Solid Fuels Combustion and Gasification: Modeling, Simulation, and Equipment Operations Second Edition [2 ed.] 1420047493, 9781420047493

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Second Edition

Solid Fuels Combustion and Gasification Modeling, Simulation, and Equipment Operations

Second Edition

Solid Fuels Combustion and Gasification Modeling, Simulation, and Equipment Operations

Marcio L. de Souza-Santos

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2010 by Taylor and Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number: 978-1-4200-4749-3 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright. com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Souza-Santos, Marcio L. de. Solid fuels combustion and gasification : modeling, simulation, and equipment operations / Marcio L. de Souza-Santos. -- 2nd ed. p. cm. -- (Mechanical engineering) Includes bibliographical references and index. ISBN 978-1-4200-4749-3 (alk. paper) 1. Fuel--Combustion. 2. Coal gasification. 3. Fuel--Combustion--Equipment and supplies. I. Title. TP319.S68 2010 621.402’3--dc22 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

2009050557

Dedication To the 200 years of Charles Darwin, one among the greatest liberators of humanity. The history of science—by far the most successful claim to knowledge accessible to humans—teaches that the most we can hope for is successive improvement in our understanding, learning from our mistakes, an asymptotic approach to the Universe, but with the proviso that absolute certainty will always elude us. Carl Sagan (The Demon-Haunted World: Science as a Candle in the Dark)

Contents Preface.......................................................................................................................ix Acknowledgments .................................................................................................. xiii Nomenclature ........................................................................................................... xv Chapter 1

Basic Remarks on Modeling and Simulation ....................................... 1

Chapter 2

Solid Fuels .......................................................................................... 19

Chapter 3

Equipment and Processes ................................................................... 43

Chapter 4

Basic Calculations .............................................................................. 79

Chapter 5

Zero-Dimensional Models .................................................................99

Chapter 6

Introduction to One-Dimensional Steady-State Models .................. 125

Chapter 7

Moving-Bed Combustion and Gasiication Model ........................... 141

Chapter 8

Chemical Reactions .......................................................................... 167

Chapter 9

Heterogeneous Reactions ................................................................. 185

Chapter 10 Drying and Devolatilization ............................................................. 203 Chapter 11 Auxiliary Equations and Basic Calculations.................................... 235 Chapter 12 Moving-Bed Simulation Programs and Results ............................... 271 Chapter 13 Bubbling Fluidized-Bed Combustion and Gasiication Model ........ 299

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viii

Contents

Chapter 14 Fluidization Dynamics ..................................................................... 315 Chapter 15 Auxiliary Parameters Related to Fluidized-Bed Processes ............. 343 Chapter 16 Bubbling Fluidized-Bed Simulation Program and Results .............. 361 Chapter 17 Circulating Fluidized-Bed Combustion and Gasiication Model ..... 411 Chapter 18 Circulating Fluidized-Bed Simulation Program and Results........... 421 Appendix A The Fundamental Equations of Transport Phenomena .................... 433 Appendix B Notes on Thermodynamics .............................................................. 439 Appendix C Possible Improvements on Modeling Heterogeneous Reactions ...... 455 Appendix D Improvements on Various Aspects ................................................... 463 Appendix E Basics on Turbulent Flow ................................................................. 467 Appendix F Classiication of Modeling for Bubbling Fluidized-Bed Equipment ........................................................................................ 473 Appendix G Basics on Techniques of Kinetics Determination ............................ 479 Index ...................................................................................................................... 483

Preface The general perception is to equate solid fuels with pollution. That is understandable, but not necessarily correct or inevitable. There is no doubt that the burning of coal in thermoelectric stations raises justiied concerns regarding the emission of pollutants. The abundance of coal, lack of alternative resources, and economic pressures have led many countries to continue and even increase the application of this fuel for power generation. Evidently, that picture is not unchangeable, irst because special attention has been given to biomasses because of their renewability and overall zero carbon dioxide generation aspects, and second because of the development of new techniques. These new techniques have allowed much lower emissions of sulfur and nitrogen oxides into the atmosphere, while raising eficiency. A good part of that success is due to process optimization through mathematical modeling and simulation. Therefore, it is not surprising that the number of professionals and graduate students entering ields related to combustion and gasiication of solid fuels is increasing. However, unlike focused researchers, many of these new professionals are not interested in deep consideration based on exhaustive literature review of specialized texts on the subject. These publications are important, but most assume an audience of accomplished mathematical modelers and are not preoccupied with presenting the details of how it is possible to start from fundamental and general equations and arrive at a inal model for an equipment or process. Those just starting out in the ield are also generally not interested in the other extreme, i.e., simple and mechanistic description of equipment design procedures or instruction manuals for application of commercial simulation packages. They tend to have a few main preoccupations: • Being acquainted enough with the fundamental phenomena taking place in the equipment or processes • Gaining knowledge of basic procedures of modeling and simulation of equipment and systems • Developing mathematical models and simulation programs to predict the behavior of this equipment or these processes, mainly for cases when no commercial simulators are available • Applying the available tools when simulators are commercially available, depending on the conditions of time and resources (in this case, one should be able to properly set the conditions asked as input by the simulator, evaluate the applicability of possible solutions, and choose among various alternatives) • Using simulation programs to improve operations of existing equipment or for optimized design of new equipment • Building conidence for decision making regarding process improvements and investments ix

x

Preface

Experience shows that a good route to acquiring real and testable understanding of a subject in the area of processing is to develop models and computer simulators. Developing a successful simulator is an accomplishment in which a student can justly take pride, since it represents the accumulated knowledge of the subject. It is important to emphasize the need for simple models. Of course, there are several levels at which models can be built. Nonetheless, one should be careful with too simple or too complex ones. The low extreme usually provides only supericial information, while the other usually takes years to develop and involves considerable computational dificulties because of convergence problems and incoherencies. In the present text, model complexity is extended only as far as needed to achieve a reasonable representation of that equipment. For instance, the examples are limited to two dimensions, and most of the models are based on a one-dimensional approach. This may sound simplistic; however—as demonstrated in the text—the level of detail and usefulness of results from such simulations are not negligible. In addition, this book can also be used as an introduction for more complex models. The text is designed to be useful to graduate students, engineers, and professionals with degrees in any exact science and has been applied as such for many years in courses at all levels. Although the main concern of this book is with modeling combustion and gasiication processes, it is also an introduction to mathematical modeling and simulation. The basic methods illustrated here can be used for modeling a wide range of processes and equipment commonly found in industry. Operations of boilers, furnaces, incinerators, gasiiers, or any other equipment dealing with combustion or gasiication phenomena involve a multitude of simultaneous processes, such as heat, mass, and momentum transfers; chemical kinetics of several reactions; drying; pyrolysis, etc. Those should be coherently combined to allow reasonable simulation of industrial units or equipment. To help accomplish that, the text describes combustion and gasiication processes in some detail. Although the basic concepts of thermodynamics and transport phenomena can be found in several texts, the respective fundamental equations are included. Thus, the need to consult other texts has been minimized. Most of the time, the concepts usually learned in engineering courses are suficient. In view of the practical approach, several correlations and equations are taken from the literature without a preoccupation with mathematical demonstrations. References are provided and should be consulted by those interested in further details. The main strategy of the book is to teach by example. Besides the signiicant fraction of industrial equipment operating with suspensions of pulverized solid fuels, the speciic cases of moving and luidized beds have been selected because they have several qualities: • They cover a good fraction of processes found in industry involving combustion or gasiication of solid fuels. In the particular case of luidized beds, the fraction of equipment using that technique has continuously increased. Actually, conventional boilers and furnaces operating with suspensions have been retroitted to luidized beds.

Preface

• They allow easy-to-follow examples on how simplifying assumptions regarding the operation of real industrial equipment can be set. • They permit relatively quick introduction of fundamental equations without the need for too complex treatments. • They provide simple examples of modeling and show how those examples can be put together in order to write a simulation program. To summarize, the book intends to accomplish the following objectives: • Show several constructive and operational features of equipment dealing with combustion and gasiication of solid fuels, such as coal, biomass, solid residues, etc. • Present basic aspects of solid and gas combustion phenomena • Introduce the fundamental methodology to formulate a mathematical model of the above equipment • Demonstrate possible routes from model to workable computer simulation program • Show comparisons between simulations and real operations • Illustrate interpretations of simulation results that may be applied as tools for improving the performance of existing industrial equipment or for optimized design of new equipment It is organized as follows: Chapter 1 presents a few generally applicable notions concerning modeling and simulation. Chapter 2 shows the main characteristics of solid fuels, such as coals and biomasses. Chapter 3 introduces basic concepts of solid-gas systems and the main characteristics of combustion and gasiication equipment. Chapter 4 provides formulas and methods to allow initial calculations regarding solid fuel processing. Chapter 5 describes the fundamental equations of zero-dimensional models with the objective of allowing veriication of overall relations between inputs and outputs of any general process, including combustors and gasiiers. Chapter 6 introduces a very basic and simple irst-dimension model of a gas reactor. Of course, it is not the intention to present any model for lames. That is beyond the scope of this introductory book. However, it is useful to introduce standard considerations regarding mathematical modeling and the application of mass, energy, and momentum transfer equations. Chapter 7 describes the irst example of a model for solid fuel combustion and gasiication equipment. The case of a moving-bed combustor or gasiier is used for this model. Chapters 8 and 9 introduce the methods to compute gas–gas and gas–solid reaction rates. Chapter 10 introduces the modeling of drying and pyrolysis of solid fuels.

xi

xii

Preface

Chapter 11 presents auxiliary and constitutive equations and methods that may be used to build a computer program to simulate the model described in the previous chapter. Chapter 12 shows how to put together all the information previously described in order to build a workable simulation program. The chapter also presents comparisons between simulations and real operations of a moving-bed gasiier. Chapter 13 repeats the approach used in Chapter 7, but now having in mind bubbling luidized-bed combustors and gasiiers. Chapters 14 and 15 provide correlations and constitutive equations needed to complete the simulation of bubbling luidized bed combustors and gasiiers. Chapter 16 shows the strategy of assembling a simulation program based on the model presented for bubbling luidized beds, as well as comparisons between simulations and real operations of boilers and gasiiers. Chapter 17 repeats the approach used in Chapter 13 for the case of circulating luidized-bed combustors and gasiiers. Chapter 18 is similar to Chapter 16, but now using examples related to circulating luidized bed equipment. Almost all chapters include exercises. They are intended to stimulate the imagination and build conidence in solving problems related to modeling and simulation. The relative degree of dificulty or volume of work involved on solutions is indicated by the increasing number of asterisks. Problems with four asterisks usually require good training in solving differential equations or demand considerable work.

Acknowledgments I had the pleasure of working with several colleagues who helped in this endeavor. In particular, I am grateful to Alan B. Hedley (University of Shefield, United Kingdom), Francisco D. Alves de Souza (Institute for Technological Research, São Paulo, Brazil), and former colleagues at the Institute of Gas Technology (Chicago). I am also grateful to my family for their support and motivation, Dr. Alex Moutsoglou (South Dakota State University) and CRC Press personnel for pointing out corrections and making suggestions, the University of Campinas for its support, and Prof. John Grace and Dr. Xuantian Li (University of British Columbia) for valuable information regarding the operations of a circulating luidized-bed gasiier.

xiii

Nomenclature a A ai â ae b B c C COC1(j) COC2(j) COF(j) COT(j) COV(j) d Dj dP E i f F f514 fair f bexp ffc ffr fm fmoist ftur fV g G

general parameter or coeficient (dimensions depend on the application), ratio between the radius of the nucleus and the original particle, or Helmoltz energy (J kg–1) area (m2) or ash (in chemical reactions) parameters or constants (dimensionless) activity coeficient (dimensionless) air excess (dimensionless) exergy (J kg–1) coeficient, constant, or parameter (dimensions depend on the application) speciic heat at constant pressure (J kg–1 K–1) constant or parameter to be deined in each situation coeficient of component j in the representative formula of char (after drying and devolatilization of original fuel) (dimensionless) coeficient of component j in the representative formula of coke (due to tar coking) (dimensionless) coeficient of component j in the representative formula of original solid fuel (dimensionless) coeficient of component j in the representative formula of tar (dimensionless) coeficient of component j in the representative formula of volatile fraction of the original solid fuel (dimensionless) diameter (m) diffusivity of component j in the phase or media indicated afterwards (m2 s–1) particle diameter (m) activation energy of reaction i (J kmol–1) factor or fraction (dimensionless) mass low (kg s–1) total mass fractional conversion of carbon air excess (dimensionless) expansion factor of the bed or ratio between its actual volume and volume at minimum luidization condition (dimensionless) mass fraction conversion of ixed carbon fuel ratio factor used in reactivity calculations (dimensionless) mass fraction of particles of kind m among all particles present in the process (dimensionless) mass fractional conversion of moisture (or fractional degree of drying) factor related to turbulent luidization regime (dimensionless) mass fractional conversion of volatiles (or degree of devolatilization) acceleration of gravity (m s–2) or speciic Gibbs function (J/kg) mass lux (kg m–2 s–1) xv

xvi

h H HHV i I jj k0i K0i ki Ki kt l L Lgrate LHV LT M Mj n NAr NBi nCP nCV nG Nj NNu NPe NPr NRe nS NSc NSh nSR p pj P q  Q r

