Principles of Gas-Solid Flows (Cambridge Series in Chemical Engineering)

Principles of Gas-Solid Flows

Gas-solid flows are involved in numerous industrial processes and occur in various natural phenomena. This authoritative book addresses the fundamental principles that govern gas-solid flows and the application of these principles to various gas-solid flow systems. The book is arranged in two parts: Part I deals with basic relationships and phenomena, including particle size and properties, collision mechanics of solids, momentum transfer and charge transfer, heat and mass transfer, basic equations, and intrinsic phenomena in gas-solid flows. Part II discusses the characteristics of selected gas-solid flow systems such as gas-solid separators, hopper and standpipe flows, dense-phasefluidizedbeds, circulatingfluidizedbeds, pneumatic conveying systems, and heat and mass transfer in fluidization systems. As a comprehensive information source on gas-solid flows, this text will be useful to a broad range of engineers and applied scientists - chemical, mechanical, agricultural, civil, environmental, aeronautical, and materials engineers, as well as atmospheric and meteorological scientists.

Cambridge Series in Chemical Engineering

Editor Arvind Varma, University of Notre Dame Editorial Board Alexis T. Bell, University of California, Berkeley John Bridgwater, University of Cambridge L. Gary Leal, University of California, Santa Barbara Massimo Morbidelli, Swiss Federal Institute of Technology, Zurich Stanley I. Sandier, University of Delaware Michael L. Shuler, Cornell University Arthur W. Westerberg, Carnegie-Mellon University

Principles of Gas-Solid Flows LIANG-SHIH FAN The Ohio State University

CHAO ZHU The Ohio State University

CAMBRIDGE

UNIVERSITY PRESS

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 2RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521581486 © Liang-Shih Fan and Chao Zhu 1998 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1998 This digitally printed first paperback version 2005 A catalogue recordfor this publication is available from the British Library Library of Congress Cataloguing in Publication data Fan, Liang-Shih. Principles of gas—solid flows / Liang-Shih Fan, Chao Zhu. p. cm. Includes bibliographical references (p. ). ISBN 0-521-58148-6 1. Two-phase flow. 2. Gas flow. 3. Bulk solids flow. I. Zhu, Chao, 1961- . II. Title. TA357.5.M84F36 1997 531'.163-dc20 96-41141 CIP ISBN-13 978-0-521-58148-6 hardback ISBN-10 0-521-58148-6 hardback ISBN-13 978-0-521 -02116-6 paperback ISBN-10 0-521-02116-2 paperback

To Lois Liang-Yen, Liang-Fu, and Jeannine Liang-Chi and Zhichi Zhu, Zhu Ruan, Baolan Shi, and Victor

Contents

Preface Part I

page xv Basic Relationships

1

Size and Properties of Particles 1.1 Introduction 1.2 Particle Size and Sizing Methods 1.2.1 Equivalent Diameters of a Nonspherical Particle 1.2.2 Particle Sizing Methods 1.3 Particle Size Distributions and Averaged Diameters 1.3.1 Density Functions 1.3.2 Typical Distributions 1.3.3 Averaged Diameters of a Particulate System 1.4 Material Properties of Solids 1.4.1 Physical Adsorption 1.4.2 Deformation and Fracture 1.4.3 Thermal Properties 1.4.4 Electrical Properties 1.4.5 Magnetic Properties 1.4.6 Material Densities 1.4.7 Optical Properties Nomenclature References Problems

3 3 3 4 10 17 18 19 23 24 25 28 32 35 37 38 39 40 42 43

2

46 46 47 47 48 49 50 52 53 59 63 72 72 74

Collision Mechanics of Solids 2.1 Introduction 2.2 Stereomechanical Impact 2.2.1 Collinear Impact of Spheres 2.2.2 Planar Impact of Spheres 2.3 Theory of Elastic Contact of Solids 2.3.1 General Relations of Stresses in a Solid Medium in Equilibrium 2.3.2 Concentrated Force at a Point in an Infinite Solid Medium 2.3.3 Force on the Boundary of a Semiinfinite Solid Medium 2.3.4 Hertzian Theory for Frictionless Spheres in Contact 2.3.5 Theories for Frictional Spheres in Contact 2.4 Collision of Elastic Spheres 2.4.1 Normal Collision of Elastic Spheres 2.4.2 Collision of Frictional Elastic Spheres vii

viii

Contents 2.5

Collision of Inelastic Spheres 2.5.1 Onset of Plastic Deformation 2.5.2 Restitution Coefficient Nomenclature References Problems 3

