000

2500 (b)

CFD Subgrid fT = 0.05 Subgrid fT = 0.1 Subgrid fT = 0.2

1500

2000

2500

T∞ (K)

Figure 10.11 (a) Temperature difference ΔTs and (b) the specific carbon consumption rate ṁ ′′ in the combustion atmosphere, without pyrolysis, for a 2 mm particle at different ambiC ent temperatures T∞ and blending factors fT (fC = 0.1, Rein = 10).

advanced pyrolysis model (e.g., the CPD model) in order to increase the accuracy of the volatiles release.

10.2 Novel Intrinsic Submodel for Gasification of a Moving Char Particle

Analysis of recent publications devoted to the modeling of large-scale entrainedflow gasifiers or combustors shows that significant progress has been achieved in the development and validation of macroscale models for chemically reacting particulate flows in gasifiers or combustors and their numerical implementation in many CFD codes, for example, see the reviews [15, 53] and more recent works [54–56]. However, the submodels for char oxidation/gasification used in the macroscale simulations correspond to the so-called surface-based reaction submodels, which are based on the models developed by Baum and Street [17] and Smith [57]. The main assumption of these models is that the heterogeneous reactions occur at the external particle surface. On the other hand, few intrinsic-based submodels used in many commercial CFD programs are formulated for kinetically and pore-diffusion-controlled regimes, where the reaction rate is corrected using an effectiveness factor and the Thiele modulus [58, 59]. The main drawback of such submodels is their limitation to predict correctly the carbon consumption rates when the effectiveness factor approaches zero. Moreover, the influence of particle velocity on the particle oxidation/gasification rate is not well introduced in existing intrinsic-based submodels. Motivated by this fact, this work is devoted to the development and validation of an intrinsic-based submodel for the gasification of a spherical char particle moving in a hot atmosphere. As a first step, we have chosen a CO2 /N2 atmosphere, which allows us to avoid homogeneous reactions for simplicity. The first part of this work is based on the description of the submodel we have developed. In the second part we introduce a validation of the model against a comprehensive CFD-based model, where the Navier–Stokes equations coupled with the energy and species conservation equations are used to solve the

10.2

Novel Intrinsic Submodel for Gasification of a Moving Char Particle

problem by means of the pseudo-steady-state approach. For validation, a comprehensive CFD-based model for a porous carbon particle is used to understand the interplay between the Stefan flow coming out of particle due to heterogeneous reactions on the surface of pores and external flow, which characterizes the relative particle velocity. Particle porosity is modeled using the so-called grain model introduced by Szekely et al. [60]. 10.2.1 Model Formulation

Before we proceed with description of an intrinsic-based mathematical model for a single porous carbon particle moving in a hot CO2 gas, we introduce the mass conservation law of the char particle. Generally, the mass of a chemically reacting char particle changes as a result of the heterogeneous reaction on the particle surface and inside the pores of the particle. This effect can be formulated mathematically as follows: ṁ P =

dmP dV d� = V P P + �P P . dt dt dt ⏟⏟⏟ ⏟⏟⏟ ṁ C,V

(10.28)

ṁ C,S

Here, �P is the density of the char particle. The first term on the right side of Eq.(10.28) characterizes the change of particle density due to carbon conversion inside the porous particle. The second term defines the decrease of particle size due to the heterogeneous reaction on the particle surface. The density and the volume of a spherical char particle can be defined as follows: ( ) 4 �P = �g,P � + (1 − �) �C = � �g,P − �C + �C , VP = π rP3 , 3

(10.29)

where � is the density and the porosity (void fraction) of the char particle, �g,P is the density of gas inside the pores, �C is the density of pure carbon, and rP is the radius of the particle. Inserting Eqs.(10.29) into Eq.(10.28), we obtain the volumetricbased and surface-based carbon mass flows: ṁ C,V = VP

( ) d� d�P d� = VP �g,P − �C ≈ −VP �C , dt dt dt

(10.30)

ṁ C,S = �P

dVP 4 d ( 3) r = �P π dt 3 dt P

(10.31)

Finally, the particle radius and particle porosity can be calculated taking integrals from Eqs.(10.30) and (10.31) as follows: √ ṁ C,V Δt 3 ṁ C,S Δt 3 , �t+Δt = �t + , (10.32) − rP, t+Δt = 3 rP, t 4 π �P V P �C

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where ( ṁ C,V = VP

MC ′′′ � k S Y CO2 ,P MCO2 g,P R

)

( , ṁ C,S = AP

MC � k Y MCO2 g,S R CO2 ,S

)

(10.33) Here, the reaction constant kR for the reaction C + CO2 → 2 CO

h0R (298K) = 14.4 MJ kg−1 C

(10.34)

is derived from the Arrhenius expression kR = AR T nR exp(−EA ∕Ru TP )

(10.35)

where AR is the pre-exponential factor, nR is the temperature exponent, EA is the activation energy of the reaction, and Ru is the universal gas constant. In this work, we use the kinetic data measured for coal by Libby and Blake [44]: AR = 4.605 m (s K)−1 , nR = 1, EA = 1.751 × 105 J (kg K)−1 . 10.2.1.1 Total Carbon Consumption Rate

The final expression for the total carbon consumption rate can be obtained by inserting Eqs. (10.33) into Eq. (10.28), leading to ) ( MC VP ′′′ Y CO2 ,P � k Y S . (10.36) 1+ ṁ P = AP MCO2 g,S R CO2 ,S Ap YCO2 ,S Considering a spherical particle, this equation converges to the following form: ) ( MC rP ′′′ Y CO2 ,P � k Y . (10.37) 1+ S ṁ P = AP MCO2 g,S R CO2 ,S 3 YCO2 ,S This form of ṁ P allows us to classify the reaction regimes for a porous particle using

Y CO2 ,P YCO2 ,S

ratio1) : Y

• Regime I – CO2 ,P ≈ 1: the overall reaction rate is controlled by the kinetics. Y CO2 ,S

–The concentration of the gaseous reactant is spatially uniform within the porous solid. —All the surface area is available for reaction and the particle reacts at a spatially uniform rate. —The internal zone of particle reacts faster than the surface. Thus, a so-called cenosphere can be developed at the final stage of gasification. Y

• Regime III – CO2 ,P ≈ 0: the overall rate is controlled by external mass transfer Y CO2 ,S

and gasification takes place at the outer surface of the shrinking solid. –The solid structure is unimportant and the expressions for nonporous system can be utilized. r ′′′ 1) Here, we assume that P S >> 1, which is true for particles with rP > 10 μm at the beginning of a 3 reaction.