Nomenclature

enthalpy (J kg–1) height (m) high heat value (J kg–1) inclination relative to the horizontal position (rad) variable to indicate the direction of mass low concerning a control volume (+1 entering the control volume; –1 leaving the control volume) mass lux of component j due to diffusion process (kg m–2 s–1) preexponential coeficient of reaction i (s–1; otherwise, unit depends on the reaction) preexponential equilibrium coeficient for reaction i (unit depends on the reaction and notation) kinetic coeficient of reaction i (s–1; otherwise, unit depends on the reaction) equilibrium coeficient for reaction i (unit depends on the reaction and notation) speciic turbulent kinetic energy (m2 s–2) mixing length (m) coeficient used in devolatilization computations (dimensionless) length of grate (m) low heat value (J kg–1) length of tube (m) mass (kg) molecular mass of component j (kmol/kg) number of moles Archimedes number (dimensionless) Biot number (dimensionless) number of chemical species or components number of control volumes number of chemical species or components in the gas phase mass lux of component j referred to a ixed frame of coordinates (kg m–2 s–1) Nusselt number (dimensionless) Peclet number (dimensionless) Prandtl number (dimensionless) Reynolds number (dimensionless) number of chemical species or components in the solid phase Schmidt number (dimensionless) Sherwood number (dimensionless) number of streams index for the particle geometry (0 = planar, 1 = cylindrical, 2 = sphere) partial pressure of component j (Pa) pressure (Pa) energy lux (W m–2) rate of energy generation (+) or consumption (–) of an equipment or system (W) radial coordinate (m)

xvii

Nomenclature

R R RC Rcond Rh R heat ri Rj R kind,j

R M,G,j R M,S,j RQ RR s S t T T* Te u U ureduc v V x X

equipment radius (m) universal gas constant (8314.2 J kmol–1 K–1) rate of energy transfer to (if positive) or from (if negative) the indicated phase due to convection (W m–3 [of reactor volume or volume of the indicated phase]) rate of energy transfer to (if positive) or from (if negative) the indicated phase due to conduction (W m–3 [of reactor volume or volume of the indicated phase]) rate of energy transfer to (if positive) or from (if negative) the indicated phase due to mass transfer between phases (W m–3 [of reactor volume or volume of the indicated phase]) heating rate imposed to a process (K/s) rate of reaction i (for homogeneous reactions: kg m–3 s–1; for heterogeneous reactions: kg m–2 s–1) rate of component j generation (if positive) or consumption (if negative) by chemical reactions (kg m–3 s–1; in molar basis [~], the units are kmol m–3 s–1) rate of component j generation (if positive) or consumption (if negative) by chemical reactions (units vary according to the kind of reaction: if the subscript indicates homogeneous reactions, the units are kg m–3 [of gas phase] s–1; if heterogeneous reactions, kg m–2 [of external are of reacting particles] s–1) total rate of production (or consumption if negative) of gas component j (kg m–3 [of gas phase] s–1) total rate of production (or consumption if negative) of solid-phase component j (kg m–3 [of reacting particles] s–1) rate of energy generation (if positive) or consumption (if negative) due to chemical reactions (W m–3 [of reactor volume or volume of the indicated phase]) rate of energy transfer to (if positive) or from (if negative) the indicated phase due to radiation (W m–3 [of reactor volume or volume of the indicated phase]) entropy (J kg–1 K–1) cross-sectional area (m2) or, if no index, cross-sectional area of reactor (m2) time (s) temperature (K) reference temperature (298 K)  ratio between activation energy and gas constant E  (K) R –1 velocity (m s ) gas supericial velocity (m s–1) or resistance to mass transfer (s m–2) reduced gas velocity (dimensionless) speciic volume (m3 kg–1) volume (m3) coordinate or distance (m) elutriation parameter (kg/s)

( )

xviii

xj y  Y  W wj z Z

Nomenclature

mole fraction of component j (dimensionless) coordinate (m) or dimensionless variable rate of irreversibility generation at a control volume (W) rate of work generation (+) or consumption (–) by an equipment or system (W) mass fraction of component j (dimensionless) vertical coordinate (m) compressibility factor (dimensionless)

GREEK LETTERS α αm β γ γB Γ δ  0, i ∆G ε ε’ εt ζ η θ Θ ι λ Λ φ µ νij Ξ ρ ρp ρj σ σv τ υ Φ χ

coeficient of heat transfer by convection (W m–2 K–1) relaxation coeficient related to momentum transfer involving solid phase (m s–1) coeficient (dimensionless) or mass transfer coeficient (m s–1) area of particles per volume of reactor or volume of indicated phase as index (m2/m3) area of bubbles per volume of reactor (m2/m3) rate of ines production due to particle attrition (kg s–1) unit vector (m) variation of Gibbs function related to reaction i (J kmol–1) void fraction (dimensionless) emissivity (dimensionless) dissipation rate of speciic turbulent kinetic energy (m2 s–3) particle porosity (m3 of pores/m3 of particle) eficiency or effectiveness coeficient angular coordinate solid particle friability (m–1) eficiency coeficient thermal conductivity (W m–1 K–1) parameter related to mass and energy transfer particle sphericity viscosity (kg m–1 s–1) or chemical potential (J kg–1) stoichiometry coeficient of component j in reaction i chemical component formula density (unit depends on the reaction and notation) apparent density of particle (kg m–3) mass basis concentration of component j (kg m–3) (in some situations, component j can be indicated by its formula) Stefan-Boltzmann constant (W m–2 K–4) standard deviation for distributed energy devolatilization model shear stress tensor (Pa) tortuosity factor Thiele modulus number of atoms of an element (irst index) in a molecule of a chemical component (second index)

Nomenclature

ψ ω ϖ Ω

xix

mass transfer coeficient (s–1 if between two gas phases; kmol m–2 s–1 if between gas and solid) Pitzer’s acentric factor (dimensionless) Air ratio (dimensionless) parameter of the Redlich-Kwong equation of state (dimensionless) or (only in Appendix E) a parameter related to mass and energy transfers

OTHER ∇ ∇2

gradient operator Laplacian operator

SUPERSCRIPTS → _ ∪ ~ ^ ′ ″ ′″ s

vector or tensor time averaged luctuation or perturbation in molar basis relative concentration (dimensionless) number fraction area fraction volume fraction for particles smaller than the particles whose kind and the level are indicated in the subscript

SUBSCRIPTS Numbers as subscripts may represent sequence of variables, chemical species, or reactions. In the particular case of chemical species, the number would be equal to or greater than 19. In the case of reactions, it will be clear when the number indicates reaction number. The numbering for components and reactions is shown in Tables 7.1 through 7.5. 0 a A air app aro ash av b B bexp bri

at reference or ideal condition at the nucleus–outer shell interface shell or residual layer air apparent (sometimes, this index does not appear and should be understood, as for instance ρp = ρp,app) aromatic ash average value based on exergy bubble related to the expansion of the bed bridges

xx

bulk c C car char CIP COF, COT, COV cond CP CSP CV cy d D daf dif dist E ental entro eq exer f F fc l fuel G h H het hom i I iCO iCV iSR j J K l L lam

Nomenclature

bulk critical value convection contribution or, in some obvious situations, carbon carbonaceous solid char coated inert particle see above (under “Nomenclature”) conduction contribution chemical component coke shell particle control volume or equipment cyclone drying or dry basis bed dry and ash-free basis diffusion contribution distributor emulsion relative to enthalpy relative to entropy equilibrium condition relative to exergy formation at 298 K and 1 atm freeboard ixed carbon luid related to fuel gas phase transfer of energy due to mass transfer related to the circulation of particles in a luidized bed or, in some obvious situations, hydrogen related to heterogeneous (or gas–solid) reactions related to homogeneous (or gas–gas) reactions reaction i (numbers are described in Chapter 7) as at the feeding point component number control volume (or equipment) number stream number chemical component (numbers are described in Chapter 7) related to the internal surface or internal dimension related to the recycling of particles, collected in the cyclone, to the bed chemical element at the leaving point or condition laminar condition

Nomenclature

m M max mb mf min moist mon mtp N O orif p P per plenum pores Q r R real S sat sit SR T tar to tr tur U v V W X Y ∞

xxi

physical phase (carbonaceous solid, m = 1; limestone or dolomite, m = 2; inert solid, m = 3; gas, m = 4) mass generation or transfer maximum condition minimum bubbling condition minimum luidization condition minimum condition moisture or water monomers metaplast nucleus, core, or, in some obvious situations, nitrogen at the external or outside surface, or, in some obvious situations, oxygen oriices in the distributor plate particle (if no other indication, property of particle is related to apparent value; see Equation 3.12) at constant pressure peripheral groups average conditions in the plenum below the distributor plate or device related to particle pores chemical reaction at reduced condition related to radiative heat transfer related to real or skeletal density of solid particles solid phase, particles, or, if indicated for a property (such as density), apparent particle density at saturation condition immobile recombination sites stream terminal value or tubes tar mixing-takeover value transition to fast luidization turbulent condition unreacted-core model related to devolatilization volatile wall exposed-core model or related to elutriation of particles related to entrainment of particles at the gas phase far from the particle surface

A

The Fundamental Equations of Transport Phenomena

The fundamental equations of mass (global and species), momentum, and energy conservation are well known and extensively published [1–4]. It is not the objective of the present book to repeat the deductions of these equations. The general forms of fundamental equations use vector and tensor notations. Although these forms are elegant, they have to be rewritten in the scalar form for each direction to be applicable to models and for easier computation. Actually, the forms presented here are simpliications. For instance, the equations for momentum transfer assume Newtonian luid behavior, constant density, and constant viscosity. The reader may ind this dificult to accept, mainly in cases of combustion processes where gases are involved. However, these differential equations are not going to be applied for sizable control volumes. As the text indicates, the differential balances are always used to set the system of differential equations for numerical solutions, where the equations are solved for a very small control volume. The solutions for variables (such as temperature, pressure, velocity, and concentration) are applied to set the boundary conditions for the next small volume. Hence, such assumptions usually are good approximations, as long numerical methods are applied to small, inite volumes. If not, the reader should consult classical references [1–4] to extract the necessary differential equations. Figures A.1 and A.2 illustrate the rectangular, cylindrical, and spherical coordinate systems. The various equations for continuity, mass, energy, and momentum transfers are listed for rectangular coordinates in Tables A.1, A.2, and A.3. A few additional important notes regarding the above tables are listed below: • In the equations of mass transfer, the diffusion coeficient (Dj) should be understood as the diffusivity of component j into the other at the same control volume. If a mixture of components is involved, the form shown here is an approximation. More precise deinitions and calculations of diffusion coeficients may be found in the literature [1–3, 5, 6]. • The equations for energy transfer consider the heat luxes only in terms of conduction. However, heat transfer by radiation may take place. Unfortunately, there is not a simple way to include that phenomenon in the above equations because radiative transfer depends not just on the properties 433

434

Solid Fuels Combustion and Gasification z (x, y, z) or (r, θ, z)

r

y θ

x

FIGURE A.1 System of rectangular (x, y, z) and cylindrical (r, θ, z) coordinates.

of the emitting body but also on the receiving part and the characteristics of the gas layer between them. • In the mass transfer equations, R M,j stands for the uniform rate of generation (+) or consumption (–) per unit of volume throughout the entire control volume. z (x, y, z) or (r, θ, φ)

θ

r

y φ

x

FIGURE A.2 System of spherical coordinates (r, θ, ϕ).

The Fundamental Equations of Transport Phenomena

435

TABLE A.1 Equations in Rectangular Coordinates Type Total mass continuity Species j continuity (ρ and Dj constants)

Equation

Equation Number

∂ρ ∂(ρu x ) ∂(ρu y ) ∂(ρu z ) + + + =0 ∂t ∂x ∂y ∂z

A.1

∂ρ j ∂ρ ∂ρ ∂ρ + u x j + u y j + uz j ∂t ∂x ∂y ∂z

A.2

 ∂2ρ ∂2ρ ∂2ρ  = D j  2j + 2j + 2j  + R M, j ∂y ∂z   ∂x Energy conservation (Newtonian, ρ and λ constants)

 ∂T ∂T ∂T ∂T  ρc  + ux + uy + uz ∂x ∂y ∂z   ∂t

A.3

2 2 2   ∂2 T ∂2 T ∂2 T  ∂u   ∂u   ∂u  = λ  2 + 2 + 2  + 2µ   x  +  y  +  z     ∂x   ∂y   ∂z   ∂y ∂z   ∂x   2 2 2  ∂u   ∂u ∂u  ∂u ∂u   ∂u + µ  x + y  +  x + z  +  y + z   + R Q   ∂y ∂y   ∂x   ∂z ∂x   ∂z  

Momentum conservation, x direction (Newtonian, ρ and µ constants)

Momentum conservation, y direction (Newtonian, ρ and µ constants)

Momentum conservation, z direction (Newtonian, ρ and µ constants)

 ∂u ∂u ∂u ∂u  ρ  x + u x x + u y x + uz x  ∂x ∂y ∂z   ∂t =−

 ∂2 u ∂P ∂2 u x ∂2 u x  + µ  2x + + + ρgx ∂x ∂y 2 ∂z 2   ∂x

 ∂u ∂u ∂u ∂u  ρ  y + u x y + u y y + uz y  ∂x ∂y ∂z   ∂t =−

A.5

 ∂2 u ∂2 u y ∂2 u y  ∂P + + µ  2y +  + ρg y ∂y ∂y 2 ∂z 2   ∂x

 ∂u ∂u ∂u ∂u  ρ  z + u x z + u y z + uz z  ∂x ∂y ∂z   ∂t =−

A.4

A.6

 ∂2 u ∂P ∂2 uz ∂2 uz  + µ  2z + + + ρgz ∂z ∂y 2 ∂z 2   ∂x

• In the energy transfer equations, RQ is the rate of generation (+) or consumption (–) of energy per unit of volume. For instance, if an endothermic or exothermic chemical reaction or electrical heating is involved, the term should be added to the equations. It is important to stress that the energy must be uniformly delivered or consumed throughout the entire control volume. Heat transfers due to localized processes should be accounted for by the other terms in the equations. The energy addition provided by work is already included in the form of the viscous

436

Solid Fuels Combustion and Gasification

TABLE A.2 Equations in Cylindrical Coordinates Type

Equation

Total mass continuity Species j continuity (ρ and Dj constants)

Energy conservation (Newtonian, ρ and λ constants)

Equation Number

∂ρ 1 ∂(ρru r ) 1 ∂(ρu θ ) ∂(ρu z ) + + + =0 ∂t r ∂r r ∂θ ∂z

A.7

∂ρ j ∂ρ u ∂ρ j ∂ρ + ur j + θ + uz j r ∂θ ∂t ∂r ∂z

A.8

 1 ∂  ∂ρ j  1 ∂2ρ j ∂2ρ j  = Dj  +  + R M, j + 2 r 2 ∂z 2   r ∂r  ∂r  r ∂θ  1 ∂  ∂T  1 ∂2 T ∂2 T   ∂T ∂T u θ ∂T ∂T  = λ ρc  + ur + + uz + 2 2  + 2 r r ∂θ ∂r ∂z  ∂z   ∂t  r ∂r  ∂r  r ∂θ

A.9

2 2 2   1  ∂u    ∂u     ∂u  + 2µ   r  +   θ + v r   +  z      ∂z    r  ∂θ   ∂r   2 2 2    1 ∂u r  ∂u 1 ∂u z  ∂  u   ∂u    ∂u + z + r + + µ  θ + + r  θ    + RQ  ∂r  r    ∂z   ∂r  r ∂θ   ∂z r ∂θ  

Momentum conservation, r direction (Newtonian, ρ and µ constants) Momentum conservation, θ direction (Newtonian, ρ and µ constants) Momentum conservation, z direction (Newtonian, ρ and µ constants)

 ∂u ∂u ∂u  u ∂u u2 ρ  r + u r r + θ r − θ + uz r  θ ∂ ∂ ∂ ∂z  t r r r  =−

 ∂  1 ∂(ru r )  1 ∂2 u r 2 ∂u ∂P ∂2 u r  + 2 − 2 θ + + µ   + ρgr 2  ∂r r ∂θ ∂z 2   ∂r  r ∂r  r ∂θ

 ∂u u ∂u uu ∂u ∂u  ρ  θ + ur θ + θ θ + r θ + uz θ  r ∂θ r ∂r ∂z   ∂t =−

A.11

 ∂  1 ∂(ru θ )  1 ∂2 u θ 1 ∂P 2 ∂u ∂2 uθ  + 2 + 2 r + + µ   + ρgθ 2  r ∂θ ∂z 2  r ∂θ  ∂r  r ∂r  r ∂θ