78 78 80 83 85 85

Momentum Transfer and Charge Transfer 3.1 Introduction 3.2 Particle-Fluid Interactions 3.2.1 Drag Force 3.2.2 Basset Force 3.2.3 Saffman Force and Other Gradient-Related Forces 3.2.4 Magnus Effect and Force Due to Rotation of a Sphere 3.3 Interparticle Forces and Field Forces 3.3.1 Van der Waals Force 3.3.2 Electrostatic Force 3.3.3 Collisional Force 3.3.4 Field Forces 3.4 Motion of a Single Particle 3.4.1 Basset, Boussinesq, and Oseen (BBO) Equation 3.4.2 General Equation of Motion 3.5 Charge Generation and Charge Transfer 3.5.1 Static Electrification of Solids 3.5.2 Charge Transfer by Collision Nomenclature References Problems

87 87 87 87 88 95 97 101 101 103 104 105 107 107 108 111 111 119 123 126 128

4

130 130 130 131 133 138

Basic Heat and Mass Transfer 4.1 Introduction 4.2 Heat Conduction 4.2.1 Heat Transfer of a Single Sphere in a Quiescent Fluid 4.2.2 Heat Conduction in a Collision of Elastic Spheres 4.3 Convective Heat Transfer 4.3.1 Dimensional Analysis of Forced Convection in a Single-Phase Flow 4.3.2 Heat Transfer of a Single Sphere in a Uniform Flow 4.3.3 Thermal Convection in Pseudocontinuum One-Phase Flow 4.4 Thermal Radiation 4.4.1 Single-Particle Scattering 4.4.2 Radiant Heating of a Particle 4.4.3 General Considerations in Radiation with a Particle Cloud 4.4.4 Radiation Through an Isothermal and Diffuse Scattering Medium 4.5 Mass Transfer 4.5.1 Diffusion and Convection 4.5.2 Mass and Heat Transfer Analogy

138 138 141 142 143 148 150 154 156 156 157

Contents

ix

Nomenclature References Problems

159 161 162

5

B asic Equations 5.1 Introduction 5.1.1 Eulerian Continuum Approach 5.1.2 Lagrangian Traj ectory Approach 5.1.3 Kinetic Theory Modeling for Interparticle Collisions 5.1.4 Ergun Equation 5.1.5 Summary 5.2 Modeling of Single-Phase Flows 5.2.1 General Transport Theorem and General Conservation 5.2.2 Governing Equations 5.2.3 Kinetic Theory and Transport Coefficients 5.2.4 Modeling for Turbulent Flows 5.2.5 Boundary Conditions 5.3 Continuum Modeling of Multiphase Flows 5.3.1 Averages and Averaging Theorems 5.3.2 Volume-Averaged Equations 5.3.3 Volume-Time-Averaged Equations 5.3.4 Transport Coefficients and Turbulence Models 5.3.5 Boundary Conditions of Particle Phase 5.4 Trajectory Modeling of Multiphase Flows 5.4.1 Deterministic Trajectory Models 5.4.2 Stochastic Trajectory Models 5.5 Kinetic Theory Modeling for Collision-Dominated Dense Suspensions 5.5.1 Dense-Phase Transport Theorem 5.5.2 Hydrodynamic Equations 5.5.3 Collisional Pair Distribution Function 5.5.4 Constitutive Relations 5.6 Equations for Flows Through Packed Beds 5.6.1 Darcy's Law 5.6.2 Straight Capillaric Model 5.6.3 Ergun's Equation 5.7 Dimensional Analysis and Similarity 5.7.1 Scaling Relationships for Pneumatic Transport of Dilute Suspensions 5.7.2 Scaling Relationships for Fluidized Beds Nomenclature References Problems

164 164 164 165 166 166 167 167 167 169 170 174 179 182 182 189 193 196 205 205 206 208 210 211 213 215 217 222 223 224 225 230

6

244 244 244 245

Intrinsic Phenomena in a Gas-Solid Flow 6.1 Introduction 6.2 Erosion and Attrition 6.2.1 Ductile Erosion and Brittle Erosion