10.2

• Regime II –0
Tliquid ) within and directly above the slag bath. Coal particles are rapidly converted, and the remaining ash is liquefied, which consequently flows down into the slag bath. Because free oxygen is present in this lower slagging fluidized bed, exothermic reactions dominate the heat balance. The upper slagging fluidized bed is characterized by an absence of free oxygen, leading to predominately endothermic gasification reactions. Fresh coal is fed into this zone, where temperatures are still above the ash liquid point. Here, the generated heat from the slag bath bed is used to promote endothermic reactions. Because no free oxygen is present at the coal feeding stage, there is no risk of oxygen flowing into the coal feeding line. As the height increases, the temperature decreases and, finally, the upper nonslagging zone emerges. Outlet temperatures below the obstruction/fouling point are reached. In the transition between the slagging zone and nonslagging zone, a temperature range is passed where particles are sticky and tend to form

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Raw gas

Raw gas

Particle separation

Particle separation

Non-slagging fluidized bed T > Tobstruction

Freeboard T < Tobstruction Recycle

Recycle

Feedstock

Transition zone Upper spouted bed Core T > T fluid Annulus T > T obstruction

Feedstock

Upper slagging fluidized bed T > Tliquid

Secondary gasification agent

Lower slagging fluidized bed T > Tliquid

Annulus flow Primary gasification agent

Lower spouted bed Core T > T fluid Annulus T > T liquid

Core flow Gasification agent

Slagbath T > Tfluid (a)

Slag

Slagbath T > T fluid (b)

Slag

Figure 11.1 Principal scheme of (a) a fluidized-bed gasifier with slag bath and (b) a multistaged spouted bed gasifier with slag bath.

agglomerates and clusters. These agglomerates fall down into the slag bath where complete conversion is guaranteed. In conclusion, this reactor is characterized by a multistaged conversion with very high local temperatures above 2000 K to ensure that carbon conversion is complete even for low-reactive feedstock. On the other hand, gas outlet temperatures below the critical ash sinter point are ensured. Hence, the thermodynamically optimal temperature is approached while carbon conversion is maximized. Because the produced ash tends to form agglomerates, fly ash carryover is markedly reduced. Consequently, a higher content of fines in the feedstock is acceptable compared to what is standard in conventional fluidized-bed gasifiers. With fly ash, carbon loss is reduced because the total amount of particles released at the top is reduced. On the other hand, the carbon content of the discharged fly ash particles must not drop below ∼60% to maintain the positive effects of carbon on raw-gas outlet and heat exchanger surfaces. 11.2.1.2 Multistaged Spouted Bed with Slag Bath

This approach combines two spouted bed zones, one on top of the other, with a bottom slag bath for internal post-gasification. The principal scheme of this type of reactor is shown in Figure 11.1b. The advantage over the gasifier presented above is the increased operational flexibility due to staged fluidization. The oxygen-containing gasification agent is fed at several levels. The lower nozzles

11.2

Trends in Gasifier Design

point toward the bottom product. The resulting oxidation reactions cause temperatures above the ash fluid point and thus the generation of a liquid slag bath. As the nozzles are arranged in a boxer alignment, emerging flames conjoin to form a jet flowing vertically upwards. This jet penetrates through the bed and thus generates a distinct lean core flow surrounded by a higher loaded annulus backflow. At an appropriate level, where free oxygen is consumed, secondary gasification agents are added. These generate a second jet, and consequently a second spouted bed emerges. It is important to note that the central jets from the two spouted beds do not conjoin into a single central upward-flowing core. The lower spout has a tendency to close on its own because of slagging conditions in the hot central zone and the resultant fluidization behavior of the slag produced. At the top of the upper spouted bed, the central jet erupts into a fountain. Above, gas and entrained fly ash flow uniformly upwards. In this freeboard-like region, temperatures are below the ash’s sintering point. Fresh feedstock is added at the top end of the upper annulus-flow region. The core flow is dominated by exothermic reactions leading to temperatures above the ash fluid temperature in both spouted-bed cores. The dense annulus flow protects the walls from high temperatures. Because of the increased particle volume fraction in this region, the oxygen-to-fuel ratio is drastically reduced compared to that in the core zones. Hence, endothermic reactions dominate, which in turn causes reduced temperatures. As a result, the lower annulus region has temperatures below the ash liquid point. The upper annulus region operates below the obstruction temperature. If the gasifier tends to produce sticky particles in the transition region between the upper spouted bed and the freeboard, particles collide and form clusters. Those fall down into the annulus flow region or the slag bath. Compared to bubbling fluidized beds, this approach has higher fluidization velocities. Hence, an increase in operational flexibility is achieved because, even in part load operation, the gas velocities are sufficient to maintain the central spout. Thus, the fluidization principle can be sustained over a broader gas-flow range compared to bubbling fluidized beds. The following advantages are combined:

• complete carbon conversion due to hot reaction zones and slag bath, even for low-reactive feedstock;

• protection of walls from high temperatures by the dense annulus flow; • gas outlet temperatures below ash sintering point due to multistaged conversion;

• reduced fly ash carryover due to spontaneous ash agglomeration; • ability to gasify feedstock of different particle sizes and reactivity. 11.2.1.3 Internal Circulating Fast Fluidized-Bed Gasifier (INCI)