 ∂u ∂u ∂u  u ∂u ρ  z + u r z + θ z + uz z  ∂r ∂z  r ∂θ  ∂t =−

A.10

A.12

 1 ∂  ∂u z  1 ∂2 uz ∂2 u z  ∂P + + µ  + ρgz 2 + 2 r ∂z ∂z 2   r ∂r  ∂r  r ∂θ

dissipation terms (those involving the momentum transfer or shear stress tensors). • Finally, the Laplacian operator, used in momentum transfer in spherical coordinates, is given by ∇2 =

1 ∂  2 ∂ 1 ∂  ∂ 1 ∂2 . r θ sin + + ∂θ  r 2 sin 2 θ ∂φ2 r 2 ∂r  ∂r  r 2 sin θ ∂θ 

(A.19)

The Fundamental Equations of Transport Phenomena

437

TABLE A.3 Equations in Spherical Coordinates Type Total mass continuity Species j continuity (ρ and Dj constants)

Equation

Equation Number

1 ∂(ρu θ sin θ) 1 ∂(ρu φ ) ∂ρ 1 ∂(ρr 2 u r ) + + + =0 r sin θ r sin θ ∂φ ∂t r 2 ∂r ∂θ

A.13

u φ ∂ρ j ∂ρ j ∂ρ u ∂ρ j + ur j + θ + r ∂θ r sin θ ∂φ ∂t ∂r

A.14

 1 ∂  2 ∂ρ j  ∂2ρ j  ∂ρ  1 1 ∂  = Dj  2 r + 2 sin θ j  + 2 2  + R M, j    2 ∂ ∂ r r ∂ ∂ θ θ  r sin θ   r sin θ ∂φ    r

 ∂T u φ ∂T  ∂T u θ ∂T Energy ρc  + ur + + sin θ θ ∂φ  ∂ ∂ ∂ r t r r  conservation (Newtonian,  1 ∂  2 ∂T  1 ∂2 T  1 ∂  ∂T  sin θ  + 2 2 r + 2 = λ 2  ρ and λ    ∂θ  r sin θ ∂φ2   r ∂r  ∂r  r sin θ ∂θ  constants) 2 2 2   1 ∂u φ u r u θ cot θ    1 ∂u θ u r   ∂u  + 2µ   r  +  + + +  +   r  r r    r ∂θ  r sin θ ∂φ   ∂r  

A.15

2 2        r ∂  u θ  + 1 ∂u r  +  1 ∂u r + r ∂  u φ      ∂r  r  r ∂θ  ∂r  r     rsinθ ∂φ    + µ  + RQ 2      uφ  ∂ u sin θ ∂ 1 θ  +  +       r ∂θ  sinθ  rsinθ ∂φ   

 ∂u u θ2 + u φ2 u φ ∂u r  ∂u u ∂u ρ  r + ur r + θ r − + Momentum A.16  r r sin θ ∂φ  ∂t ∂r r ∂θ conservation,  r direction  ∂P 2 2 ∂u 2 ∂u 2 2 ∂u φ  =− + µ  ∇ 2 u r − 2 u r − 2 θ − 2 θ − 2 u θ cot θ − 2 + ρgr (Newtonian, ∂r r r ∂θ r ∂θ r r sin θ ∂φ   ρ and µ constants)  ∂u u φ ∂u θ u φ2 cot θ  ∂u u ∂u uu ρ  θ + ur θ + θ θ + r θ + − A.17 Momentum  ∂t ∂r r ∂θ r r sin θ ∂φ r  conservation,  θ direction  1 ∂P 2 ∂u u 2 cos θ ∂u φ  =− + µ  ∇ 2 uθ + 2 r − 2 θ − 2 2 + ρgθ (Newtonian, r ∂θ r ∂θ r sin θ r sin θ ∂φ   ρ and µ constants)   ∂u φ u φ ∂u φ u φ u r u φ u θ ∂u φ u θ ∂u φ ρ cot θ + + + + + ur A.18 Momentum r ∂θ r sin θ ∂φ r ∂r ∂t r   conservation, ϕ direction  uφ 1 ∂P 2 ∂u r 2 cos θ ∂u θ  =− + µ  ∇ 2 uφ − 2 2 + 2 + 2 2 + ρgφ (Newtonian, r sin θ ∂φ r sin θ r sin θ ∂φ r sin θ ∂φ   ρ and µ constants)

438

Solid Fuels Combustion and Gasification

REFERENCES 1. Bird, R.B., Stewart, W.E., and Lightfoot, E.N., Transport Phenomena, John Wiley, New York, 1960. 2. Slattery, J.C., Momentum, Energy, and Mass Transfer in Continua, Robert E. Krieger, New York, 1978. 3. Brodkey, R.S., The Phenomena of Fluid Motions, Dover, New York, 1967. 4. Luikov, A.V., Heat and Mass Transfer, Mir, Moscow, 1980. 5. Poling, B.E., Prausnitz, J.M., and O’Connell, J.P., The Properties of Gases and Liquids, 5th Ed., McGraw-Hill, New York, 2000. 6. Reid, C.R., Prausnitz, J.M., and Poling, B.E., The Properties of Gases and Liquids, 4th Ed., McGraw-Hill, New York, 1987.

on B Notes Thermodynamics CONTENTS B.1 B.2 B.3 B.4 B.5

Heat and Work .............................................................................................. 439 Chemical Equilibrium Equation ................................................................... 441 Speciic Heat ................................................................................................. 443 Correction for Departure from Ideal Behavior ............................................. 443 Generalized Zero-Dimensional Stable Models ............................................446 B.5.1 Second Law.......................................................................................446 B.5.1.1 Isenthalpic, Isentropic ........................................................ 447 B.5.2 Eficiencies........................................................................................448 B.5.2.1 Based on Enthalpy .............................................................448 B.5.2.2 Based on Entropy ...............................................................449 B.5.2.3 Based on Exergy ................................................................449 B.5.3 Summary .......................................................................................... 452 B.6 Heating Values .............................................................................................. 452 B.7 Representative Formation Enthalpy of a Solid Fuel ..................................... 453 References .............................................................................................................. 454 The objectives of this appendix are as follows: • To ensure a uniform notation regarding concepts used at various points in the book • To present some deinitions and methods also applied to various situations in the main text

B.1

HEAT AND WORK

Throughout this text, a given portion of matter will be called a system if no mass low crosses the system boundary. However, in the case of a control volume (CV), mass can low into or out from it. In thermodynamics, the most important entities are interactions and properties. Heat and work are interactions between a system and its surrounding environment. Temperature, pressure, volume, energy, entropy, enthalpy, exergy, etc., are properties of a system. In addition, temperature and pressure are intensive properties and thus do not depend on the amount of mass in the system. However, extensive properties, such as energy, do. If the value of an extensive property is divided by the mass in the system, the result is the respective speciic value, which is now called an intensive 439

440

Solid Fuels Combustion and Gasification

property. Throughout the book, the value of a property should be understood as the value of its intensive or speciic value. Heat interaction can occur between a system and its surroundings only if, in the absence of adiabatic walls, temperature difference exists between them. Work is an interaction taking place between a system and its surroundings if the only effect could be represented by the lifting or by the resisted descent of a weight in the environment. Resisted descent excludes free falls in a gravitational ield. This is so because in classical thermodynamics, work is understood as useful mechanical work. No useful work can be drawn from free fall, since no resistance is imposed to it. If there were a resistance, it could, for instance, be used to lift another weight or perform other forms or work, such as turning an electrical generator shaft. Therefore, the values of accelerations for resisted downward movements vary between 0 (constant-velocity fall) and values smaller than the characteristic of local gravitational ield (g). Properties of a system do not require reference to the condition of the environment surrounding it. In opposition, heat or work can be measured or evaluated only if processes involving exchanges between the system and its environment takes place. Measurement of intensity or effects of those processes might allow quantiication of heat and work interactions. On the other hand, the values of properties—for instance, temperature or pressure—are not functions of particular processes between the system and its environment. Heat interaction takes positive values if the internal energy of the involved system increases as a consequence of that sole interaction. On the other hand, heat interactions would acquire negative values if, acting alone, they provoke decreases in the internal energy of the system involved. Work interaction assumes positive values if the internal energy of the system involved decreases as a consequence of that sole interaction. On the other hand, work interactions would acquire negative values if, acting alone, they provoke increases in the internal energy of the system involved. It is also important to stress the character of system-environment interactions, represented by heat and work. The process are usually referred to as heat transfer and work transfer, respectively. However, one should be careful because they are just interactions or processes, and the system involved cannot accumulate heat or work. Also, it makes no sense to talk about temperature transfer, volume transfer, entropy transfer, or even exergy transfer.* They are properties, and properties cannot be transferred between a system and its environment. Properties can, however, experience modiications of their values due to heat and work interactions. In addition, the clear distinction between properties and interactions is necessary to preserve the mathematical rigor. For instance, unlike interactions, properties can be differentiated exactly. Properties are point functions, whereas interactions are line functions. As line functions, heat and work cannot present exact derivatives. This is because the amounts of heat or work interactions depend on the process path, which is given by sequence of states experienced by the system. *

Some authors insist on talking about entropy or exergy transfers. These are very relaxed concepts and might lead to mistakes.

441

Notes on Thermodynamics

As mentioned before, a CV differs from a system because matter can enter and leave the CV. This does not prevent the application of above concepts. Heat and work interactions can be observed between a CV and its environment. Property values can be assigned to a CV. The mass in the CV might vary, and if so, the values of intensive properties should be obtained by dividing the respective extensive value by the instantaneous mass held within the CV. The rate at which heat interaction takes place between the CV and the environment can be measured by using the energy balance equation for a CV (Chapter 5, Equation 5.3 or 5.4). The same can be said for rate at which work (or power) interaction occurs. To facilitate the dialogue, the text below may, on several occasions, refer to heat and work transfers.

B.2

CHEMICAL EQUILIBRIUM EQUATION

From classical thermodynamics, the equilibrium for an isolated system is attained when the Gibbs function reaches a minimum, or  dG

T,P

= 0.

(B.1)

The Gibbs function for a system, where a given number of components (nCP) or chemical species are found, is a function of its properties, such as temperature (T), pressure (P), and composition. Then, it is possible to write  =G  (T, P, n1, n 2 ,…, n CP ). G

(B.2)

For a given temperature and pressure, the total derivative of the Gibbs function is  dG

T ,p

n CP n CP   ∂G = ∑ dn j = ∑ µ jdn j .  j=1  ∂n j  T ,p ,n ≠ n j=1 j k

(B.3)

The equilibrium requires the Gibbs function to reach a minimum, and therefore Equation B.3 leads to n CP

∑ µ dn j

j

= 0.

(B.4)

j =1

On the other hand, when equilibrium of reaction i is established, the following relationship among all involved chemical species j is dn j = constant. ν j, i

(B.5)

442

Solid Fuels Combustion and Gasification

The reader should remember that the stoichiometry coeficients are positive for products and negative for reactants. Equations B.4 and B.5 provide n CP

∑ν

j, i

µ j = 0.

(B.6)

j =1

The chemical potential of speciic molar Gibbs function is deined by µ j = g j (T, P ) = h j (T) − Ts j (T, P ),

(B.7)

which, for ideal gaseous components, can be written as x P  µ j = h j (T) − T s0, j (T) − R ln j  . P0  

(B.8a)

g 0, j = h j (T) − Ts0, j (T),

(B.8b)

Since

Equation B.8a can be written as  ln µ j = g 0, j (T) + RT

x jP . P0

(B.9)

Here, the 0 subscript indicates the reference pressure (usually 1 atmosphere). Thus, using Equation B.6, the equilibrium imposes n CP

∑v j =1

j, i

x jP    g 0, j (T) + RT ln P  = 0. 0  

(B.10)

Let us call* n CP

 0 = v j, ig 0, j (T) ∆G ∑ j =1

T T n CO       c = ∑ ν j, i   h f + ∫ c jdT  − T s0, j + ∫ j dT   . T   j =1   298 298  

(B.11)

Therefore, it is now possible to write ν j ,i  nCP  P  ∆G −  0 = ln ∏ x νj j,i    . RT  P0    j=1

*

(B.12)

The formation enthalpies, absolute entropy, and polynomial coeficients to compute speciic heats of some substances can be found below.

443

Notes on Thermodynamics

The equilibrium coeficient K is deined as n CP

ν j ,i   nCP ν  P  ν j,i  nCP ν   P  ∑  ∆G j=1 0 j ,i j ,i K i = exp  −   = ∏ x j   =  ∏ x j    .  P0   RT  j=1   P0   j =1

B.3

(B.13)

SPECIFIC HEAT

Table B.1 provides relations between temperature and speciic heat of a few substances. The objective here is just to provide a few values to apply in examples. That table was compiled from several sources [1–9]. Of course, similar and more complete tables can be found in almost any manual for engineers or other professionals of the exact sciences. In any case, it is advisable to verify the limits of temperature for which the relations are applicable. If the temperature surpasses the upper or lower limit, adopt the value computed at that limit for higher or lower temperatures, respectively. This is called the saturation procedure. Note that the forms of the functions used to estimate the speciic heat of solids, liquids, and gases (ideal) are similar, thus facilitating the building of computational routines for processes where several phases and their mixtures may be involved.