230 232 236 239 242

Contents 6.2.2 Locations of Erosive Wear 6.2.3 Mechanisms of Attrition 6.3 Thermodynamic Properties of a Gas-Solid Mixture 6.3.1 Density, Pressure, and Equation of State 6.3.2 Internal Energy and Specific Heats 6.3.3 Isentropic Change of State 6.4 Pressure Waves Through a Gas-Solid Suspension 6.4.1 Acoustic Wave 6.4.2 Normal Shock Wave 6.5 Instability 6.5.1 Wave Motions in Stratified Pipe Flows 6.5.2 Continuity Wave and Dynamic Wave 6.6 Particle-Turbulence Interaction Nomenclature References Problems

Part II

247 252 254 254 257 258 259 259 265 270 270 280 285 288 292 293

System Characteristics

7

Gas-Solid Separation 7.1 Introduction 7.2 Separation by Rotating Flow 7.2.1 Mechanism and Type of Rotary Row Dust Separators 7.2.2 Flow Field in a Cyclone 7.2.3 Collection Efficiency of Cyclones 7.3 Electrostatic Precipitation 7.3.1 Mechanism of an Electrostatic Precipitator 7.3.2 Migration Velocity and Electric Wind 7.3.3 Collection Efficiency of Electrostatic Precipitators 7.4 Filtration 7.4.1 Mechanisms of Filtration and Types of Filters 7.4.2 Pressure Drop in a Filter 7.4.3 Collection Efficiency of Fabric Filters 7.5 Gravity Settling and Wet Scrubbing 7.5.1 Gravity Settling Chambers 7.5.2 Mechanisms of Scrubbing and Types of Scrubbers 7.5.3 Modeling for Scrubbing and Collection Efficiency Nomenclature References Problems

297 297 297 297 300 303 309 310 311 312 314 314 315 319 320 321 323 324 328 330 331

8

333 333 333 334 336 337

Hopper and Standpipe Flows 8.1 Introduction 8.2 Powder Mechanics in Hopper Flows 8.2.1 Mohr Circle for Plane Stresses 8.2.2 Mohr-Coulomb Failure Criterion and Coulomb Powders 8.2.3 Static Stress Distributions in Standpipes and Hoppers

Contents

xi

8.2.4 Stress Distribution in a Steady Hopper Flow 8.2.5 Flowability of Powders in Hopper Design 8.3 Hopper and Standpipe Flow Theories 8.3.1 Moving Bed Flows in a Feed Hopper 8.3.2 Standpipe Flows 8.3.3 Hopper-Standpipe-Discharger Flow 8.3.4 Multiplicity of Steady Standpipe Flows 8.3.5 Leakage Flow of Gas in a Standpipe 8.4 Types of Standpipe Systems 8.4.1 Overflow and Underflow Standpipes 8.4.2 Inclined Standpipe and Nonmechanical Valves Nomenclature References Problems

340 342 346 346 350 354 357 359 361 361 364 366 368 369

9

371 371 371 372 374 376 378 378 380 381 381 382 388 389 392 396 396 398 399 400 401 402 403 403 404 406 407 408 408 408 409 411 415

Dense-Phase Fluidized Beds 9.1 Introduction 9.2 Particle and Regime Classifications and Fluidized Bed Components 9.2.1 Classification of Fluidized Particles 9.2.2 Fluidization Regimes 9.2.3 Components in a Dense-Phase Fluidized Bed 9.3 Minimum Fluidization and Particulate Fluidization 9.3.1 Minimum Fluidization 9.3.2 Particulate Fluidization 9.4 Bubbling Fluidization 9.4.1 Onset of Bubbling 9.4.2 Single Bubble in a Fluidized Bed 9.4.3 Bubble/Jet Formation and Bubble Coalescence and Breakup 9.4.4 Bubble/Jet Size and Bubble Rise Velocity 9.4.5 Gas Row Division and Bed Expansion 9.5 Turbulent Fluidization 9.5.1 Regime Transition and Identification 9.5.2 Determination of Transition Velocity 9.5.3 Hydrodynamic Characteristics 9.6 Entrainment and Elutriation 9.6.1 Mechanisms of Solids Ejection into the Freeboard 9.6.2 Correlations and Modeling 9.7 Slugging 9.7.1 Shapes and Rise Velocities of Single Slugs 9.7.2 Continuous Slugging 9.8 Spouted Beds 9.8.1 Onset of Spouting 9.8.2 Maximum Spoutable Bed Depth and Spout Diameter 9.8.3 Fountain Height 9.8.4 Gas Flow Distribution Nomenclature References Problems