The following section describes a new gasifier principle combining several technical, constructional, and fluid mechanical aspects from different mature gasifiers. While this section mostly focuses on aspects that are relevant for future numerical simulations, some additional features are described in Chapter 1. This

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Raw gas

Particle separation

Upper fast fluidized bed T < Tobstruction

Recycle

Jetting fluidized bed Tjet > Tfluid Trecirc. cell > Tfluid

Primary gasification agent

Lower, turbulent fluidized bed T < Tsinter

Feedstock inlet

Fixed bed T < Tsinter Secondary gasification agent Ash agglomerates Figure 11.2 Principal scheme of an internal circulating gasifier (INCI).

coal gasifier is currently developed at the TUBAF, Germany. An atmospheric pressure, 10 kg h−1 test unit started operation in 2013. A scheme is presented in Figure 11.2. The system is intended to convert high-ash coal fines that accrue during the exploitation of such coals. Hence, it was designed to gasify particles smaller than 500 �m. A modeling-based comparison with conventional coal gasifiers indicated that its performance is superior, especially for a high-ash (e.g., 25.3 wt%, dry basis) coal [12]. The gasifier applies the idea of a staged conversion. As also found for other new approaches, a post-gasification zone (dry fixed bed) is present in the lower part of the gasifier. Hence, the carbon content of the discharged bottom product is minimized (60%) are fed into the freeboard. This is known from the hot operation mode of conventional HTW gasifiers, but here it is developed further. The new feature is to increase the temperature in the freeboard above the ash sintering point in order to force ash agglomeration. Agglomerates fall onto the fixed bed and are post-gasified, as explained for the INCI approach. Even though the fluidization regime is different from that of the multistaged spoutedbed principle or INCI principle, the result is comparable. The outlet temperature is between 1000 and 1100 ◦ C, which is optimal from a thermodynamic point of view and close to the critical obstruction temperature. Thus, the outlet temperature needs to be monitored and adjusted carefully to avoid fouling in downstream units. Moreover, the carbon content of the bottom product is minimized ( Tfluid

Quenchwater Quench chamber T < Tsinter

Raw gas

Slag Figure 11.3 Principal scheme of a hybrid wall gasifier.

supply is interrupted temporary. This obviates the need for a permanent ignition burner and all the associated drawbacks. Nevertheless, a start-up burner for heating and pressurization purposes is required. However, the particle residence time is intended to be ∼1 s at gas velocities around 2 m s−1 . The partially converted coal enters the second section, which has a larger diameter. This is necessary to counteract an increase in the gas velocities due to gas production and the feeding of the secondary gasification agent. The oxygen-containing secondary gasification agent is added through radially distributed nozzles. The temperatures are increased above the ash fluid point, and the remaining carbon from the first section is converted. Feeding oxygen into the second stage requires a distinct safety concept because this approach has inherent risks. The lower section is equipped with a water jacket wall (comparable to the Lurgi fixed-bed

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gasifiers). Liquid slag cools at the wall and subsequently solidifies. In this way, a protective and self-renewing layer is generated. A small fraction of liquid slag flows down onto the solidified layer. The remaining slag is transported with the gas flow and directed into a water quench chamber. Here, the raw gas is cooled down and the slag solidifies to generate a solid slag bed, which can be discharged. The particle residence time is intended to be ∼2 s in the lower section in order to ensure that carbon conversion is complete. The gas velocities remain at ∼2 m s−1 and the temperatures are 50–100 K above the ash fluid temperature. The partial pressures of the reactive gas component are comparatively high because of the high total pressure. Consequently, carbon conversion is comparatively fast, which is advantageous if a short reactor is the main goal. Nonetheless, single-pass carbon conversion could be less than 99%. Thus, this approach features the separation and recycling of unconverted carbon from the bottom slag. The flow through both sections does not involve recirculation zones because of the very high particle loads. As a consequence, only a small fraction of the slag is transported to the walls and flows down onto the solidified slag layer. Thus, even in the lower part of the second section comparatively high fractions of (liquefied) ash are found in the gas flow. This is important for the radiative heat loss at the intersection between the cold quench chamber and the hot gasification chamber. Usually, provisions are made for considerable necking in the intersection zone. This is required to limit the heat loss in the lower part of the gasification chamber due to radiation. The main risk is the inadvertent cooling down of the slag below the ash liquid temperature and plugging the outlet of the gasification chamber. However, with a highly loaded flow, the heat transport is increased compared to the dilute flows from conventional entrained-flow gasifiers. Consequently, the risk of unacceptable radiative heat loss is reduced. Accordingly, necking is not needed, which is very advantageous from a constructional point of view. To summarize this approach, the use of an (ultra-) dense phase transport reactor is suggested. This has the potential for many simplifications compared to the conventional entrained flow, and the quench chamber allows for significant constructional simplifications. A comparable concept is being developed by Pratt and Whitney Rocketdyne (PWR). Their goal is also to design a small, compact, dense-phase gasifier for the same reasons as mentioned above. Since 2009, an 18 tpd (tons per day) pilot plant has been undergoing testing at the Gas Technology Institute, USA [17]. Their unique technical solutions for dry feedstock pressurization, feedstock mix injection, and spray quench are described elsewhere [7, 18, 19]. Since PWR was acquired by Gen Corp. in 2013, this compact gasifier is now being marketed and further developed as Aerojet Rocketdyne.1)

1) This news was announced on the Website www.rocket.com.