B.4 CORRECTION FOR DEPARTURE FROM IDEAL BEHAVIOR When conditions depart from ideal, the Redlich-Kwong equations may be used for corrections of thermodynamic property values. Of course, several other methods can be found in the literature [7, 10, 11]. Nonetheless, the Redlich-Kwong-Soave method is among most commonly applied and is more than enough for this introductory text. By this method, enthalpy and entropy can be computed by [11]  ( Z − 1) h − h 0 = a − a 0 + T(s − s0 ) + RT

(B.14)

v − C2 C1 v + C2  v ln s − s0 = R ln − + R ln . 3/ 2 v  2C 2 T C2 v0

(B.15)

and

Here, the index 0 indicates the ideal value. The Helmoltz energy is given by  ln a − a 0 = −RT

v − C2 C1 v + C2  v ln − − RT ln . 1/ 2 2C 2 T v C2 v 0

(B.16)

Other parameters necessary for computations are as follows: P=

 2f Ω RTC RT − a v − C2 Ω b v (v + C2 )

(B.17)

444

TABLE B.1 Various Parameters and Properties for Chemical Speciesa Coefficients for Equation:  = a + a T + a T2 + a 4 C P 1 2 3 T2 Chemical Species

a1 0.11376 × 103 0.16336 × 102 0.20682 × 102 0.10213 × 103 0.48362 × 102 0.40936 × 102 0.59267 × 102 0.23607 × 102 0.39278 × 102 0.41385 × 102 0.71242 × 102 0.55852 × 102 0.64642 × 102 0.95505 × 102 0.10812 × 103 0.88627 × 102 0.12982 × 103

a2 0.15108 × 10–1 0.60972 × 10–2 0.17765 × 10–1 0.28002 × 10–1 0.57497 × 10–2 0.20272 × 10–1 0.12590 × 100 0.49622 × 10–1 0.56445 × 10–1 0.83145 × 10–1 0.59924 × 10–1 0.90085 × 10–1 0.11326 × 100 0.12143 × 100 0.17362 × 100 0.12074 × 100 0.20429 × 100

a3 –0.16144 × 10–5 –0.64762 × 10–6 –0.18696 × 10–5 –0.29650 × 10–5 –0.60058 × 10–6 –0.21335 × 10–5 –0.13250 × 10–4 –0.52248 × 10–5 –0.59491 × 10–5 –0.87601 × 10–5 –0.63223 × 10–5 –0.94938 × 10–5 –0.11936 × 10–4 –0.12802 × 10–4 –0.18300 × 10–4 –0.12735 × 10–4 –0.21533 × 10–4

a4 –0.34642 × 107 –0.83634 × 106 0.51256 × 105 –0.25145 × 107 0.66959 × 106 0.58490 × 105 0.36325 × 106 –0.21280 × 106 –0.10745 × 107 –0.11449 × 107 –0.20813 × 107 –0.16060 × 107 –0.21008 × 107 –0.28842 × 107 –0.34173 × 107 –0.36688 × 107 –0.41081 × 107

Molecular Mass (kg kmol–1) 101.961 12.011 40.080 100.089 56.079 72.140 136.138 16.043 28.054 30.069 46.069 42.080 44.096 58.123 72.150 78.113 86.177

h 0f

g 0f

–1

(MJ kmol ) –1676.8 0 0 –1207.7 –635.51 –482.74 –1433.6 –74.86 52.50 –84.78 –23.48 20.43 –103.92 –126.23 –146.54 82.98 –167.31

s 0 (MJ kmol ) (kJ kmol–1 K–1) –1

–1583.3 0 0 –1129.5 –604.16 –477.71 –1321.2 –50.85 68.39 –33.00 –168.18 62.76 –23.49 –17.17 –8.37 129.75 –0.25

50.95 5.74 41.74 92.95 39.77 56.52 10.68 186.44 219.43 229.64 282.78 267.12 270.09 310.33 349.18 269.38 388.66

Solid Fuels Combustion and Gasification

Al2O3 (s) C (s) Ca (s)b CaCO3 (s) CaO (s) CaS (s)b CaSO4 (s) CH4 (g) C2H4 (g) C2H6 (g) C2H6O ethanol (g) C3H6 propene (g) C3H8 Propane (g) C4H10 n-butane (g) C5H12 n-pentane (g) C6H6 benzene (g) C6H14 n-hexane (g)

Specific Heat in kJ kmol–1 K–1, and Temperature in K

a b c

0.28448 × 102 0.36299 × 102 0.25310 × 102 0.31096 × 102 0.28166 × 102 0.66581 × 102 0.29805 × 102 0.19393 × 102 0.71520 × 102 0.41807 × 102 0.59267 × 102 0.30519 × 102 0.27599 × 102 0.40332 × 102 0.27883 × 102 0.37741 × 102 0.31119 × 102 0.54295 × 102 0.42129 × 102 0.21282 × 102 0.48453 × 102

0.23633 × 10–2 0.20352 × 10–1 0.82575 × 10–2 0.24898 × 10–1 0.14667 × 10–1 0.22762 × 10–1 0.15288 × 10–1 0.16217 × 10–1 0.73652 × 10–1 0.92938 × 10–2 0.12590 × 100 0.24586 × 10–1 0.64315 × 10–2 0.85968 × 10–2 0.29838 × 10–2 0.19836 × 10–1 0.31088 × 10–3 0.19658 × 10–1 0.12388 × 10–1 0.12803 × 10–1 0.65251 × 10–2

–0.24877 × 10–6 –0.21455 × 10–5 –0.86850 × 10–6 –0.26225 × 10–5 –0.15433 × 10–5 –0.23946 × 10–5 –0.16093 × 10–5 –0.17067 × 10–5 –0.77634 × 10–5 –0.98241 × 10–6 –0.13250 × 10–4 –0.25893 × 10–5 –0.67677 × 10–6 –0.90872 × 10–6 –0.31384 × 10–6 –0.20910 × 10–5 –0.33884 × 10–7 –0.20789 × 10–5 –0.13078 × 10–5 –0.13475 × 10–5 –0.69305 × 10–6

0.42919 × 104 –0.44910 × 106 0.10601 × 106 –0.22130 × 106 0.10023 × 106 0.20325 × 106 –0.55732 × 104 0.63535 × 105 –0.15288 × 107 –0.59270 × 106 0.36325 × 106 –0.18315 × 106 0.32370 × 105 –0.54386 × 106 0.38452 × 105 –0.42985 × 106 –0.16342 × 106 –0.13874 × 107 –0.53787 × 106 0.36941 × 105 –0.89371 × 106

28.010 44.010 2.0158 27.026 18.015 18.015 34.076 24.305 84.314 40.304 120.363 17.030 30.006 46.006 28.013 44.013 31.999 60.084 64.059 65.380 81.379

–110.60 –393.78 0 134.82 –241.98 –286.02 –20.93 0 –1113.7 –601.64 –1262.6 –46.22 90.31 33.49 0 82.06 0 –911.55 –297.10 0 –350.85

–137.24 –394.64 0 124.35 –228.76 –237.40 –33.83 0 –1030.0 –569.32 –1148.3 –16.72 86.64 51.58 0 104.18 0 –857.25 –300.41 0 –320.88

197.68 213.82 130.61 201.85 188.85 70.13 205.82 32.70 65.73 26.96 91.44 192.76 210.72 240.32 191.62 220.02 205.17 41.87 248.23 41.66 43.67

Notes on Thermodynamics

CO (g) CO2 (g) H2 (g) HCN (g) H2O (g) H2O (l)c H2S (g) Mg (s) MgCO3 (s)b MgO (s) MgSO4 (s) NH3 (g) NO (g) NO2 (g) N2 (g) N2O (g) O2 (g) SiO2 (s) SO2 (g) Zn (s) ZnO (s)

Unless otherwise indicated, correlations are valid from 298 to 1500 K (of course, if no decomposition occurs). Interval: 298–1000 K. Interval: 298–500 K.

445

446

Solid Fuels Combustion and Gasification

Z=

Ω v C2f − a v − C2 Ω b (v + C2 )

(B.18)

Ωa R 2 Tc2.5 Pc

(B.19)

 c Ω b RT Pc

(B.20)

C1 =

C2 = Ωa =

1 1/ 3

9(2

Ωb =

− 1)

(B.21)

21/ 3 − 1 3

(B.22)

 RT . P

(B.23)

v 0 =

The Soave form for factor f has been adopted: 1 [1 + (0.480 + 1.574ω − 0.176ω2 )(1 − Tr0.5 )]2 . Tr

f=

(B.24)

Extensive tables with critical constants and Pitzer’s acentric factor (ω) can be found in the literature [10, 11].

B.5

GENERALIZED ZERO-DIMENSIONAL STABLE MODELS

The treatment shown here may be applied to almost any sort of equipment in order to provide a irst zero-dimensional model.

B.5.1

SECOND LAW

The application of Equation 5.4 (Chapter 5) may present dificulties in some situations: for instance, when the state of at least one exiting stream from a CV is not known, nor is the power generated or consumed by the CV. In such situations, the second law of thermodynamics might help. The second law applied to each CV of a process formed by nCV equipment or parts operating at steady state can be written as [1, 2] nSR

∑F

s I

iSR iSR iSR , iCV

iSR =1

+∑ j

j Q  i = 0, 1 ≤ i CV ≤ n CV , +Y CV Tj

(B.25)

where the second term represents heat exchanges with the environment at various discrete points j of the control surface (CS), which is at temperature Tj. Rigorously, that term is given by an integral over that surface, or

447

Notes on Thermodynamics

 δQ . T CS

∫

(B.26)

The symbol δ is used here to indicate the inexact differential. The last term on the left side of Equation B.25 is the rate of irreversibility generated in the CV. Since the rate of irreversibility generation is seldom known, the application of Equation B.25 is useful to establish limiting or ideal conditions. On the other hand, as work is not involved, its application is ideal to set those conditions for systems involving power input or output, such as compressors, pumps, turbines, etc. For example, an ideal turbine could be thought of as working under adiabatic and reversible conditions, as noted below. B.5.1.1 Isenthalpic, Isentropic Ideal representations of equipment or process are very useful and allow veriication of the limiting operational conditions or maximum possible performance. After that, the application of eficiency coeficients permits determination of the performance of real operations. Limiting or ideal conditions require few assumptions. For the above classes of equipment, the usual are: a) Adiabatic operation. Therefore, heat transfer through the CS is 0, or  =0 Q

(B.27)

b) No work is involved or consumed, or  =0 W

(B.28)

c) Kinetic and potential energy are negligible compared with the rates of enthalpy among inputs and outputs, and   u i2SR + gz iSR  I iSR ,i CV = 0, 1 ≤ i CV ≤ n CV F ∑ i SR  i SR =1   2 d) Reversible operations, where  =0 Y nSR

(B.29)

(B.30)

Of course, equipment operations may it only a few of these requirements. For instance, reactors cannot follow the requirement of Equation B.30 when chemical reactions are present. however, it is possible to devise the following two main ideal models, applicable to a considerable group of industrial equipment or systems: a) Isenthalpic model. For this, steady-state operation and the requirements given by Equations B.27 through B.29 are assumed. Therefore, Equation 5.4 (Chapter 5) becomes nSR

∑F

iSR

iSR =1

h iSR I iSR , iCV = 0.

(B.31)

448

Solid Fuels Combustion and Gasification

b) Isentropic model. For this, steady-state operation and the requirements given by Equations B.27 and B.30 are assumed. Therefore, Equation 5.4 (Chapter 5) becomes nSR

∑F

s I

iSR iSR iSR , iCV

= 0.

(B.32)

iSR =1

It is important to stress that isenthalpic here means no overall variation of enthalpy. This differs from the usual concept that classiies isenthalpic process simply as those in which the enthalpy of input and output streams are equal. An example is during throttling of a luid through a valve, which also follows Equation B.31. Clearly, most processes follow those requirements approximately. No process is perfectly adiabatic, and all involve some sort of irreversibility. Thus, the following can eventually be assumed: nSR

∑

i SR =1

FiSR h iSR I iSR ,i CV >>

  u i2SR F + gz iSR  I iSR ,i CV . ∑ i SR  i SR =1   2 nSR

(B.33)

Equipment approaching the isenthalpic model is here called near-isenthalpic, and equipment approaching the isentropic model is called near-isentropic. Among the near-isenthalpic types are combustors, gasiiers, heat exchangers, mixers, valves, separators, distillation columns, etc. Among the near-isentropic types are compressors, pumps, turbines, etc. Therefore, all industrial equipment operating under steady-state (or even nearsteady-state) conditions can be classiied between the two limits: isenthalpic and isentropic. How well equipment approaches those limits can be veriied by the application of eficiencies.

B.5.2

EFFICIENCIES

B.5.2.1 Based on Enthalpy The eficiency for near-isenthalpic equipment can be given by nSR

∑F

iSR

h iSR I iSR , iCV − (1 − ηental ,iCV )H ref = 0.

(B.34)

iSR =1

This may, alternatively, be written on a molar basis. Comments on the meaning of the present eficiency are found in Chapter 5. Apart from those presented in Chapter 5, some examples of commonly assumed eficiencies follow: • Heat exchangers: between 0.96 and 0.98, with Href as the enthalpy variation found in any stream passing through the exchanger times its mass low rate • Mixers and splitters: around 0.99, with Href as the mass low of the entering streams multiplied by their respective mass low rates

449

Notes on Thermodynamics

• Combustors: around 0.97, with Href given by the combustion enthalpy of fuel times the mass low rate of its injection into the equipment • Boilers: between 0.95 and 0.97, with Href deined similarly as for the case of combustor B.5.2.2 Based on Entropy Given the above development, it is opportune to introduce a deinition similar to that in Equation B.34 for near-isentropic process, or nSR

∑F

s I

iSR iSR iSR , iCV

+ (1 − ηentro, iCV )Sref = 0.

(B.35)

iSR =1

Here, the parameter Sref is the reference entropy. As an example for turbines, it could be set as the entropy of the entering gas stream. Despite their generality and easy applicability, the above deinitions are not commonly found in the literature. The usual method requires special deinitions and considerations for each type of equipment. For instance, turbine eficiency is deined by the ratio between the power rate (actually produced by the turbine) and the power that would be obtained from the turbine if operating isentropically. For compressors and pumps, that ratio should be inverted. A generalization is possible by using the following relation:   nSR C * − η F h I =0 ∑ i SR i SR i SR ,i CV ental ,i CV  ∑ Fi SR h i SR I i SR ,i CV   isoentropic  iSR =1 i SR =1 nSR

(B.36)

where the isentropic path could be obtained from nSR

∑F

i SR =1

* i SR i SR i SR ,i CV

s I

= 0.

(B.37)

The asterisk indicates the value computed under isoentropic conditions, i.e., obeying Equation B.37. The exponent C varies for each kind of equipment; for turbines, it is equal to 1, and for compressors and pumps it is equal to –1. The usual values for eficiencies are as follows: • • • •

Compressors, between 0.80 and 0.90 Turbines, between 0.80 and 0.92 Pumps, between 0.60 and 0.90 Nozzles or ejectors, around 0.95

B.5.2.3 Based on Exergy Despite their usefulness for calculations involving turbines, compressors, and pumps, the deinitions of eficiency given by Equations B.35 through B.37 are of little or no use for processes involving combustion or gasiication. However, deinition based on exergy can be very useful, even for those processes. Again, let us conine our deinitions and deductions to systems operating under steady-state regimes.

450

Solid Fuels Combustion and Gasification

Exergy (sometimes called availability) is a thermodynamic property and therefore cannot be exchanged between two systems. Its value may change only because of heat and work exchanges between a system and others around it. If it is a CV, it can also be modiied because of mass injection or exhaustions. For a system, it is deined as the maximum theoretical work obtainable from it, and it is given by the following [1–3, 5]: b = (h − h 0 ) − T0 (s − s0 ) +

u2 + gz. 2

(B.38)*

Here, h, s, u, and z are the average enthalpy, entropy, velocity, and height, respectively, in relation to a reference (usually sea level). The index 0 refers to the conditions found in the ambient atmosphere at temperature (T0) and pressure (P0), called the dead state. Therefore, a system that could reach the dead state would be incapable of providing any useful work. Equation B.38 can also be applied to individual streams. This allows the application for a CV. In this case, exergy is the maximum power output from the CV and can be obtained by the exergy contributions from all entering and leaving streams, or i = W CV ,max

nSR

∑F

iSR

b iSR I iSR , iCV .