xii

Contents

10

Circulating Fluidized Beds 10.1 Introduction 10.2 System Configuration 10.3 Flow Regimes and Transitions Between Regimes 10.3.1 Flow Regimes and Regime Diagrams 10.3.2 Regime Transition Determination 10.3.3 Operable Fluidization Regimes 10.4 Hydrodynamic Behavior in a Macroscale 10.4.1 Axial Profiles of Cross-Sectional Averaged Voidage 10.4.2 Radial Profiles of Voidage and Solids Flux 10.4.3 Overall Solids Holdup 10.5 Local Solids Flow Structure 10.5.1 Transient Nature of Solids Flow 10.5.2 Characterization of Intermittent Solids Flow 10.6 Mathematical Models of Fast Fluidization 10.6.1 Models Based on the Concept of Cluster 10.6.2 Models Based on the Core-Annular Flow Structure 10.6.3 Models Based on the Axial Profiles of Solids Holdup 10.6.4 Two-Phase Flow Models and Computational Fluid Dynamics Nomenclature References Problems

421 421 422 423 423 425 429 438 438 440 442 443 444 445 447 447 448 451 451 453 455 459

11

461 461 461 461 462 464 464 466 467 469

Pneumatic Conveying of Solids 11.1 Introduction 11.2 Classifications of Pneumatic Conveying Systems 11.2.1 Horizontal and Vertical Transport 11.2.2 Negative- and Positive-Pressure Pneumatic Conveyings 11.2.3 Dilute Flow Versus Dense Flow 11.2.4 Flow Regimes and Regime Transitions 11.3 Pressure Drop 11.3.1 General Pressure Drop in One-Dimensional Flow 11.3.2 Drag Reduction 11.3.3 Pressure Drop and Acceleration Length in Developing Regions 11A Critical Transport Velocities 11.4.1 Minimum Transport Velocity 11.4.2 Pick-up Velocity 11.5 Rows in Bends 11.5.1 Single-Phase Flow in a Curved Pipe 11.5.2 Paniculate Flow in a Bend 11.6 Fully Developed Dilute Pipe Flows 1 L6.1 Basic Equations and Boundary Conditions 11.6.2 Characteristic Relations 11.6.3 Temperature Distributions of Phases Nomenclature References Problems

472 474 475 476 478 478 481 482 483 487 489 494 496 498

Contents

xiii

12

499 499 499 500 502 503 506 512 512 513 518 521 521 524 525 526 526 527 527 527 532 532 535 537

Heat and Mass Transfer Phenomena in Fluidization Systems 12.1 Introduction 12.2 Suspension-to-Surface Heat Transfer 12.2.1 Heat Transfer Modes and Regimes 12.2.2 Film Model 12.2.3 Single-Particle Model 12.2.4 Emulsion Phase/Packet Model 12.3 Heat Transfer in Dense-Phase Fluidized Beds 12.3.1 Particle-to-Gas and Bed-to-Gas Heat Transfer 12.3.2 Bed-to-Surface Heat Transfer 12.3.3 Effect of Operating Conditions 12.4 Heat Transfer in Circulating Fluidized Beds 12.4.1 Mechanism and Modeling 12.4.2 Radial and Axial Distributions of Heat Transfer Coefficient 12.4.3 Effect of Operating Parameters 12.5 Heat Transfer in Spouted Beds 12.5.1 Gas-to-Particle Heat Transfer 12.5.2 Bed-to-Surface Heat Transfer 12.6 Mass Transfer in Multiparticle Gas-Solid Systems 12.6.1 Mass Transfer in Dense-Phase Fluidized Beds 12.6.2 Mass Transfer in Circulating Fluidized Beds Nomenclature References Problems Appendix: Index