11.3

Future Gasifier Simulations

11.3 Future Gasifier Simulations 11.3.1 Requirements of Proposed Future Gasifiers

The presented patents in Section 11.2.1 and their respective approaches are only a small selection of registered patents devoted to improving fluidized-bed gasifiers. All these systems have some or all of the following key features in common:

• Combination of several flow regimes • Multistaged conversion • Several injection levels for gasification agents • High outlet velocities (20–100 m s−1 ) from gasification agent nozzles • Local zones where temperatures are above the ash sintering point and/or ash fluid point

• Incorporation of agglomeration as operating principle • Internal post gasification in moving bed or slag bath • Ability to handle a broad spectrum of particle sizes and high ash content. A systematization of these approaches is shown in Table 11.2. Here, all the discussed patents are summarized, where the focus is on applied fluidization regimes in the different zones of these multistaged reactors. For the bottom zone (zone 1), one can distinguish between systems with a dry fixed bed or a slag bath. Because Table 11.2 Systematization of prospective fluidized-bed gasifiers. FB, Fluidized bed; SB, spouted bed; JFB,–jetting fluidized bed; FFB, fast fluidized bed; Ag., agglomerating; Slag., slagging. Reactor zones from bottom to top Zone 1 Zone 2 Zone 3 Zone 1 Zone 2 Zone 3 Zone 4 Zone 1 Zone 2 Zone 3 Zone 4

Bottom zone Main reaction zone Dry Slag Dry Ag. Slag. Slag. Ag. bottom bath FB FB FB SB JFB FB with slag bath –Patent 1 X X X Multistaged SB with slag bath –Patent 2 X X X X Internal circulating gasifier –Patent 3 X

Ag. FFB

X X X Agglomerating FB –Patent 4

Zone 1 Zone 2 Zone 3

Dry FFB

X X X

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only the post gasification is done in this bottom zone, the remaining zones are assumed to be the main reaction zone (zones 2–4). From Table 11.2, it can be seen that each patent combines different fluidization regimes and temperature regimes. As a result of the multistaged conversion and combination of different flow regimes, future gasifiers will have zones of very different particle loads. There will be moving beds, dense fluidized beds, core annulus flow structures, lean and fast fluidized beds, and freeboards with particle volume fractions close to zero, all merged in the same vessel. Consequently, these systems will have regions where particle–particle interaction needs to be considered. From this point of view, the primary requirement of future gasifier simulations is that they reflect particle–particle interactions. This can be realized via discrete element models (DEMs) or an Euler–Euler approach. In the latter case, both the gas phase and solids phase are modeled as interpenetrating continua. Instead of resolving each particle or groups of particles in a Lagrangian framework, here the solids phase obeys Navier–Stokes-type equations. This requires an Eulerian framework. van der Hoef [20] presents an overview of possibilities to simulate dense gas–solid systems. Several results from the literature show how Euler–Euler-based simulations are applied to gas–solid interacting systems [21, 22], such as dense fluidized beds [23], spouted beds [24, 25], fast fluidized beds [26, 27], and jetting fluidized beds [28, 29]. All the presented simulations show a good agreement with the respective measurements. They prove the ability of Euler–Euler-based simulations to predict the behavior of systems with a broad range of local void fractions. So far, the subgrid models discussed in the previous chapters have applied a Lagrangian framework to describe the particle phase. They need to be adapted to be applicable to Euler–Euler-type simulations. A second important topic is the area of validity of the developed subgrid models in terms of validated Re number ranges. Future gasifiers can be expected to operate with a broad particle size spectra and a wide range of relative velocities between the gas phase and solids. The presence of several particle sizes results from the requirement to feed coarser particles with coal fines and from the formation of clusters/agglomerates in the reactor. Furthermore, the comparatively high slip velocities are a general feature of fluidized beds and especially fast fluidized beds (circulating fluidized beds), which results in comparatively high Re numbers [30]. Additionally, several nozzles for gasification agents can be found at several heights in the proposed gasifiers. These nozzles should have outlet velocities of 20–100 m s−1 . Consequently, Re numbers will range from close to zero (for fine particles with very low slip velocities) up to 104 for larger particles (e.g., 5 mm) and high slip velocities in the jetting regions (e.g., 40 m s−1 ). Thus, at these Re numbers the validity of developed subgrid models needs to be confirmed. A third central aspect expected of future gasifier simulations is to reflect changes to the solid state of particles, namely softening and liquefaction. The resulting agglomeration and slag droplet behavior needs to be considered. As explained previously, several patented gasifiers will operate in different temperature zones.

11.3

Future Gasifier Simulations

Both elevated gas-outlet temperatures and locally restricted hot zones cause temperatures above the ash softening and ash fluid points. Hence, a dry, nonsticky ash, as known from the conventional HTW fluidized-bed gasifier, can no longer be assumed to be present. Particle clustering, as observed in the U-gas fluidizedbed gasifier [13] and the Kellogg–Rust–Westinghouse (KWR) fluidized-bed gasifiers [11], is an essential operating principle for several recently patented gasifiers. Besides agglomeration, the liquefaction of solid particles must also be incorporated in future simulations. This includes the flow behavior of highly viscous slag droplets, taking the influence of surface tension into account. However, there is a general tendency in modeling to focus exclusively on the carbon content in coal particles because this is the main reacting species. This might be sufficient to predict the carbon conversion rate and thus the overall thermodynamic performance of the gasifier, but many operational issues are disregarded. As explained by Laugwitz [16], undesired ash and slag behavior is often a reason for unplanned gasifier downtime. Moreover, blockages in the ash and slag discharge system cause fairly long gasifier stand times. From this point of view, it is absolutely necessary to incorporate the ash/slag and its properties in future simulations. Finally, an evaluation of proposed gasifier concepts shows not only the need to incorporate fluidized or entrained particles but additionally also to simulate the bottom post-gasification zone. It does not seem practical to simulate these moving beds or slag baths in a stand-alone simulation. The coupling between the amount of settling solid mass and the generated gases in the post-gasification zone, the resulting fluidization velocities, and again the amount of falling particles requires an integrated simulation approach (see Figure 11.4). Stand-alone simulations would require an iterative solution between these two systems. Convergence would require immense computation time and resources, if convergence could be achieved at all. Assuming, for example, that a dry moving-bed post-gasification zone must be incorporated in the gasifier simulation, there would be an additional flow regime in the reactor being investigated. Besides bubbling fluidization, jetting fluidization, and fast fluidization, there would also be a flow through a fixed bed. Thus, Amount of settling solids

Gas velocity out of internal postgasification zone

Amount of gas produced in internal post-gasification zone

Figure 11.4 Link between internal post-gasification zone and fluidized bed.