(B.39)

iSR =1

In general, for a CV iCV, the following equation is valid: i = W CV

nSR

∑F

i SR =1

i SR

 T    biSR I iSR ,i CV + ∑  1 − 0  Q j − Yi CV . Tj  j 

(B.40)

One should resist the temptation to apply terms such as “balance of exergy” simply because, unlike energy, exergy is a property that is not conserved in a process. The same applies to entropy. As seen, an adiabatic CV (or well-insulated machine, for instance) going through a reversible process would develop the maximum possible power output given by the overall computation of exergy. The deinitions of exergetic eficiencies can be set as the ratio between the actual power output and the maximum possible one. The eficiencies based on exergy are also called second law eficiencies, whereas the ones based on enthalpies are called irst law eficiencies. A list of exergetic eficiency deinitions for the more common cases follows: • Turbines:  W ηexer , iCV =  iCV WiCV ,max

*

(B.41)

If chemical reactions are involved, the value as stated by Equation B.33 should be added to the chemical availability or chemical exergy. More details can be found elsewhere [40].

451

Notes on Thermodynamics

It should be remembered that the irst law eficiency for a turbine is given by the ratio between the actual power output from the turbine and the power output of an equivalent isentropic turbine. • Pumps and compressors: ηexer , iCV =

i W CV ,max i W

(B.42)

CV

• Heat exchangers:

ηexer , iCV

   ∑ FiSR I iSR b iSR  iSR  COLD =−    ∑ FiSR I iSR b iSR   iSR  HOT

(B.43)

Here, the indexes COLD and HOT indicate the cold and hot streams (or legs) entering and leaving the heat exchanger. • Combustors and gasiers: ηexer , iCV = −

(Fb)leaving −stream    ∑ FiSR b iSR   iSR entering −stream

(B.44)

Here, the leaving stream is represented by the gas from the combustor or gasiier. All enthalpies used to compute exergy should include the formation and sensible values. Typical exergy eficiencies for luidized-bed combustors and gasiiers vary from 20% to 30%. • Alternative generalization. Similarly to the eficiencies based on enthalpy and entropy, exergy around the CV could be set by accounting for the eficiency through the following: nSR

∑F

iSR

 CV b iSR I iSR , iCV − (1 − ηexer ,iCV )Bref = W

(B.45)

iSR =1

For instance, in the case of turbines, Bref can be named as the total exergy of entering streams. For compressors and pumps, Bref could be set by the total exergy of the leaving stream. For heat exchangers and mixers, the power involved is usually 0 and Bref is the total entering exergy and variation of exergy of the hot side or leg, respectively. From these, the applications to other classes of equipment can easily be devised. If the mechanical or electrical power input or output is known, Equation B.45 may replace Equations B.34 and B.35. For near-isenthalpic equipment, such as boilers (not including pumping sections), combustors, or gasiiers, this is not a problem because the mechanical power input is 0. In the case of combustors and gasiiers, deinition by Equation B.44 leads to the same values as Equation B.45. Those interested in deeper aspects of exergy may consult the literature [5].

452

B.5.3

Solid Fuels Combustion and Gasification

SUMMARY

The method by which to solve the zero-dimensional stable model for equipment involving work or power input/output in order to obtain, for instance, the composition and temperature of the exiting stream follows the method described in Section 5.5 (Chapter 5) added to Equation B.35 (or B.36) and a value for the eficiency based on entropy. Another route is to use Equation B.45 and the respective value for eficiency.

B.6 HEATING VALUES The heating value has two different deinitions. One is the low heating value (LHV), which is measured by the energy released in the form of heat when 1 mass unit of fuel is burned with a stoichiometric amount of oxygen, with both fuel and oxygen initially at 298 K and 1 bar. The equipment to do so is called a calorimeter; in it, a given mass of fuel (usually dry) is burned in a chamber surrounded by a water jacket. The variation of water temperature is measured during the process. As the amount of water in the jacket is known, the enthalpy variation of the water divided by the mass of the fuel in the chamber gives the LHV. During the process, the hydrogen content of the fuel is transformed to steam. If the steam in the internal chamber is allowed to condense, some extra energy is released to the water in the surrounding jacket. This leads to a further increase in the temperature of water in the jacket. This extra amount of energy is added to the LHV to obtain the high heating value (HHV). Actually, heating values are forms to express combustion enthalpy. The formula for estimating the LHV from the HHV is L HV,d = H HV,d −

1 h vap wH . 2

(B.46)

Here, wH is the mass fraction of hydrogen (dry basis) in the fuel (from proximate analysis), and hvap is the enthalpy of vaporization at 101.325 kPa, or 2.258 × 106 J/kg. In the absence of speciic information, the following formulas (see Chapter 8, Table 8.2, for notation) allow estimation of the high heating values (in J/kg) [12]: • For coals: • If wO,daf < 1.1 × 10 –3 H HV ,d = 2.326 × 10 5 (144.5w C,daf + 610.0 w H ,daf − 62.5w O ,daf + 40.5wS,daf )(1 − wash ,d )

(B.46a)

• If wO,daf ≥ 1.1 × 10 –3 H HV ,d = 2.326 × 10 5 (144.5w C,daf + 610.0 w H ,daf − (65.9 − 31.0 w O ,daf )w O ,daf + 40.5wS,daf )(1 − wash ,d )

(B.46b)

453

Notes on Thermodynamics

• For wood or other biomasses: H HV ,d = 4.184 × 10 5 (81.848w C,d + 263.38w H ,d − 28.645(w O ,d + w N ,d ) − 3.658wash ,d + 0.16371)

B.7

(B.46c)

REPRESENTATIVE FORMATION ENTHALPY OF A SOLID FUEL

A method by which to calculate a representative formation enthalpy of solid fuel is introduced here. It may facilitate computations in combustion and gasiication because the fuel can be treated as if it is a pure substance in overall energy balances. If pure carbon (graphite) is inserted into a calorimeter to determine its combustion enthalpy or heating value, the following total combustion would occur: C + O2 → CO2 . If the reactants and products are at 298 K, the released heat would be the deinition of heating value as well, and it would be given by L HV,C =

(

)

1  h f ,C + h f ,O2 − h f ,CO2 . MC

(B.47)

Of course, formation enthalpies of carbon and oxygen are 0. Also, as no water is produced from that reaction, the high heating value is equal to the low heating value. The same experience for a general solid fuel, under the complete combustion, or (see Chapter 8, Tables 8.1 and 8.2, for notation) a a   a CH a531 Oa551 N a546 Sa563 A a824 +  1 + 531 − 551 + 546 + a 563  O2 → CO2 4 2 2   +

a 531 H 2O + a 546 NO + a 563SO2 + a 824 A , 2

(B.48)

leads to a similar relation, or L HV,d , fuel =

1 M fuel

    h f ,fuel − ∑ ν jh f,j  .   j= prod

(B.49)

In combustion and gasiication processes, it is more convenient to work with the low heating value because water is usually kept as vapor. It should be noted that LHV is determined using dried fuel, or on a dry basis. However, ash should be present. Equation B.49 serves as a deinition of a representative formation enthalpy for the fuel, which in mass basis is given by h f, fuel = L HV ,d ,fuel +

1 M fuel

824

∑ ν h

j f,j

j= 514

.

(B.50)

454

Solid Fuels Combustion and Gasification

The stoichiometric coeficients of products in the last term of B.50 are those on the left side of Reaction B.48, including ash. The representative molecular mass of the fuel is simply given by the coeficients of components in its formula: M fuel =

824

∑aM. j

j

(B.51)

j = 514

Equations B.49 and B.50 are consistent in any situation. For instance, for the case of pure carbon (or graphite), they lead to a fuel formation enthalpy equal to 0. Even in the case of pure ash (or exhausted fuel and modeled here by SiO2), the heating value is 0 as well on the right side of B.49. The coeficients aj in the fuel representative formula are easily determined from the ultimate analysis (wj, dry basis), as aj =

wd, j M 514 , wd,514 M j

j = 514, 531, 546, 551, 563, 824.

(B.52)

REFERENCES 1. Moran, M.J., and Shapiro, H.N., Fundamentals of Thermodynamics, 3rd Ed., John Wiley, New York, 1996. 2. van Wylen, G.J., and Sonntag R.E., Fundamentals of Classical Thermodynamics, John Wiley, New York, 1973. 3. Kestin J., A Course in Thermodynamics, Vols. I and II, Hemisphere, New York, 1979. 4. Perry, J.H., Green, D.W., and Maloney, J.O., in Perry’s Chemical Engineers Handbook, 7th Ed., Perry, J.H., Ed., McGraw-Hill, New York, 1997, 12–1 to 12–90. 5. Szargut, J., Morris, D.R., Steward, F.R., Exergy Analysis of Thermal, Chemical and Metallurgical Processes, Hemisphere, New York, 1988. 6. Beaton, C.F., and Hewitt, G.F., Physical Property Data for the Design Engineer, Hemisphere, New York, 1989. 7. Alemasov, V.E., Dregalin, A.F., Tishin, A.P., and Khudyakov, V.A., Computation methods, in Thermodynamic and Thermophysical Properties of Combustion Products, Glushko, V.P., Ed., Vol. I, Viniti, Moscow, 1971. 8. Yaws, C.L., Hood, L.D., Gorin, C., Thakore, S., and Miller, J.W., Jr., Correlation constants for chemical compounds, Chemical Engineering, 16, 79–87, 1976. 9. Yaws, C.L., Schorr, G.R., Shah, P.N., and Miller, J.W., Jr., Correlation constants for chemical compounds, Chemical Engineering, 22, 153–161, 1976. 10. Poling, B.E., Prausnitz, J.M., and O’Connell, J.P., The Properties of Gases and Liquids, 5th Ed., McGraw-Hill, New York, 2000. 11. Reid, C.R., Prausnitz, J.M., and Poling, B.E., The Properties of Gases and Liquids, 4th Ed., McGraw-Hill, New York, 1987. 12. Ergun S., Coal classiication and characterization, in Coal Conversion Technology, Wen, C.Y., and Lee. E.S., Eds., Addison-Wesley, Reading, MA, 1979, pp. 1–53.

Improvements C Possible on Modeling Heterogeneous Reactions CONTENTS C.1 General Mass Balance for a Particle ............................................................ 456 C.2 Generalized Energy Balance for a Particle .................................................. 456 C.2.1 Example of an Application for Unexposed-Core Particle ................ 458 C.2.1.1 Mass Balance ..................................................................... 458 C.2.1.2 Energy Balance ..................................................................460 Reference ............................................................................................................... 461 The solutions for the concentration proiles of a single component j within a particle are presented in Chapter 9. There, it was assumed that each individual reaction responsible for the consumption of component j inside the particle did not suffer interference from other simultaneous reactions taking place in the same physical space. The effect of various simultaneous reactions upon the concentration proile of component j was obtained by the simple summation of reaction rates ri involving j (Equation 9.2). The other approximation was to assume an isothermal particle. Although that approach is used by almost all models in the area of gas–solid process, it may lead to important deviations. The classic criteria to evaluate whether the isothermal hypothesis is reasonable or not uses the Biot number, given by N Bi =

αVp A pλ p

(C.1)

where α is the convective heat transfer coeficient at the particle surface. If NBi is smaller than 0.1, the hypothesis of a near-isothermal particle is reasonable. Considering a typical coal moving-bed gasiier, the following values could be found: • Thermal conductivity around 0.26 W m–1 K–1 • Convective heat transfer from 10 to 100 W m–2 K–1 455

456

Solid Fuels Combustion and Gasification

Assuming particles with an almost spherical shape, the isothermal hypothesis would be reasonable for diameters below 1.5 cm. For charcoal, this number could be something around 0.7 cm. On the other hand, it is not uncommon to ind cases of moving-bed reactors operating with particle dimensions above those critical values. Therefore, sizable variations of temperature within such particles are expected. In addition, as the reaction rates strongly depend on temperature, one should anticipate more or less serious deviations from reality from models assuming isothermal particles. This appendix presents tentative solutions for the proiles of a component j inside a reacting particle where several simultaneous reactions take place. In addition, the problem of nonisothermal particles is going to be addressed as well. This might shed some light on the validity, or lack thereof, of the isothermal particle assumption. These considerations may be useful for further improvement in the models presented in the main text.

C.1

GENERAL MASS BALANCE FOR A PARTICLE

It has been seen that Equation 9.6 (Chapter 9) represents the continuity of species. If M simultaneous reactions are involved in the process, the general balance for the species j is written as M

∇ 2 y j + ∑ ν ijΦ 2ijy nj = 0.

(C.2)

i =1

Here, is also assumed that the n-order reactions can be written in the form shown by Equation 9.3. The Thiele modulus is still given by Equation 9.8. Now assume the following notation: • ν+ij = νij when νij ≥ 0, or j produced by reaction i. • νij− = νij when νij ≤ 0, or j consumed by reaction i. Therefore, Equation C.2 can be written as ∇ 2 y j = − ∑ ( ν+ij − νij− ) Φ ij2 y nj .

(C.3)

i

C.2

GENERALIZED ENERGY BALANCE FOR A PARTICLE

From the general energy balance for reacting system [1], it is possible to write    (C.4) ∇ • q = ∑ h j (∇ • jj ) − R j  .   j The heat transfer lux vector is given by  q = − λ ∇T

(C.5)

Possible Improvements on Modeling Heterogeneous Reactions

457

and the mass lux vector by    j = −ρ D j∇x j + x j N  m 1 − Mm ∑ j  M av m 

 . 

(C.6)

Here, the diffusivity coeficient should be understood as the diffusivity of component j in the medium present at the local position. In porous media, it is given by the effective diffusivity, as seen in Chapters 9 and 11. Let it be assumed that for cases of combustion and gasiication processes, the individual molecular mass could be approximated to the average in the gas mixture. Also, for a condition where the global molar concentration almost constant, Equation C.6 becomes  j = −D j∇ρ j. (C.7) j Then, Equation C.4 provides λ∇ 2 T = ∑ h jD j∇ 2ρ j − ∑ h j ∑ v ij k i (ρ j − ρ j,eq )n . j

j

(C.8)

i

Using the following dimensionless variable: θ=

T − T∞ . T∞

(C.9)

Equation C.8 can be written as ∇ 2 θ = ∑ Ψ 2j ∇ 2 y j − ∑ ∑ ( ν+ij + νij− ) Ω2ij y nj j

j

(C.10)

i

where:  h D  Ψ j =  j j (ρ j − ρ j,eq )    λT∞

1/ 2

(C.11)

and Ω2ij =

h jrA2 k i D h (ρ − ρ j,eq ) 2 Φ ij. (ρ j − ρ j,eq )n = j j j λT∞ λT∞

(C.12)

At this point, some important conclusions can be drawn: • The temperature proile of a reacting particle is inluenced by two basic terms: the energy transfer due to the mass (with respective enthalpy) transported by diffusion—which is represented by the irst term on the right side of Equation C.10—and an energy transfer due to the enthalpy modiication due to the reactions—which is represented by the second term on the right side of Equation C.10. • As expected, highly conductive particles lead to smoother temperature proiles. This can be seen by looking at the dimensionless coeficients Ψ and Ω

458

Solid Fuels Combustion and Gasification

(Equations C.11 and C.12). Depending on the magnitude, this could justify the hypothesis of an isothermal particle. • As anticipated, the problem deviates from the isothermal approximation for larger particles. That is due to the inluence of rA in the dimensionless parameter Ω. • The problem also runs away from the isothermal approximation for faster exothermic and endothermic reactions far from equilibrium, as veriiable by the combined inluence of coeficients kj, the difference (ρ j − ρ j,eq), and the enthalpy (h j). It should not be forgotten that the resulting inluence is given by the summation of the contributions from various reactions. In fact, for regions of the reactor where fast exothermic reactions surpass the endothermic (or vice versa), it is possible to neglect the irst right-side term of Equation C.10 compared with the second. On the other hand, if the exothermic and endothermic reactions balance each other out, leading to an almost zero overall energy release, the hypothesis of an isothermal particle is valid.