Summary of Scalar, Vector, and Tensor Notations

540 545

Preface

Gas-solid flows are involved in numerous industrial processes and occur in various natural phenomena. For example, in solid fuel combustion, gas-solid flows are involved in pulverized coal combustion, solid waste incineration, and rocket propellant combustion. Gas-solid flows are encountered in pneumatic conveying of particulates commonly used in pharmaceutical, food, coal, and mineral powder processing. Fluidization is a common gas-solid flow operation with numerous important applications such as catalytic cracking of intermediate hydrocarbons, and Fischer-Tropsch synthesis for chemicals and liquid fuel production. Gas-solid flows occur in gas-particle separations, as exemplified by cyclones, electrostatic precipitators, gravity settling, and filtration operations. Fine powder-gas flows are closely associated with material processing, as in chemical vapor deposition for ceramics and silicon production, plasma coating, and xerography. In heat transfer applications, gassolid flows are involved in nuclear reactor cooling and solar energy transport using graphite suspension flows. Solid dispersionflowsare common in pigment sprays, dust explosions and settlement, and nozzle flows. The natural phenomena accompanied by gas-solid flows are typified by sand storms, moving sand dunes, aerodynamic ablation, and cosmic dusts. The optimum design of the industrial processes and accurate account of the natural phenomena that involve gas-solid flows as exemplified previously require a thorough knowledge of the principles governing these flows. This book is intended to address basic principles and fundamental phenomena associated with gas-solid flows, as well as characteristics of selected gas-solid flow systems. It covers the typical range of particle sizes of interest to gas-solid flows, i.e., 1 ^m-10 cm, recognizing that flow characteristics for submicrometer particles are also of great industrial importance. The book features a systematic account of important theories or models concerning particle mechanics as well as fluid dynamics from their origins of the development. The physical interpretation and limitations in application of these theories or models are emphasized. Various intrinsic phenomena underlying the gas-solid flow systems are also illustrated. The book is aimed as a textbook for seniors and graduate students who are interested in general or specific topics of gas-solid flows. In addition, it can be used as a reference for researchers and practitioners who are interested in the general field of multiphase flow. It is written with multidisciplinary engineering readers in mind. Specifically, it will be of benefit to chemical and mechanical engineering readers as well as readers in other engineering disciplines, including agricultural, civil, environmental, pharmaceutical, aeronautical, mining, and atmospheric and meteorological sciences. The book contains two parts; each part comprises six chapters. Part I deals with basic relationships and phenomena of gas-solid flows while Part II is concerned with the characteristics of selected gas-solid flow systems. Specifically, the geometric features (size and size distributions) and material properties of particles are presented in Chapter 1. Basic particle sizing techniques associated with various definitions of equivalent diameters of particles are also included in the chapter. In Chapter 2, the collisional mechanics of solids, based primarily on elastic deformation theories, is introduced. The contact time, area, and xv

xvi

Preface

force of colliding particles are discussed using theories of elastic collision, which are important to the formulation of the momentum, heat, and charge transfer processes involving collisions of solids. Chapter 3 is devoted to the momentum and charge transfer of gas-solid flows. Various forces in gas-solid flows due to gas-particle interactions, particle-particle interactions, and external fields are delineated. Equations for single-particle motion, based on a force balance analysis, are derived. Basic mechanisms of charge generation in gassolid flows are also introduced in the chapter, along with a detailed discussion of charge transfer mechanism by particle collisions. Chapter 4 deals with fundamental concepts and theories of heat and mass transfer in gas-solid flows. Highlights include thermal radiation of the particulate phase and heat conduction in collisions of elastic particles. Chapter 5 presents four basic modeling approaches of gas-solid flows, namely, continuum modeling of multiphase flows or multifluid modeling, trajectory modeling, kinetic theory modeling for collision-dominated dense suspensions, and the Ergun equation for flow through a packed bed of particles. In this chapter, the hydrodynamic equations of single-phase flows are first discussed. Here, basic concepts of kinetic theory of gas and turbulence models are introduced as a preamble to discussion of these basic modeling approaches. In contrast to the k—e turbulence model for single-phase flows, the k—e—k p model is introduced with the continuum approach of gas-solid flows to account for gas-solid turbulence interactions. Chapter 6 focuses on the discussion of intrinsic phenomena in gas-solid flows, such as erosion and attrition, acoustic wave and shock wave propagation through a gas-solid suspension flow, thermodynamic properties of a gas-solid mixture, flow instability, and gas-solid turbulence interactions. Chapter 7 is concerned with gas-solid separations. The basic separation methods introduced in this chapter include cyclone, filtration, electrostatic precipitation, gravity settling, and wet scrubbing. Chapter 8 deals with hopper flows and standpipe flows, which are commonly encountered in the bulk solids handling and transport processes. In order to understand the fundamental hopper and standpipe flow characteristics, some basic concepts of powder mechanics are illustrated. Chapter 9 introduces the general concept of gas fluidization. Specifically, the chapter addresses dense-phase fluidization, which represents a gas-particle operation of enormous industrial importance. Various operating regimes including particulate fluidization, bubbling/slugging fluidization, and turbulent fluidization are discussed along with spouting. The fundamental properties of bubble, cloud, and wake and the intrinsic bubble coalescence and breakup and particle entrainment phenomena are illustrated. Chapter 10 continues the discussion of fluidization under higher-velocity conditions which are characterized by fast fluidization. Fast fluidization is conducted in a riser of a circulating fluidized bed system where solid particles are circulating in a loop. This chapter illustrates the interactive relationship of gas-solid flows in a loop situation by considering the flow behavior of the individual loop components and their effects on the overall gas-solid flow characteristics. Chapter 11 is concerned mainly with the dilute transport or pipe flow of gas-solid suspensions. Some pertinent phenomena such as drag reduction are discussed. Fully developed pipe flow and gas-solid flow in a bend are also illustrated. Chapter 12 describes transport phenomena underlying heat and mass transfer in fluidized systems. Transport models and empirical correlations are introduced to allow heat and mass transfer properties in various fluidized systems to be quantified. An appendix which summarizes the scalar, vector, and tensor notations presented in the text is provided. Throughout the text, unless otherwise noted, the correlation equations presented are given in SI units. Common notations used across the chapters such as superficial gas velocity