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the particle models applied (e.g., the Eulerian framework) must be adapted to be able to cover such a broad range of fluidization regimes. Spouted-bed simulations have demonstrated the feasibility of modeling such systems with different flow regime zones [24]. 11.3.2 Additional Fundamental Aspects of Future Numerical Simulations

Apart from future simulation requirements that are strongly connected with proposed future gasifier concepts, additional aspects of a more fundamental nature must be considered in the future. These aspects mainly address various simplifications made during gasifier simulations as well as gas and solid material properties and reaction kinetics. Most of the simplifications concern the devolatized char particles themselves. Usually, char particles are assumed to be spherical, nonporous solids composed of pure carbon and a certain ash fraction that is not well defined. Additionally, Euler–Euler-based simulations in most cases only feature one particle size. This is because each additional particle size requires the definition of an additional Euler phase. Consequently, computational costs increase proportionally. The sphericity assumption can easily be corrected by applying a sphericity factor (Wadell factor). It accounts for the increased surface of an irregularly shaped particle in comparison to a sphere with the same volume [31]. The corrected surface affects the drag forces and reaction rates. However, there is no information about the actual shape included in the sphericity factor. On the other hand, drag forces in the real process are affected by the actual shape of a particle. In this respect, the model’s informative value is limited and thus additional submodels need to be developed to predict, for example, drag forces, conversion rates, or particle radiation for an irregularly shaped particle. Even for Euler–Euler-based simulations, several particle sizes must be considered in the future. This is especially necessary for the agglomerating gasifiers presented in Section 11.2.1. There should be at least two or three sizes of fresh feedstock to reflect the segregation effects in fluidized beds and the potential of dust carryover. One additional size must reflect coarse agglomerates. What is most important is to account for particle porosity. It is well known that most of the reactive surface is provided by the inner structures (pores) of a char particle [32]. Moreover, these inner structures change during particle conversion, leading to complex effects determining the reactive and accessible surface area. If these structures and their development are incorporated into CFD models, more realistic simulations of char reaction rates at each time step are possible. If such advanced particle models are applied, suitable kinetics will be required. For this, kinetics can be used only where entailed measurements are evaluated under the assumption of porous particles. Furthermore, the porosity and inner surface must be known depending on carbon conversion. In addition to the particle size, shape, and porosity, the material properties are highly simplified. There are powerful models to simulate pyrolysis, for example, distributed activation energy model (DAEM) [33], chemical percolation

11.3

Future Gasifier Simulations

devolatilization (CPD) [34], functional group–devolatilization–vaporization– crosslinking (FG–DVC) [35], and FLASHCHAIN [36]. They can model complex pyrolysis products and devolatilization rates and have been successfully applied to simulations [37]. Nevertheless, the survival of certain pyrolysis products, for example, tars or other long-chain hydrocarbons, is not considered in most CFD simulations. In this respect, the path of unconverted pyrolysis products into the produced raw gas would be a field for further investigation in terms of CFD modeling. Because this issue is important for downstream gas cleaning, there is a great interest in predicting trace-gas components. Besides pyrolysis, there might be a need to improve the modeling of effects during the heating and drying phase of the fed particles. As demonstrated by Lee [38], particles might tend to burst. This may be due to several reasons. Both thermal shocks and the increase in pressure inside the particle due to gas production (moisture evaporation, devolatilization) are discussed in the literature. Fractionation results in small particle sizes and increased total outer surface areas. If suitable experimental data are available, then describing the amount and size of the fragments generated will be an easy task from a programming point of view. Another more complex and expensive approach would be to include mechanical stress models of solids into the simulation [39]. The most important improvement in terms of material properties would be a proper simulation of ash behavior. This refers to both ash liquefaction steps and ash compounds that can evaporate and accumulate in the reactor (e.g., sodium, potassium, chlorine, and phosphor). The ash and slag liquefaction behavior is of vital importance for many aspects of gasifier operation. Slagging gasifiers operate at temperatures above the ash fluid point, and so slag solidification inside the gasification chamber must be avoided. Some locations in the reactor are prone to slag cooling, but in general it is difficult to predict or explain where, when, or why slag solidifies at the reactor walls. CFD simulations could be a powerful tool to optimize reactor geometry and operation in terms of undesired slag solidification. For this, detailed information about slag viscosity, density, surface tension, and heat capacity at different temperatures (and the given pressure) need to be incorporated into the simulation. The resulting knowledge about the coating thickness of solidified slag layers and films of draining liquid slag would be of high value. Issues around the slag flow concern slag flow both down the walls and in the outlet region. In particular, the transition region from the gasification chamber to the cooling sections (e.g., quench chamber) is found to account for unplanned reactor downtimes due to solidified slag [16]. In addition to slag flow and slag agglomeration issues, there is a third concern. Fly ash can be entrained with the gas, which leads to the condensation and deposition of alkali and earth alkali components at the heat exchanger surfaces. The resulting fouling and degradation account for a significant amount of unplanned downtime [16]. An emerging issue connected with evaporated ash compounds is the accumulation of these components in the reactor and accordingly in downstream units and gray water systems. Moreover, slag foam generation can be caused by the evaporation of ash compounds out of the slags. This phenomenon is mainly reported for metallurgical application

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and glass production but has been observed in gasifiers as well [40]. Slag foams have altered flow conditions compared to original slags because of their different density and surface tension. This aspect could also be incorporated into future simulations of high-ash coal gasification. Another issue in terms of gasifier simulation is the modeling of thermal radiation. Not only the radiant energy transfer but also the absorption, emission, and scattering of the participating media are in most cases modeled by applying one of the following models [41]:

• • • • •

discrete ordinates model discrete transfer radiation model P-1 radiation model Rosseland radiation model surface-to-surface radiation model.