C.2.1

EXAMPLE OF AN APPLICATION FOR UNEXPOSED-CORE PARTICLE

Let us take as an example a particle in slab form (Figure C.1). Here, the physicalchemical properties are assumed to be constants, and only irst-order reactions are present. C.2.1.1 Mass Balance For the shell, where no reaction occurs, Equation C.2 becomes d2y j = 0, a < x ≤ 1 dx 2

(C.13)

where the dimensionless variables x and a are given by Equations 9.4 and 9.11 (Chapter 9), respectively. The solution of the above equation is y j = a1, jx + a 2, j , a < x ≤ 1.

(C.14) Shell

rA

r rN

Nucleus

FIGURE C.1

Scheme of a lat particle.

Possible Improvements on Modeling Heterogeneous Reactions

459

For the core, the solution is y j = ∑  A −j sinh(xΦ ij ν −ij ) + B−j cosh(xΦ ij ν −ij )  i + A +j sin(xΦ ij ν ij+ ) + B+j cos(xΦ ij ν +ij )  , 0 ≤ x < a. 

(C.15)

Since at the center of the particle (x = 0) the total mass transfer is zero, the derivatives of concentration are also zero at that point. The reader can verify that the solution acquires the form y j = ∑  B−j cosh(xΦ ij ν −ij ) + B+j cos(xΦ ij ν +ij ) , 0 ≤ x < a.   i

(C.16)

The coeficients a1,j, a2,j, B−j , and B+j can be obtained by the following interface conditions D jN

dy j dx

= D jA x→a −

dy j dx

.

(C.17)

x→a +

That is valid at the interface between the nucleus and the layer of spent material or shell, and dy j dx

= x → l−

D jG N Sh, j 1 − y j (1)  D jA

(C.18)

where N Sh, j =

β jG rA D jG

(C.19)

is valid at particle surface. Moreover, the concentration values at the core-shell interface should be at equilibrium, or y j (x → a − ) = γ jcore−shell y j (x → a + ).

(C.20)

The equilibrium parameter γ is obtained from experimental determinations and should be a function of temperature, pressure, and concentration itself. The same is valid for the gas-shell interface: y j (x → 1− ) = γ jgas−shell y j (x → 1+ ).

(C.21)

Therefore, the four boundary conditions above provide the means to determine the four constants. It should be noted that the treatment using equilibrium correlation is more rigorous than the approximation used in Chapter 9, where the equality of the concentration at the core-shell interface was assumed.

460

Solid Fuels Combustion and Gasification

Finally, it should be observed that the coeficients B−j and B+j need to be positive, and therefore, from Equation C.16: • If j is being consumed, the concentration proile is a hyperbolic cosine. This would provide a decreasing concentration of j toward the particle center. • If j is being produced, the concentration proile is a cosine, and the concentration increases from the particle center toward its surface. C.2.1.2 Energy Balance It is possible to obtain the temperature proile in the regions where fast dominant exothermic (or endothermic) reactions take place. In this case, the energy balance becomes d 2θ = − ∑ ∑ ( ν+ij − νij− ) Ωij2 y j . dx 2 j i

(C.22)

For the shell region, the solution given by Equation C.14 can be used, and the above equation leads to   x3 x2 θ = − ∑ ∑ ( νij+ − νij− ) Ωij2  a 1, j + a 2, j + a 3 x + a 4  , a < x ≤ 1. 6 2   j i

(C.23)

For the nucleus, the solution C.16 is substituted into C.22, leading to  B−j dθ = − ∑ ∑ ( νij+ − νij− ) Ωij2  ∑ sinh xΦ ij νij− dx  j Φ ij νij− j i 

(

+

 sin xΦ ij νij+ + a 5  , 0 ≤ x < a.  νij+ 

(

B+j Φ ij

)

) (C.24)

As the temperature proile is similar at the center of the slab (x = 0), the constant a5 is 0. The second integration leads to

(

 Λ 2j B−j θ = − ∑ ∑ ( ν+ij − νij− )  ∑ − cosh xΦ ij νij− j i  j νij −

Λ 2j B+j ν

+ ij

(

)

 cos xΦ ij νij+ + a 6  , 0 ≤ x < a. 

) (C.25)

Here, a new dimensionless parameter appears and is readily given from Equation C.12 as Λ 2j =

Ω2ij D jh j (ρ j − ρ j,eq ) . = λT∞ Φ 2ij

(C.26)

Possible Improvements on Modeling Heterogeneous Reactions

461

It should be noted that Λj does not depend on the particular reaction i. This parameter is very useful for the analysis on the inluence of various properties and conditions regarding the temperature proile. The constants a3, a4, and a6 can be obtained from the following: • The equality of temperature obtained by Equations C.23 and C.25 at the core-shell interface (x = a), or θ(x → a − ) = θ(x → a + )

(C.27)

• The equality of heat lux at that interface, or λN

dθ dθ = λA dx x →a − dx x →a +

(C.28)

• The heat transfer at the surface of the particle dθ λ = G N Nu [1 − θ(1)] dx x → l− λ A

(C.29)

where N Nu =

α SG rA λG

(C.30)

REFERENCE 1. Bird, R.B., Stewart, W.E., and Lightfoot, E.N., Transport Phenomena, John Wiley, New York, 1960.

on D Improvements Various Aspects CONTENTS D.1 Rate of Particle Circulation in the Fluidized Bed ........................................ 463 D.2 Improvements in the Simulator for Fluidized-Bed Equipment ....................465 References ..............................................................................................................466 Here, a few possible or actually implemented improvements to the present simulation program for luidized-bed equipment are described.

D.1

RATE OF PARTICLE CIRCULATION IN THE FLUIDIZED BED

A procedure that generalizes Soo’s work [1–3] and allows the computations of circulation rates of individual particle species in a multispecies luidized bed is proposed here. The set of equations for this generalization is described below. • Continuity for the gas phase: 1 ∂(ru r ) ∂u z + =0 r ∂r ∂z

(D.1)

The reader should note that both axial (uz ) and radial (ur) velocities are functions of the radial and axial directions. • Continuity of solid species m: 1 ∂(ru m , r ) ∂u m ,z + =0 r ∂r ∂z

(D.2)

• Momentum (z-component) for the solid species m: α m (u z − u m ,z ) +

µm ρm

 1 ∂ ∂u m ,z ∂ 2 u m ,z  r +   ∂z 2   r ∂r ∂r

M

− g + ∑ α m ,1 (u1,z − u m ,z ) = 0

(D.3)

1=1

• Averaged momentum of the mixture (gas and solid phases): M ∂P + u 0,z ∑ fm [ρm α m + (ρm + ρ)g] = 0 ∂z m =1

(D.4)

463

464

Solid Fuels Combustion and Gasification

The boundary conditions are as follows: • At z = 0 (base of the bed), 0 ≤ r ≤ R: u = u0,z , um,z= 0, and P = P0. • At z = zD (top of the bed), 0 ≤ r ≤ R: um,z= 0 (no particle escape from the bed). Therefore, this is a strong approximation. ∂u • At r = 0 (center line of the bed), 0 ≤ z ≤ zD: ur = 0, um,r = 0, and m ,z = 0. ∂r • At r = R (wall), 0 ≤ z ≤ zD: ur = 0, um,r = 0, uz = 0, and um,z = um,z(R,z). The axial or vertical velocity at the wall is assumed to be a “slip” velocity, which can be computed for a semiempirical correlation. Initially, the following generalized series solutions for the velocity ields are assigned:  πz  r u z = u 0 ,z + sin   ∑ Cn    z D  n =1  R  ur = −

n −1

 πz  πR C  r cos   ∑ n   zD  z D  n =1 n + 1  R 

 πz   r u m ,z = sin   ∑ a m ,n    R  z D  n =1

(D.5) n

(D.6)

n −1

(D.7) n

u m ,r

a  r  πz  πR =− cos   ∑ m ,n   . zD  z D  n =1 n + 1  R 

(D.8)

Similarly to the approach used before [1, 2], the coeficients Cn and am,n are determined through the application of the solutions of Equations D.5 through D.8 into Equations D.1 through D.4 and their respective boundary conditions. These forms are very convenient because they automatically satisfy the continuity equations. They also satisfy most of the boundary conditions, which are also generalized from the work of Soo [2]. However, the above series are not completely consistent with the momentum equations, and only an approximate solution is possible. The complete solution of the above equations is not within the scope of the present text. However, some hints or suggestions for possible improvements are noted here. For instance, the momentum equations used by Soo did not include the convective terms. The simpliication has probably been the cause of some poor comparisons with experimental values for various conditions. To improve the generalization, the inclusion of convective terms in the momentum equations can be tried as well. In order to apply the proposed series of solutions to the new system and to minimize deviations from the exact solution of this boundary value problem, the weighted residual method—in particular, the method of orthogonal collocation [4–6]—might be used. The number of collocation points can be varied to seek a compromise between precision and computational times. In both cases, the computational algorithm should be developed.

465

Improvements on Various Aspects

An averaged rate of circulation of particles at each axial position (z) would be provided by the integral average velocity ields in the radial direction, or G H,m = ρm (u m+ ,z )av =

2ρm B2m

Bm +1

∫

u m+ ,z r dr.

(D.9)

Bm

Here, Bm and Bm+1 are the roots of the um,z function in the radial direction at each height z in the bed. The integral should be computed only for the regions where um,z is positive (which have been indicated by the plus sign) or in the upward direction. The integration for regions of negative um,z would lead to the same value for the average circulation because of the hypothesis that um,z(zD,r) is null, or no particle escape from the bed. Therefore, the circulation rates would be computed at close intervals throughout the bed height. This would allow approximate uncoupling from the energy and mass balances, which include all transfers, production, and consumption due to chemical reactions, as found in furnaces and gasiiers. Until now, no experimental determination of circulation rates of individual different species of particles could be found in the literature. Only this would allow conirmation of the above solutions. However, overall checks of the model and comparisons with experimental data or semiempirical correlation can be made using the total average of the circulation. Future experimental research may ill this gap. In addition, it must be remembered that the circulation rate of each species has a signiicant inluence on various parameters—for instance, individual and average temperature proiles.

D.2

IMPROVEMENTS IN THE SIMULATOR FOR FLUIDIZED-BED EQUIPMENT

In the years since the release of its irst version [7, 8], the Comprehensive Simulator of Fluidized and Moving Beds (CSFMB*; previously known as CSFB) has evolved. The latest versions [8–11] have received a great number of improvements, among the most important of which are the following: • Application of the functional group model (see Chapter 10) for calculations of fuel devolatilization, and computation of total tar release using the DISKIN model. • Introduction of rigorous overall balance based on formation enthalpy of fuel (see Appendix B). • Inclusion of more data regarding kinetics of heterogeneous reactions for a wide range of solid fuels. • Extensive improvements on calculations of physical and chemical properties of each chemical species as well mixtures. A complete data bank that can also compute deviations from ideality was developed. It is capable of computing properties at a wide range of temperatures and pressures. • General improvements on computations of equilibrium of reactions using Gibbs formation enthalpies and the above-mentioned data bank. *

See http://www.csfmb.com.

466

Solid Fuels Combustion and Gasification

• • • •

Application to updraft and downdraft moving-bed reactors. Application to circulating luidized-bed equipment. Possibility of simulating luidized-bed reactors consuming liquid fuels. Simulation of large boilers where several tube banks may be immersed in the bed or freeboard. Those banks could be linked or not. • Allowing simulation of units with external heating, such as is provided by electrical blankets around the equipment. • Introduction of routines to compute several interesting engineering design parameters. Among them are the following: • Ability to account for several intermediate gas injections at any point of the bed and freeboard. The injected gas can have any desired composition. • Erosion rate of tubes immersed in the bed. • Heat transfers to the jacket surrounding the bed or freeboard. The jacket can operate with water or any mixture of gases, and those can be diverted for use elsewhere or reinjected into any point in the bed or freeboard. • Circulation rates of individual solids at each point of the bed. • Ability to simulate processes in which fuel or any other solid feeding is made using slurries (mixtures with liquid water). This method is being used in large pressurized boilers and gasiiers to avoid the usual problems with solid feeding against a pressurized environment.

REFERENCES 1. Soo. S.L., Note on motions of phases in a luidized bed, Powder Technology, 45, 169– 172, 1986. 2. Soo, S.L., Average circulatory motion of particles in luidized beds, Powder Technology, 57, 107–117, 1989. 3. Soo, S.L., Particulates and Continuum, Multiphase Fluid Dynamics, Hemisphere, New York, 1989. 4. Finlayson, B.A., The Method of Weighted Residuals and Variational Principles, Academic Press, New York, 1972. 5. Villadsen, J., and Michelsen, M.L., Solution of Differential Equation Models by Polynomial Approximations, Prentice-Hall, New Jersey, 1978. 6. de Souza-Santos, M. L., Analytical and Approximate Methods Applied to Transport Phenomena, CRC Press, New York, 2007. 7. de Souza-Santos, M.L., Modelling and Simulation of Fluidized-Bed Boilers and Gasiiers for Carbonaceous Solids, PhD thesis, University of Shefield, Shefield, United Kingdom, 1987. 8. de Souza-Santos, M.L., Comprehensive modelling and simulation of luidized-bed boilers and gasiiers, Fuel, 68, 1507–1521, 1989. 9. de Souza-Santos, M.L., A new version of CSFB, Comprehensive Simulator for Fluidized Bed Equipment, Fuel, 86(12–13), 1684–1709, 2007. 10. de Souza-Santos, M.L., Comprehensive simulator (CSFMB) applied to circulating luidized bed boilers and gasiiers, The Open Chemical Engineering Journal, 2, 106–118, 2008. 11. de Souza-Santos, M.L., CSFB applied to luidized-bed gasiication of special fuels, Fuel, 88, 5, 826–833, 2009.

on Turbulent E Basics Flow CONTENTS E.1 Momentum Transfer ..................................................................................... 467 E.2 Heat and Mass Transfers............................................................................... 470 E.3 Reaction Kinetics .......................................................................................... 470 References .............................................................................................................. 472 The basic aspects of turbulent low are shown here as an aid for the discussions that appear throughout the book.