Preface

xvii

and particle Reynolds number are unified. The solution manual for homework problems is available. Interested instructors are urged to contact the publisher about it. The book is intended to be used in various ways depending on the specific information that the readers desire. While the material is described in a logical sequence chapter by chapter, each chapter is presented with extensive cross-references and maintains reasonable independence. Thus, readers who wish to have a quick grasp of a specific subject may go directly to the relevant chapters. It is important to note that gas-solid flow is a rapidly developing field of research, and that the physical phenomena of gas-solid flows are so complex that a comprehensive understanding of the phenomena is far from complete. Therefore, the present text is also intended to provide readers with ample fundamental concepts to allow them to follow through new developments in the field. We would like to express our sincere thanks to the following colleagues who have reviewed the text and provided constructive suggestions and overviews: Professor R. S. Brodkey, Professor R. Clift, Professor J. F. Davidson, Dr. R. Davis, Professor N. Epstein, Professor J. R. Grace, Dr. K. Im, Professor B. G. Jones, Professor D. D. Joseph, Dr. C.-H. Lin, Dr. P. Nelson, Dr. S. L. Passman, Professor R. Pfeffer, Professor M. C. Roco, Professor S. L. Soo, Dr. B. L. Tarmy, Professor U. Tiiziin, and Professor L.-X. Zhou. We are grateful to Dr. E. Abou-Zeida, Dr. P. Cai, Mr. S. Chauk, Dr. T. Hong, Dr. P.-J. Jiang, Professor J. Kadambi, Dr. T. M. Knowlton, Dr. S. Kumar, Dr. R. J. Lee, and Dr. J. Zhang for their valuable technical assistance in providing information which was incorporated in the text. Special thanks are due to Mr. R. Agnihotri, Dr. D.-R. Bai, Dr. H.-T. Bi, Dr. A. GhoshDastidar, Mr. E.-S. Lee, Dr. S.-C. Liang, Mr. J. Lin, Mr. T. Lucht, Mr. X.-K. Luo, Dr. S. Mahuli, Mr. J. Reese, Mr. S.-H. Wei, Dr. J. Zhang, Mr. T.-J. Zhang, and Mr. J.-P. Zhang, who have read part of the text and have provided valuable comments. The outstanding editorial assistance of Dr. T. Hong and Dr. K. M. Russ is gratefully acknowledged. Thanks are also extended to Dr. E. Abou-Zeida and Mrs. Maysaa Barakat for their excellent drawing of the figures. The inquisitive students in the Chemical Engineering 801 course, Gas-Solid Flows, and the 815.15 course, Fluidization Engineering, taught by the senior author in the Department of Chemical Engineering at the Ohio State University have provided important feedback about the text. Their input is indeed extremely helpful. Financial assistance to this writing project provided by the members of the Ohio State University/Industry Consortium Program on Fluidization and Particulates Reaction Engineering, including Shell Development Co., E. I. duPont de Nemours & Co., Hydrocarbon Research Inc., Exxon Research & Engineering Co., Texaco Inc., and Mitsubishi Chemical Co., is deeply appreciated.