An overview of radiation in particle-loaded flows is given in [42]. However, all simulations would benefit from improved data regarding the radiation-related material properties of the present optically thick mixture of particles and raw gas components. Even for gases (and particles) at atmospheric pressures, the property databases are incomplete. At elevated pressures, most gas species present in a gasification gas are even more poorly described regarding their radiation properties. The final topic for improved gasifier simulations is reaction kinetics. Even though some aspects from recent research have found their way into CFD simulations, they are still far from the state of the art. It is generally agreed that future gasifiers will be operated at higher pressures compared to today’s operating pressures of 20–40 bar. Besides, with the general requirement to have gas-phase material properties at higher pressures, there is a need for high-pressure reaction kinetics [43, 44]. This is especially true for heterogeneous reactions. An Australian research group has generated extended kinetic datasets for different ranks of coals at 25 bar [45]. In the open-source literature, 25 bar is the highest pressure at which complete kinetic datasets for different coals have been reported. Future simulations will need data from measurements conducted at up to 100 bar. Moreover, it is clear that kinetic data are valid only for the investigated coal. Within limited ranges, the information can be transferred to comparable coals but usually an individual set of kinetics is required. Thus, future gasifier simulations will need kinetics for low-grade, high-ash coals in particular. Apart from the coal type, the development of the reactive char surface as a function of the conversion rate is important and needs to be measured individually. As already mentioned, the reactive surface is strongly connected to the inner pore structures and their development during conversion. Future subgrid models must account for the inner pore structures. There is need to focus, in particular, on pore-diffusion phenomena. In addition to this, the type of kinetic expression can be improved by adopting more complex Langmuir–Hinshelwood-type kinetics instead of Arrhenius-type kinetics. This is a reasonable way to account for the so-called inhibition effects. This requires both a suitable model and measured

References

kinetic expressions. The inhibition effects reduce the rate of a certain reaction if the respective reactive sites are blocked by the inhibiting species [43]. This phenomenon is especially important in gasification environments, as e.g. H2 and CO are known to act as inhibitors for the Boudouard reaction and the heterogeneous water-gas reaction respectively. To summarize the suggested improvements, they are categorized for purposes of convenience into

• solid particles –particle–particle interaction –several particle sizes –non-sphericity –porosity –breakup • ash/slag –liquidation steps –density, viscosity, heat capacity, surface tension during all steps of liquefaction –coating thicknesses of solidified slag layers; film thickness of draining slag films –agglomeration –evaporating ash species (fouling at downstream units, slag foam) • radiation –gas phase/particle phase –elevated pressures • kinetics –high-pressure kinetics for homogeneous reactions –high-pressure kinetics for heterogeneous reactions (Arrhenius-type, Langmuir–Hinshelwood-type), especially for low-grade high-ash coals –kinetics for porous particle models. Because there is a strong need for improved CFD simulations of coal gasifiers, it is obvious that there is plenty of potential for future research.

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329

Index

a

c

adopted initial specific surface 200 advanced fluidized-bed coal gasifiers 308–309 – agglomerating fluidized bed with internal post gasification 314–315 – fluidized bed with slag bath 309–310 – internal circulating fast fluidized-bed gasifier (INCI) 311–314 – multistage spouted bed with slag bath 310–311 ambient gas composition effect 157–158 ANSYS-CFX 2 ANSYS-Fluent 2, 12–17, 76, 247, 248, 282, 297 – heterogeneous reactions setup 235 – – boundary settings 238–239 – – reactions definition 237–238 – – species and mixtures defining 236–237 – – species transport model 235–236 Arrhenius equation 175, 211, 280–281, 292 ash agglomeration 311, 313, 315

carbon burnout kinetic (CBK) model 273 carbon mass flux 232, 297 Carman–Kozeny equation 174 Cartesian grids 75–77, 80, 106, 131, 178 central difference scheme (CDS) 154 char particle combustion and gasification pore-resolved simulation 243–245 – 3D simulations under gasification conditions 264–267 – large porous particle of 2 mm – – large Reynolds numbers 259–262, 264 – – small Reynolds numbers 257–259 – model assumptions and chemistry 245–248 – – numerical scheme, discretization, and software validation 248–249 – small porous particles of 90 μm 249–256 – – gas temperature influence 256–257 chemical percolation devolatilization (CPD) 272, 276, 287, 290, 323 chemically reacting porous particle 166 coal oxidation 155 commercial gasification technologies 32 commercial software 14–17 Commonwealth Scientific and Industrial Research Organization (CSIRO) 22 – based modeling of entrained-flow gasifiers 6–8 – – mainstream computational submodels 8–9 – – review of works 13–17 – drying model 125–126 – model results 131–135 – modeling benchmark tests 17 – – Brigham Young University (BYU) reactor 20, 22

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b Baum and Street model 11 Beeman–Verlet scheme 52 Biot number 123 Boudouard reaction 251, 256–257 boundary conditions, on reacting interface 296 Brigham Young University (BYU) reactor 19–20, 22 Brinkman–Forchheimer equation 174 British coal utilization research association (BCURA) reactor 14, 18

Gasification Processes: Modeling and Simulation, First Edition. Edited by Petr A. Nikrityuk and Bernd Meyer. © 2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA.

330

Index

Commonwealth Scientific and Industrial Research Organization (CSIRO) (contd.) – – British coal utilization research association (BCURA) reactor 18 – – pressurized-entrained-flow reactor (PEFR) 22–23 Cooperative Research Centre for Coal in Sustainable Development (CCSD) 15, 22 current developments and modeling – CFD modeling benchmark tests 18 – – Brigham Young University (BYU) reactor 20, 22 – – British coal utilization research association (BCURA) reactor 18 – – pressurized-entrained-flow reactor (PEFR) 23 – CFD-based modeling of entrained-flow gasifiers 8 – – mainstream computational submodels 9–12 – – review of works 13–17 – direct numerical simulation (DNS), in particulate-flow modeling 3–6 – numerical modeling in engineering 1–3