E.1

MOMENTUM TRANSFER

As mentioned in Chapter 6, important luctuations in the velocity ield take place in turbulent lows. The usual treatment is to represent the velocity by a time-averaged value added to luctuations. For instance, for the component in x direction,  u x = ux + u x . (E.1) Here, the average velocity in a suficiently large time interval Δt would be given by ux =

1 ∆t

t + ∆t

∫

u x dt.

(E.2)

t

The time interval should be such that the value of parameter a, computed by 1 ∆t

t + ∆t

∫

 u x dt = a,

(E.3)

t

is 0 or very near 0. To illustrate the nature of turbulence, take the case of turbulent low in a pipe. Typically, three main regions can be distinguished: • A central layer of a fully developed turbulent regime • A layer very close to the tube wall, where a laminar regime is found. • A buffer zone intermediate between the above two From this, one can conclude that lows between narrow channels (as found in packed beds) can be treated—at least as a good approximation—as laminar, even if the overall low would be turbulent for an equivalent empty bed. 467

468

Solid Fuels Combustion and Gasification

When the forms given by Equation E.1 are substituted into the momentum equation, or    Du = −∇P − ∇ • τ + ρg, ρ (E.4) Dt the following is obtained [1]:     Du = −∇ P − ∇ • τ1am + ∇ • τtur + ρg. ρ Dt

(E.5)

As we have seen, a term representing the shear stress due to turbulent low is added to the former laminar representation in Equation E.4. The basic difference among all treatments of turbulent low rests on how the viscosity is related to turbulent shear stress, or τtur , xy = −µ tur , xy

∂ux . ∂y

(E.6)

Here, the tensor component xy was chosen just to illustrate the various forms. These are called closure relations or closure strategies. A few examples follow [1–4]: 1. Boussinesq’s approach. His work dates from 1877 and assumes a similar form for Newtonian stress, i.e., taking the viscosity as a value that could be experimentally determined as a constant. However, the turbulent or eddy viscosity is not a property of the luid but rather a function involving all conditions, including the position inside the low. 2. Prandtl’s approach. A similarity between the movement of molecules and the movement of pockets of luid (eddies) is assumed. The analogy led Prandtl to imagine “mixing lengths” [l] similar to the free-path of molecules, and to write µ tur , xy = −ρl 2

∂ux . ∂y

(E.7)

The problem with this formulation is that the mixing lengths are function of position as well. However, some success was achieved by describing the length as a linear function of the distance (y) from the wall, or 1 = C1y

(E.8)

where C1 is a constant. 3. von Kármán’s approach. He suggested the following form: µ tur , xy = −ρC22

(∂ux / ∂y)3 . (∂ 2 ux / ∂y 2 )2

(E.9)

In the original work, the parameter C2 was assumed to be a constant equal to 0.4. Later works found that it would be better represented by 0.36.

469

Basics on Turbulent Flow

4. Deissler’s approach. He proposed the following:   C2 u y   µ tur , xy = −ρC32 ux y 1 − exp  − 3 x   . ρµ    

(E.10)

The constant C3 was determined to be 0.124. 5. Prandtl-Kolmogorov’s approach. The following is assumed: µ tur ,xy = ρC4

k 2t εt

(E.11)

where kt is a speciic kinetic energy, and εt is the dissipation rate of that energy. For instance, the kinetic energy (in the x direction) is given by  u2 (E.12) kt = x . 2 Note that the time average is performed over the square of luctuations of velocity. Actually, the evaluation of kt is usually made through the following equation (in the x direction): ∂u ∂k t ∂k ∂  ρµ ∂k t  − ux uy x − ε t + ux t = ∂y ∂t ∂x ∂x  C5 ∂x 

(E.13)

and the dissipation of kinetics energy is given as ε t = C6

k 3t / 2 l2

(E.14)

where C5 and C6 are (usually) empirically determined constants, and l2 is a dissipation length, which should not be mistaken for the one indicated in Equation E.7. This approach brings some inconvenience because the dissipation length is as dificult to estimate as the Prandtl mixing value. 6. The k-ε approach. The attack is made by providing an equation to describe the kinetic energy dissipation rate, or 2

ε  ε  ∂u ∂ε t ∂ε ∂  ρµ ∂ε t  − C 7  t  u x u y x − C8  t  . + ux t =   ∂y ∂t ∂x ∂x  C5 ∂x   kt   kt 

(E.15)

As this model requires the solution of equations E.13 and E.15, it is also classiied as a two-equation model. This is the most commonly used approach, including in cases of combustion using pneumatic transport techniques. However, it does not represent well several such situations, particularly those with fast, swirling lows. The solution of Equations E.13 and E.15 requires speciications of boundary conditions. This is one of the awkward aspects because, rigorously speaking, each case requires experimental determination of turbulence intensity at injection positions. Estimations are possible [4].

470

E.2

Solid Fuels Combustion and Gasification

HEAT AND MASS TRANSFERS

As we have seen, turbulence affects the velocity ield unevenly and at a molecular level. Therefore, it is easy to imagine that it also affects the rate of mixing of, for instance, two reacting gases. Similarly, turbulence affects the rate of heat transfer within the lowing gas or liquid. The analytical treatment for effects of turbulence on heat and mass transfer follows principles similar to those for momentum transfer. There, the affected parameter is viscosity, which is the dissipative function for velocity distribution. As the dissipative term is the one involving the second derivatives of the main variable on directions, that part is played by thermal conductivity in heat transfers and by diffusivity in mass transfers. Looking at Equations E.5 and E.6, it is easy to verify that Newtonian representation of total shear stress—in a given direction (x)—can be written as τxy = −(µ + µ tur ,xy )

∂u x ∂y

(E.16)

where the irst viscosity is that for laminar lows. Therefore, it is possible [5] to write the Fourier’s law for turbulent low in the same direction as q x = −(λ + λ tur )

∂T . ∂x

(E.17)

For mass transfer, Fick’s law for turbulent low would become jj, x = −(D jk + D jk , tur )

∂ρ j . ∂x

(E.18)

Here, the mass transfer indexes j and k refer to the chemical components involved. Note that isotropy is assumed for thermal conductivity and diffusivity. These aspects are responsible for additional complications in modeling and simulation of combustion or gasiication in suspensions. On the other hand, heat and mass luxes are vectors, whereas shear stress is a tensor. This allows much simpler treatments for heat and mass compared with momentum. That is why it is possible to develop relatively simple empirical and semiempirical correlations representing convective heat and mass transfer coeficients well. These correlations are found in various texts [1–3, 5–8] on the subject.

E.3

REACTION KINETICS

The presence of turbulence also affects the kinetics of reactions. Suppose, for instance, two reacting gas–gas chemical species. Like  the luctuations of velocity,  luctuations of concentration (ρ j) and temperature (T) also occur. These will affect the computations of reaction rate. Just for an example, take the reaction i, assumed to be irreversible and following the rate in the classical form, or from Equation 5.15 (Chapter 5),

471

Basics on Turbulent Flow

ri =

1 dρ j ⇒ m = nCO − νm ,i = k i ∏ ρ m . ν j, i dt m =1

(E.19)

In addition, if the kinetic coeficient follows the Arrhenius form, the resulting relation would be 

ri =

E m = n CO − i 1 dρ j  ρ m− νm ,i . = b ie RT ∏ ν j,i dt m =1

(E.20)

The luctuations can be written in a fashion similar to that in Equation E.1, or  T = T+T

(E.21)

 ρ j = ρ j + ρ j.

(E.22)

and

The averages could be computed in forms similar to those in Equation E.2. Therefore, it becomes obvious that substitution of those into Equation E.20 would not maintain that form, or ri ≠ bi e

E m = n CO −i RT

∏ ρ

− νm , i m

.

(E.23)

m =1

As an example, an originally Arrhenius form for the kinetic coeficient can be given [9] by the following series:  T k = k 0 exp  − e  T

 Te   2   Te   Te2  1 + T 2 T +  2T 4 − T 3  T +      

(E.24)

where E Te =  . R

(E.25)

However, some situations of turbulence allow the form of Equation E.20 to be kept at least as a reasonable approximation, for instance, when the reaction times are much greater than the turbulent time scale, which is given by the time taken to mix (because of turbulence) two reacting components at the molecular level. Therefore, if the reaction is slow enough, the mixing period can be ignored. However, for very fast reactions, the mixing time might be the limiting factor. In these situations, the kinetics should include the turbulence effect. The realistic treatment of turbulent combustion, or of most industrial lames, should include chain mechanisms (see Chapter 8). Simulation of lames is beyond the scope of this book, but the interested reader can ind plenty of material in the literature [4, 10–13].

472

Solid Fuels Combustion and Gasification

REFERENCES 1. Bird, R.B., Stewart, W.E., and Lightfoot, E.N., Transport Phenomena, John Wiley, New York, 1960. 2. Slattery, J.C., Momentum, Energy, and Mass Transfer in Continua, Robert E. Krieger, New York, 1978. 3. Brodkey, R.S., The Phenomena of Fluid Motions, Dover, New York, 1967. 4. Smoot, L.D., and Smith, P.J., Coal Combustion and Gasiication, Plenum Press, New York, 1985. 5. Luikov, A.V., Heat and Mass Transfer, Mir, Moscow, 1980. 6. Incropera, F.P., and DeWitt, D.P., Fundamentals of Heat and Mass Transfer, 4th Ed. John Wiley, New York, 1996. 7. Schmidt, F.W., Henderson, R.E., and Wolgemuth, C.H., Introduction to Thermal Sciences, 2nd Ed., John Wiley, New York, 1984. 8. Rohsenow, W.M., and Hartnett, J.P., Handbook of Heat Transfer, McGraw-Hill, New York, 1973. 9. Libby, P.A., and Williams, F.A., Turbulent Reacting Flows, Academic Press, New York, 1994. 10. Smoot, L.D., and Pratt, D.T., Pulverized-Coal Combustion and Gasiication, Plenum Press, New York, 1979. 11. Glassman, I., Combustion, 3rd ed., Academic Press, San Diego, CA, 1996. 12. Warnatz, J., Maas, U., and Dibble, R.W., Combustion, Springer, Berlin, Germany, 1999. 13. Kuo, K.K., Principles of Combustion, John Wiley, New York, 1986.

F

Classification of Modeling for Bubbling Fluidized-Bed Equipment

CONTENTS F.1 Main Aspects ................................................................................................ 473 References .............................................................................................................. 474 Comments on various levels and aspects of modeling and simulation of equipment operating under bubbling luidized beds are presented here. Several characteristics of published models have already been discussed throughout the main text, and this work does not intend to present a thorough study. The comments below are aimed only at guiding those beginning in this ield and orienting future developments.

F.1

MAIN ASPECTS

Several works on the modeling of bubbling luidized-bed equipment have been published. Among them, one inds many with a more comprehensive approach [1–54]. Speciic aspects of a few of those are discussed in Chapters 14 and 15. As seen in Chapters 3 and 13 through 18, luidized-bed combustors and gasiiers involve a respectable number of aspects. Each of them can be attacked at several levels, which does not necessarily imply any particular degree of dificulty. Among the most important, with their respective levels, are the following: a) Phases in the bed 1. One-phase: no distinction between bubbles and emulsion 2. Two-phase: bubble and emulsion 3. Three-phase: cloud, bubble, and emulsion b) Gas low regime 1. Zero-dimensional in compartments or total 2. One-dimensional or plug-low for gas and bubble 3. Two-dimensional c) Solid low regime 1. Well-stirred 2. Well-stirred in compartments 3. Combination of well-stirred and poorly stirred 4. Two-dimensional 473

474

Solid Fuels Combustion and Gasification

d) Chemical reactions 1. Instantaneous or just equilibrium 2. Considers rates plus some sort of control e) Devolatilization 1. Not considered 2. Instantaneous 3. Homogeneous release 4. Proportional to mixing or feeding rates 5. Diffusion and kinetic controlled f) Pollutant generations 1. SOx generation and absorption 2. NOx generation and consumption g) Heat transfers 1. Between phases 2. Between phases and immersed surfaces h) Particle size distribution 1. Variations not included 2. Variations due to chemical reactions 3. Variations due to chemical reactions, attrition, and entrainment to freeboard i) Freeboard 1. Not considered 2. Considers only particle low decay 3. Considers particle low decay and chemical reactions 4. Considers all of the above plus heat transfers to tubes j) Comparisons with experimental or real operations 1. Combustors and/or boilers 2. Gasiiers k) Involved dimensions 1. One 2. Two 3. Three There are a few works with tables in which these or similar aspects are used to classify models and simulation programs [7, 43].* However, those starting in the area should be more interested in developing a simulation model to better appreciate these differences.

REFERENCES 1. de Souza-Santos, M.L., Modelling and Simulation of Fluidized-Bed Boilers and Gasiiers for Carbonaceous Solids, PhD thesis, University of Shefield, Shefield, United Kingdom, 1987. *

After that, several other models appeared in the literature, as well as improvements on former ones. The list of references does not include those works on modeling and simulation of circulating luidized bed. A few of those can be found in the reference list of Chapter 17.