PART I

Basic Relationships

CHAPTER 1

Size and Properties of Particles

1.1

Introduction

The flow characteristics of solid particles in a gas-solid suspension vary significantly with the geometric and material properties of the particle. The geometric properties of particles include their size, size distribution, and shape. Particles in a gas-solid flow of practical interest are usually of nonspherical or irregular shapes and poly dispersed sizes. The geometric properties of particles affect the particle flow behavior through an interaction with the gas medium as exhibited by the drag force, the distribution of the boundary layer on the particle surface, and the generation and dissipation of wake vortices. The material properties of particles include such characteristics as physical adsorption, elastic and plastic deformation, ductile and brittle fracturing, solid electrification, magnetization, heat conduction and thermal radiation, and optical transmission. The material properties affect the long- and short-range interparticle forces, and particle attrition and erosion behavior in gas-solid flows. The geometric and material properties of particles also represent the basic parameters affecting the flow regimes in gas-solid systems such as fluidized beds. In this chapter, the basic definitions of the equivalent diameter for an individual particle of irregular shape and its corresponding particle sizing techniques are presented. Typical density functions characterizing the particle size distribution for polydispersed particle systems are introduced. Several formulae expressing the particle size averaging methods are given. Basic characteristics of various material properties are illustrated.

1.2

Particle Size and Sizing Methods

The particle size affects the dynamic behavior of a gas-solid flow [Dallavalle, 1948]. An illustration of the relative magnitudes of particle sizes in various multiphase systems is given in Fig. 1.1 [Soo, 1990]. It is seen in this figure that the typical range of particle sizes of interest to gas-solid flows is roughly from 1 /xm to 10 cm. The particle shape affects the flowability of powders, their packing, and the covering power of pigments. Qualitative definitions for particle shapes are given in Table 1.1. The shape of particles is commonly expressed in terms of shape factors and shape coefficients [Allen, 1990]. Particles used in practice for gas-solid flows are usually nonspherical and polydispersed. For a nonspherical particle, several equivalent diameters, which are usually based on equivalences either in geometric parameters (e.g., volume) or in flow dynamic characteristics (e.g., terminal velocity), are defined. Thus, for a given nonspherical particle, more than one equivalent diameter can be defined, as exemplified by the particle shown in Fig. 1.2, in which three different equivalent diameters are defined for the given nonspherical particle. The selection of a desired definition is often based on the specific process application intended.

11 Size and Properties of Particles Table 1.1. Definitions of Particle Shape Acicular Angular Crystalline Dendritic Fibrous Flaky Granular Irregular Modular Spherical

needle-shaped sharp-edged or having roughly polyhedral shape freely developed in a fluid medium of geometric shape having a branched crystalline shape regularly or irregularly thread-like plate-like having approximately an equidimensional irregular shape lacking any symmetry having rounded, irregular shape global shape

Source: T. Allen's Particle Size Measurements, Chapman & Hall, 1990.

L— Typical particle size range covered in this book .Clay-

. Silt >L

- Aerosol reactor

J

Sand —J* Gravel J

Fluidizedbed

J

. Air pollutionMetallized L- propellant I rocket

United States Department of Agriculture (USDA) classification

Applications Pneumatic conveying

Electric mobility _ -Centrifuge sedimentation. Measurements

Microscope _ LElectronJ Imicroscopel 10-

10"

. Sieves.

10"°

10"

10"

10°

10"

1 10"

l

(Unit: m)

Figure 1.1. Magnitudes of particle sizes in gas-solid systems (after Soo, 1990).

1.2.1

Equivalent Diameters of a Nonspherical Particle

An equivalent diameter of a particle is usually defined in relation to a specific sizing method developed on the basis of a certain equivalency criterion. Several equivalent diameters of a spherical particle commonly employed are discussed in the following sections. 1.2.1.1 Sieve Diameter A sieve diameter is defined as the width of the minimum square aperture through which the particle will pass. A common sizing device associated with this definition is a series of sieves with square woven meshes. Two sieve standards, i.e., Tyler Standard and American

1.2 I Particle Size and Sizing Methods

Minimum dimension

Figure 1.2. Schematic illustration of multidimensions of a particle and its equivalent volume diameter, surface diameter, and sieve diameter. Martin's diameter • Feret's diameter «

• Projected area diameter

Figure 1.3. Schematic illustration of different particle diameters based on 2-D projected image.