d dense phase reactors. See transport reactors devolatilization 10–11 diffusion coefficient 177 diffusion kinetic single film (DKSF) 11–12, 15 dilute particulate flows 44 direct numerical simulation (DNS) 108, 110, 112, 116, 171, 217, 284 – in particulate-flow modeling 3–6 discrete element models (DEMs). See discrete particle models (DPM) discrete particle models (DPM) 43–45 distributed activation energy models (DAEMs) 272, 323 drag forces for two rotations 97, 100 dry-feed entrained flow gasifiers 35

e eddy dissipation concept (EDC) model 12 eddy dissipation model (EDM) 12–13 effective diffusion coefficient 177 effectiveness factor 181–182 energy conservation equation 126 entrained-flow processes 34 Euler–Euler model 43, 320, 322 Euler–Lagrange models 3–5, 43, 45 external heat transfer 113

f fixed-bed dry-bottom (FBDB) technology 31 fixed-bed gasifiers. See moving-bed gasifiers fixed-grid models 144 flamelet model 13 FLASHCHAIN 272, 276, 323 fluidized-bed processes 34, 36 Fourier number 109, 123 functional group–devolatilization–vaporization crosslinking (FG–DVC) 272, 323 future gasifier simulations – future numerical simulations additional fundamental aspects 322–325 – proposed future gasifier requirements 319–322

g Golia relation 87 grain model 244, 291 grain pore model 199–200

h hard-sphere models 45–46, 59 – formulation of collisions 63–65 – collision treatment in dense particulate systems 62–63 – governing equations 60–62 – illustration 65–68 Henry number 106 higly loaded compact gasifiers 315–316 – hybrid wall gasifier 316–318 homogeneous reaction layer (H-zone) 276, 278–280 homogeneous-zone single-film submodel ((H-zone model) 16

i ideal gas assumption 280 ideal gas law 177 immersed boundary (IB) method 75, 76, 80, 118 impulse conservation equation 125, 210 initial particle size and ambient gas temperature effect, on oxidation regime 158–160 integrated gasification combined cycle (IGCC) power plants 30–31 interface-marker function 147–148 interface-tracking, during char particle gasification – 3D interface tracking for porous char particle in kinetic regime – – internal surface reconstruction 196–197

Index

– – porous particle description 194–195 – – problem description 192–194 – porosity tracking for moving char particle 171–172 – – boundary conditions at particle surface 175 – – governing equations in gas phase 173 – – governing equations in porous particles 174 – – model setup 172–173 – – numerics 178–180 – – porous structure and particle shape change 176 – – reaction kinetics 175–176 – – results and discussion 180–192 – – transport properties 177 interfacial balance equations 293 internal circulation gasifier (INCI) 36, 38–39, 311–314 intrinsic heating regime 111 intrinsic solid reactivity. See char particle combustion and gasification pore-resolved simulation intrinsic submodel 291

k kinetic/diffusion model. See diffusion kinetic single film (DKSF)

l laminar and turbulent regimes 214–217 Langmuir–Hinshelwood mechanism 273 large eddy simulations (LESs) 3, 14 lattice Boltzmann method 4 Lewis number 177, 208, 234, 277, 294 Linde–Fränkel air separation process 30 linear model 108–109 linear spring–dashpot model 54 Lurgi fixed-bed dry-bottom (FBDB) technology 31

m

– model validation 282–286 – pyrolysis models 271–272 moving flame front (MFF) model 15, 219, 274–275 moving-grid-based models 144 moving particles modeling 43–47 – hard-sphere model 59 – – formulation of collisions 63–65 – – collision treatment in dense particulate systems 62–63 – – governing equations 60–62 – – illustration 67–68 – soft-sphere model 47 – – illustrative examples 56–59 – – numerical implementation 48–53 – – validation cases 53–56 multiparticle collision algorithm 62–63 – application 65–68 multiscale modeling strategy 5

n Navier–Stokes equations 43, 75–76, 118, 173, 290 Neumann boundary condition 211 noncommercial software 13–14 nonporous spherical particle heating in hot air stream – problem and model formulation 107–109 – results illustration and subgrid model 109–114 – semiempirical two-temperature subgrid model 114–116 – state of art 105–106 Nusselt number 4, 10, 77, 79–80, 86, 88–91, 96–98, 100–101, 106, 110, 112–113, 115, 121, 123–124, 129, 234, 279, 294–295

o one-film model 162–163, 165 OpenFOAM 2, 14

p

mass conservation equation 174, 210 MATLAB 2 MFIX code 2 moving-bed gasifiers 33–34 moving coal particle devolatilization and combustion subgrid model – CFD-based model 281–282 – char conversion models with homogeneous reaction 273–276 – heterogeneous char conversion models 272 – model formulation 276–281

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particle conversion 9–12 particle porosity impact and validation against experiments 217–218 particle-resolved CFD simulations – non-spherical particles 225–226 – – particle shapes 227–228 – – results 228–234 – spherical particles – – boundary condition 211–212 – – governing equations 210–211 – – numerics and software validation 212–218

331

332

Index

particle-resolved CFD simulations (contd.) – – relative particle velocity influence on oxidation regimes 219–225 – – review of literature 205–207 – – setup and model formulation 207–210 particle-source-in-cell method 8–9, 13 phenomenological flux relationships 124 point-implicit linear equation solver (Gauß–Seidel) 76 porous particle heating 116–117 – analytical model 123–124 – porosity 118–119 – problem and model formulation 117–118 – simulations results 119–123 porous particles oxidation behavior 156–157 porous sphericity 81 Prandtl number 91, 118, 221, 279 Pratt and Whitney Rocketdyne (PWR) 318 pressurized-entrained-flow reactor (PEFR) 24 pseudo-steady-state approach for carbon particle combustion and gasification – heterogeneous reactions setup in ANSYS-Fluent 235 – – boundary settings 238–239 – – reactions definition 237–238 – – species and mixtures defining 236–237 – – species transport model 235–236 – particle-resolved CFD simulations of non-spherical particles 225–226 – – particle shapes 227–228 – – results 228–234 – particle-resolved CFD simulations of spherical particles – – boundary condition 211–212 – – governing equations 210–211 – – numerics and software validation 212–218 – – relative particle velocity influence on oxidation regimes 219–225 – – review of literature 205–207 – – setup and model formulation 207–210 pyrolysis and particle conversion model 276–277