Society for Testing and Materials (ASTM) Standard, are widely used; they are introduced in §1.2.2.1.

1.2.1.2 Martin's Diameter, Feret's Diameter, and Projected Area Diameter Martin's diameter, Feret's diameter, and projected area diameter are three diameters defined on the basis of the projected image of a single particle. Specifically, Martin's diameter is defined as the averaged cord length of a particle which equally divides the projected area. Feret's diameter is the averaged distance between pairs of parallel tangents to the projected outline of the particle. The projected area diameter is the diameter of a sphere having the same projected area as the particle. These diameters are schematically represented in Fig. 1.3. The projected area diameter of a particle d\ can be related to the particle projected

11 Size and Properties of Particles area A by )

(...)

J Martin's diameter and Feret's diameter of a particle depend on the particle orientation under which the measurement is made. Thus, obtaining a statistically significant measurement for these diameters requires a large number of randomly sampled particles which are measured in an arbitrarily fixed orientation. Since Martin's diameter, Feret's diameter, and projected area diameter are based on the two-dimensional image of the particles, they are generally used in optical and electron microscopy. The principles of microscopy as a sizing method are discussed in §1.2.2.2. 1.2.1.3 Surface Diameter, Volume Diameter, and Sauter's Diameter The surface diameter, d$, volume diameter, dy, and Sauter's diameter, J32, are defined such that each of them reflects a three-dimensional geometric characteristic of an individual particle. A surface diameter is given as the diameter of a sphere having the same surface area as the particle, which is expressed by

*

=

Vx

(L2)

where S is the particle surface area. A volume diameter is the diameter of a sphere having the same volume as the particle, which is defined by dv = ( — )

(1.3)

where V is the particle volume. The Sauter's diameter or surface-volume diameter is defined as the diameter of a sphere having the same ratio of external surface to volume as the particle, which is given by

The concept of the surface diameter may be mostly used in the field of adsorption and reaction engineering, where the equivalent surface exposure area is important. The determination of the surface area depends on the method of measurements; for example, permeametry can give a much lower area than does gas adsorption. The latter often includes the contribution of pore surface area, which is accessible to the gas molecules. The determination of particle surface area by gas adsorption is given in §1.2.2.4. The fundamentals of gas adsorption are further covered in §1.4.1. The volume diameter of a particle may be useful in applications where equivalent volume is of primary interest, such as in the estimation of solids holdup in a fluidized bed or in the calculation of buoyancy forces of the particles. The volume of a particle can be determined by using the weighing method. Sauter's diameter is widely used in the field of reacting gas-solid flows such as in studies of pulverized coal combustion, where the specific surface area is of most interest.

1.2 I Particle Size and Sizing Methods 1.2.1 A Dynamic Diameter The dynamic response of a particle in gas-solid flows may be characterized by the settling or terminal velocity at which the drag force balances the gravitational force. The dynamic diameter is thus defined as the diameter of a sphere having the same density and the same terminal velocity as the particle in a fluid of the same density and viscosity. This definition leads to a mathematical expression of the dynamic diameter of a particle in a Newtonian fluid as

(1.5) dt =

where Ret is the particle Reynolds number at the terminal velocity; Co is the drag coefficient, which is a function of Ret; fi denotes the viscosity of the fluid; p and p p represent the densities of the fluid and the particle, respectively; Upt is the particle terminal velocity; g is the gravitational acceleration; and dt is the equivalent dynamic diameter. The relationship between Co and Ret for a sphere is given by Fig. 1.4 [Schlichting, 1979]. Mathematically, it can be expressed by 24 CD = —

Ket

Re

t

< 2

18 5 C D = —06 2 < Re Ret C D = 0.44

t

(1.6)

< 500

500 < Ret < 2 x 105

The three correlations in Eq. (1.6), in order from top to bottom, are known as Stokes's, Allen's, and Newton's equations, respectively. Combining these equations with Eq. (1.5),

O° 10

10

10

10

10

10

10

10

Figure 1.4. Drag coefficient for spheres as a function of Ret (from Schlichting, 1979).

11 Size and Properties of Particles the terminal velocity of a sphere is related to its diameter by

RCt

2