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q quadratic upstream interpolation for convective kinematics (QUICK) scheme 77, 212, 248, 297

r random pore model 199–200 random pore model (RPM) 145

Ranz–Marshall relation 10, 13, 87, 91, 113, 129, 279, 295 reactive core models 144–145 resolved discrete particle models (RDPM) 4–5, 44 Reynolds-averaged Navier–Stokes equations (RANS) 3, 14–15 Reynolds number 75, 77, 79, 80, 82, 83, 85, 87–89, 92, 94–96, 100–101, 107, 111, 118–120, 122–123, 131–133, 181–182, 184–185, 187, 206–208, 217, 222–225, 232, 279, 295, 298, 320 – large Reynolds numbers 259–262, 264 – small Reynolds numbers 257–259 Rhie and Chow stabilization 76 Rosin–Rammler distribution 14, 58

s second Damköhler number 180, 182, 253–255 semiglobal chemical reactions 280–281 Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) algorithm 76, 178, 212, 297 Sherwood number 130, 180, 221, 233–234, 294 shoelace algorithm 179 shrinking core model 144 shrinking particle model 144 single Nth-order reaction (SNOR) model 16 single particle heating and drying – nonporous spherical particle heating in hot air stream 105–116 – – problem and model formulation 107–109 – – results illustration and subgrid model 109–114 – – semiempirical two-temperature subgrid model 114–116 – – state of art 105–106 – porous particle heating 116–117 – – analytical model 123–124 – – porosity 118–119 – – problem and model formulation 117–118 – – simulations results 119–123 – spherical particle drying in hot air stream 124–125 – – CFD-based drying model 125–126 – – CFD-based model results 131–135 – – subgrid models and validation 127–131, 135–137 soft-sphere model 45–47 – illustrative examples

Index

– – – – – – – – – – –

– breaking dam problem 56–57 – fixed beds generation 58–59 – rotating drum 57–58 numerical implementation – collision parameters 49–50 – contact detection 50–52 – contact forces 48–49 – time integration 52–53 program flowchart 52–53 validation cases 53 – analytic solution for free-falling particle 54–55 – – free-falling particle 53–54 – – slipping sphere on rough surface 55–56 solids gasification 29–30 – derived challenges for research 40 – development trends 36–40 – historical background 30–33 – reactor types 33–36 species conservation of liquid water 126 species conservation of vapor 126 specific heat capacity 177 specific surface 183, 187, 190, 192, 198–200 spherical and nonspherical particles – code and software validation 78–80 – literature review 73–74 – model description 74–75 – – numerical scheme and discretization 75–78 – nonspherical particles – – heat and fluid flow of particles in flow direction 88–91 – – particle flow characteristics at different attack angles 91–95 – – particle orientation influence on drag forces and heat transfer 95–100 – porous particles 81–88 – – drag and Nusselt numbers for porous particles 85–88 – – geometry assumptions 81–82 – – heat and fluid flow past porous particles 82–85 spherical particle drying in hot air stream 124–125 – CFD-based drying model 125–126 – CFD-based model results 131–135 – subgrid model validation 135–137 – subgrid models 127–131 stagnant boundary layer approximation 143 STAR-CCM+ 2 Stefan flow 129–130, 185–186, 212–213, 225, 244, 245, 252–253, 255, 265, 267, 279, 291, 293, 297, 298 Stefan velocity 212

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Strongly Implicit Procedure (SIP) algorithm 76, 178 subgrid models, for particle devolatilization-combustion-gasification – moving coal particle – – CFD-based model 282 – – char conversion models with homogeneous reaction 274 – – heterogeneous char conversion models 272–273 – – model formulation 276–277 – – model validation 287–290 – – pyrolysis models 272 – novel intrinsic submodel for moving char particle gasification 290–291 – – CFD-based model 295–297 – – model formulation 291–295 – – model performance 297–300 subgrid models – new model 127–131 – results illustration 109–114 – semiempirical two-temperature model 114–116 – standard model 127 – validation 135–137 super-sampling method 76 surface Damköhler number 220–222 surface velocity 176 Sutherland–Hodgman clipping algorithm 179 syngas 30, 34 synthesis gas. See syngas

t thermal conductivity 177 Thiele modulus 180–181 time-splitting algorithm. See multiparticle collision algorithm total carbon consumption rate 292–293 transport equations, for chemical species 294 transport reactors 316, 318 trends, in gasifier design 307–308 – advanced fluidized-bed coal gasifiers 308–309 – – agglomerating fluidized bed with internal post gasification 314–315 – – fluidized bed with slag bath 309–310 – – internal circulating fast fluidized-bed gasifier (INCI) 311–314 – – multistage spouted bed with slag bath 310–311 – higly loaded compact gasifiers 315–316 – – hybrid wall gasifier 316–318 tridiagonal matrix algorithm (TDMA) 154

333

334

Index

turbulence–chemistry interactions 12–13 two-film model (TFM) 165–166, 213, 219, 273–274

u unreacted core models 144 unresolved discrete particle models (UDPM) 3–5, 44 unsteady char gasification and combustion 143–145 – advice for beginners 160–161 – analytical models – – chemically reacting porous particle 166 – – one-film model 162–163, 165 – – two-film model 165–166 – modeling approach 145–146 – – governing equations 146–148 – – initial conditions and boundary conditions 148–150

– – pore structure and interface tracking 152–153 – – reaction kinetics and transport properties 151–152 – numerics and code validation 153–155 – – results and discussion 155–160 user-defined function (UDF) 16, 17, 247, 282

v Verlet list 51–52 virtual H-zone single-film (VHZ-SF) model 275 volume-averaged model 109–110, 124, 125, 138 volume fraction of gas 147, 179

w wake flame 224, 231 Wolfram Mathematica

98, 99