Classification of Modeling for Bubbling Fluidized-Bed Equipment

475

2. de Souza-Santos, M.L., Comprehensive modelling and simulation of luidized-bed boilers and gasiiers, Fuel, 68, 1507–1521, 1989. 3. de Souza-Santos, M.L., A study on pressurized luidized-bed gasiication of biomass through the use of comprehensive simulation, in Proc. Fourth International Conference on Technologies and Combustion for a Clean Environment, Lisbon, Portugal, July 7–10, 1997, paper 25.2, Vol. II, pp. 7–13. 4. de Souza-Santos, M.L., A study on pressurized luidized-bed gasiication of biomass through the use of comprehensive simulation, in Combustion Technologies for a Clean Environment, Gordon and Breach, Amsterdam, Netherlands, 1998. 5. de Souza-Santos, M.L., Application of comprehensive simulation of luidized-bed reactors to the pressurized gasiication of biomass, Journal of the Brazilian Society of Mechanical Sciences, 16(4), 376–383, 1994. 6. de Souza-Santos, M.L., Search for favorable conditions of atmospheric luidized-bed gasiication of sugar-cane bagasse through comprehensive simulation, in Proc. 14th Brazilian Congress of Mechanical Engineering, Bauru, São Paulo, Brazil, December 8–12, 1997. 7. de Souza-Santos, M.L., Search for favorable conditions of pressurized luidized-bed gasiication of sugar-cane bagasse through comprehensive simulation, in Proc. 14th Brazilian Congress of Mechanical Engineering, Bauru, São Paulo, Brazil, December 8–12, 1997. 8. de Souza-Santos, M.L., A study on thermo-chemically recuperated power generation systems using natural gas, Fuel, 76(7), 593–601, 1997. 9. Rabi, J.A., and de Souza-Santos, M.L., Incorporation of a 2-lux model for radiative heat transfer in a comprehensive luidized bed simulator. Part I: Preliminary model equations, in Proc. ENCIT-2002, 9th Brazilian Congress on Engineering and Thermal Sciences, Caxambu, Brazil, October 15–18, 2002. 10. Rabi, J.A., and de Souza-Santos, M.L., Incorporation of a 2-lux model for radiative heat transfer in a comprehensive luidized bed simulator. Part II: Numerical results and assessment, ENCIT-2002, 9th Brazilian Congress on Engineering and Thermal Sciences, Caxambu, Brazil, October 15–18, 2002. 11. Rabi, J.A., Usage of Flux Method to Improve Radiative Heat Transfer Modelling inside Bubbling Fluidized Bed Boilers and Gasiiers, PhD thesis, Faculty of Mechanical Engineering, State University of Campinas, Campinas, São Paulo, Brazil, 2002. 12. Rajan, R.R., and Wen, C.Y., A comprehensive model for luidized bed coal combustors, AIChE J., 26(4), 642–655, 1980. 13. Costa, M.A.S., and de Souza-Santos, M.L., Studies on the mathematical modeling of circulation rates of particles in bubbling luidized beds, Power Technology, 103, 110– 116, 1999. 14. Rabi, J.A., and de Souza-Santos, M.L., Incorporation of a two-lux model for radiative heat transfer in a comprehensive luidized bed simulator. Part I: Preliminary theoretical investigations, Thermal Engineering, 3, 64–70, 2003. 15. Raman, P., Walawender, W.P., Fan, L.T., and Chang, C.C., Mathematical model for the luid-bed gasiication of biomass materials. application to feedlot manure, Ind. Eng. Chem. Process Des. Dev., 20(4), 686–692, 1981. 16. Chang, C.C., Fan, L.T., and Walawender, W.P., Dynamic modeling of biomass gasiication in a luidized bed, AIChE Symp. Series, 80(234), 80–90, 1984. 17. Tojo, K., Chang, C.C., and Fan, L.T., Modeling of dynamic and steady-state shallow luidized bed coal combustors. effects of feeder distribution, Ind. Eng. Chem. Process Des. Dev., 20(3), 411–416, 1981. 18. Weimer, A.W., and Clough, D.E., Modeling a low pressure steam-oxygen luidized bed coal gasiication reactor, Chem. Eng. Sci., 36, 549–567, 1981. 19. Overturf, B.W., and Reklaitis, G.V., Fluidized-bed reactor model with generalized particle balances. Part I: Formulation and solution, AIChE J., 29(5), 813–820, 1983.

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20. Overturf, B.W., and Reklaitis, G.V., Fluidized-bed reactor model with generalized particle balances. Part II: Coal combustion application, AIChE J., 29(5), 820–829, 1983. 21. Campbell, E.K., and Davidson, J.F., The combustion of coal in luidized beds, Inst. Fuel Symp. Series, 1:A2.1–A2.9, 1975. 22. Gibbs, B.M., A mathematical model for predicting the performance of a luidized combustion, Institute of Fuel Symposium Series, 1(1), A5.1–5.10, 1975. 23. Gordon, A.L., and Amudson, N.R., Modelling of luidized bed reactors-IV: Combustion of carbon particles, Chem. Eng. Sci., 31(12), 1163–1178, 1976. 24. Chen, T.P., and Saxena, S.C., Mathematical modeling of coal combustion in luidized beds with sulphur emission control by limestone or dolomite, Fuel, 56, 501–413, 1977. 25. Horio, M., and Wen, C.Y., An assessment of luidized-bed modeling, AIChE Symp. Series, 73(161), 9–21, 1977. 26. Horio, M., Rengarajan, P., Krishnan, R., and Wen, C.Y., Fluidized Bed Combustor Modeling, West Virginia University, Morgantown, WV, NASA Report No. NAS3-19725, 1977. 27. Gordon, A.L., Caram, H.S., and Amudson, N.R., Modelling of luidized bed reactors—V; Combustion of coal particles: an extension, Chem. Eng. Sci., 33, 713–722, 1978. 28. Horio, M., Mori, S., and Muchi, I., A model study for development of low NOx luidized-bed coal combustion, in Proc. 5th International Conference of Fluidized Bed Combustion, Washington DC, Conf-771272-p2, II, 605–624, 1978. 29. Chen, T.P., Saxena, S.C., A mechanistic model applicable to coal combustion in luidized beds, AIChE Symp. Series, 74(176), 149–161, 1978. 30. MIT Energy Laboratory, Modelling of Fluidized Bed Combustion of Coal. Final Report: “A First Order” System Model of Fluidized Bed Combustor, Vol. I, Massachusetts Institute of Technology, Report to U.S. Department of Energy, Contract No. E(49-18)2295, 1978. 31. Saxena, S.C., Grewal, N.S., and Venhatoramana, M., Modeling of a Fluidized Bed Combustor with Immersed Tubes, University of Illinois at Chicago Circle, Report to U.S. Department of Energy, No. EE-1787-10, 1978. 32. Horio, M., and Wen, C.Y., Simulation of luidized bed combustors, Part I. Combustion eficiency and temperature proile, AIChE Symp. Series, 74(176), 101–109, 1978. 33. Fan, L.T., Tojo, K., and Chang, C.C., Modeling of shallow luidized bed combustion of coal particles, Ind. Eng. Chem. Process Des. Dev., 18(2), 333–337, 1979. 34. Kayihan, F., and Reklaitis, G.V., Modeling of staged luidized bed coal pyrolysis reactors, Ind. Eng. Chem. Process Des. Dev., 19, 15–23, 1980. 35. Park, D., Levenspiel, O., and Fitzgerald, T.J., A model for scale atmospheric luidized bed combustors, AIChE Symp. Series, 77(277), 116–126, 1981. 36. Congalidis, J.P., and Georgakis, C., Multiplicity patterns in atmospheric luidized bed coal combustors, Chem. Eng. Sci., 36, 1529, 1981. 37. Glicksman, L., Lord, W., Valenzuela, J., Bar-Cohen, A., and Hughes, R., A model of luid mechanics in luidized bed combustors, AIChE Symp. Series, 77(205), 139–147, 1981. 38. Fee, D.C., Myles, K.M., Marroquin, G., and Fan, L.S., An analytical model for freeboard and in-bed limestone sulfation in luidized-bed coal combustors, Chem. Eng. Sci., 39, 731–737, 1984. 39. Chandran, R.R., and Sutherland, D.D., Performance simulation of luidized-bed coal combustors, Preprint Am. Chem. Soc., Div. Fuel Chem., 33(2), 145, 1988. 40. Azevedo, J.L.T., Carvalho, M.G., and Durão, D.F.G., Mathematical modeling of coalired luidized bed combustors, Combustion and Flame, 77, 91–100, 1989. 41. Faltsi-Saravelou, O., Vasalos, I.A., and Sim, F.B., A model for luidized bed simulation—I. Dynamic modeling of an adiabatic reacting system of small gas luidized particles, Computers Chem. Eng., 15(9), 639–646, 1991.

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42. Gururajan, V.S., Agarwal, P.K., and Agnew, J.B., Mathematical modeling of luidized bed coal gasiiers, Trans. Inst. Chem. Eng., Part A, 211–238, 1992. 43. Adánez, J., and Abánades, J.C., Modelling of lignite combustion in atmospheric luidized bed combustors. 1. Selection of submodels and sensitive analysis, Ind. Eng. Chem. Res., 31, 2286–2296, 1992. 44. Ciesielczyk, E., and Gawdizik, A., Non-isothermal luidized-bed reactor model for char gasiication, taking into account bubble growth, Fuel, 73(1), 105–112, 1994. 45. Sriramulu, S., Sane, S., Agarwal, P., and Mathews, T., Mathematical modeling of luidized bed combustion, Fuel, 75(12), 1351–1362, 1996. 46. Jensen, A., and Johnsson, J.E., Modelling of NOx emissions from pressurized luidized bed combustion—a parameter study, Chem. Eng. Sci., 52(11), 1715–1731, 1997. 47. Pre, P., Hemati, M., and Marchand, B., Study on natural gas combustion in luidized beds: modeling and experimental validation, Chem. Eng. Sci., 53(16), 2871–2883, 1998. 48. Kulasekaran, S., Linjewile, T.M., and Argawal, P.K., Mathematical modeling of luidized bed combustion 3. Simultaneous combustion of char and combustible gases, Fuel, 78, 403–417, 1999. 49. Chen, Z., Lin, M., Ignowski, J., Kelly, B., Linjewile, T.M., and Agarwal, P.K., Mathematical modeling of luidized bed combustion. 4: N2O and NOx emissions from combustion of tar, Fuel, 80:1259–1272, 2001. 50. Scala, F., and Salatino, P., Modelling luidized bed combustion of high-volatile solid fuels, Chem. Eng. Sci., 57, 1175–1196, 2002. 51. de Souza-Santos, M.L., Solid Fuels Combustion and Gasiication: Modeling, Simulation, and Equipment Operation, Marcel Dekker (CRC Press), New York, 2004. 52. de Souza-Santos, M.L., A New Version of CSFB, Comprehensive Simulator for Fluidized Bed Equipment, Fuel, 86, 12–13, 1684–1709, 2007. 53. de Souza-Santos, M.L., CSFB applied to luidized-bed gasiication of special fuels, Fuel, 88(5), 826–833, 2009. 54. Rabi, J.A., and de Souza-Santos, M.L., Comparison of two model approaches implemented in a comprehensive luidized-bed simulator to predict radiative heat transfer: results for a coal-fed boiler, International Journal on Computer and Experimental Simulations in Engineering and Science, 3, 87–105, 2008.

on Techniques of G Basics Kinetics Determination Reined experimental techniques have been used to study various aspects of kinetics involving solid fuels, and most can be classiied into two main types of laboratory methods: • Captive, or batch, in which a ixed amount of solid sample is inserted just once into the equipment for analysis at the beginning of the run • Continuous, or steady state, in which the sample is continuously fed into and withdrawn from the equipment used for the analysis Many captive methods utilize thermal gravimetric analysis (TGA; also called the TG method). The sample of ine particles is put in a small barge and into a chamber, through which a stream of gas (inert or other) lows. The temperature and pressure of the injected gas stream are controlled, and the gas leaving the chamber is continuously analyzed. A series of thermocouples measures the gas and sample temperatures, and the sample weight is recorded continuously or at intervals. Two basic processes are used: a) The isothermal method, in which the gas is kept at a constant temperature. The sample is assumed to reach that temperature in a very short time and is kept at that temperature. b) The dynamic method, or dynamic thermogravimetry, in which the gas stream is slowly heated (most of the time under a linear increase with respect to time). In any analysis, the sample of solid should be inely ground to avoid the interference of mass transport of gas components in and out of the particle interior or pores. These transports occur at rates similar to or higher than the investigated reaction kinetics. Actually, before the deinitive data are extracted to derive kinetics, several tests use samples with decreasing particle sizes. The useful data are from those whose results are no longer affected by the particle size of the sample. However, even a small sample of ine powder requires time to reach a desired temperature. Therefore, even during procedures intended to be isothermal, the sample may spend considerable time with temperatures below the intended value for the determinations. Depending on the rate or kinetics, this may lead to serious errors and therefore unacceptable results [1]. Combustion reactions are an example. Even reactions associated with devolatilizations fall into that category. On the other hand, these problems are avoided by the dynamic method, and decomposition kinetics is obtained from the TGA measurements. 479

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Three main groups of techniques are used to derive the kinetics of decomposition [1]: • Integral methods, which use the area covered by the curve in the graph of weight against time. The most commonly used are analytical solution, van Krevelen, Kissinger, Horowitz and Metzger, and Coats and Redfen. • Differential methods, which use the rate of mass loss at various times. The most commonly used are classical, multiple linear regression, Freeman and Carroll, and Vachusca and Voboril. In addition to the irst derivative of solid conversion against time, the Vachusca and Voboril method uses the second derivative as well. • Special methods that cannot be classiied as integral or differential. The most commonly applied are those by Reich, Friedman, Reich and Stivala, and Popescu and Segal. In addition, Carrasco [1] presents a critical review of TGA methods applied to pyrolysis. He also derived a general methodology for the study of thermal decomposition kinetics of materials through dynamic thermogravimetry with linear variation of temperature. Since then, almost all works in the area have tended to use dynamic methods. Another important aspect of any method relates to ensuring the correct atmosphere around the sample of solid particles. For instance, argon is commonly used in determinations of pyrolysis kinetics when inert atmosphere is required. Moreover, the low of argon should guarantee that gases from the devolatilization itself are blown away using fresh inert gas; otherwise, secondary reactions may occur between these gases and solid compounds in the sample. Actually, it is almost impossible to completely avoid such secondary reactions, but a good experimental procedure should try to minimize them. Among the most applied continuous or steady-state methods are the following: • Fluidized beds. Fuel particles are continuously fed into a vessel where a preheated low of inert or reacting gas (depending of the objective of the study) is injected through its base. The gas velocity is maintained to ensure unchanged bubbling bed luidization conditions. The temperature is kept constant by electrical resistances coating the bed. Variations of temperature are very small because the mass of sand in the bed is much larger than the mass of fuel, leading to a high thermal inertia. Compared with other usual methods, luidized beds allow relatively high heat transfer rates between gas and particles. On the other hand, because of fast heating, all volatile are released before reliable and repetitive data related to the dynamics of pyrolysis can be obtained. Moreover, as usual residence times are around a few seconds, the uniform temperature of even small particles cannot be ensured [2]. In addition, the easy formation of particle clusters in the case of a test with caking or agglomerating coals leads to additional dificulty for the application of this technique. During pyrolysis, a good part of the tar polymerizes at the surface of particles, and the plastic consistency of those layers favors agglomeration with neighboring particles. This is not

Basics on Techniques of Kinetics Determination

481

a problem at relatively high temperatures (1000 K or above) because all volatiles escape too fast, and the remaining tar is coked. Nonetheless, it represents a signiicant limitation at lower temperatures. Among early works with this technique are those of Stone et al. [3], Zielke and Gorin [4], Peters and Bertling [5], and Jones [6]. • Entrained low. Fuel particles are continuously fed into a vertical tube and carried by a gas stream. The tube could be heated or the gas could be preheated, or both. The collected particles are rapidly quenched with watercooled probes at several points along the vertical tube. This technique tries to reproduce conditions in pulverized fuel combustors (or suspension combustion) and therefore is useful for studies related to such processes. The major problem with this procedure is that some of the particles stick to the tube walls, leading to uncertainties concerning the sample mass loss due to pyrolysis. Early works using that method can be found in the available literature [7–18]. • Free fall. Similarly to the entrainment apparatus, a vertical tube is used. However, just a few particles are allowed to travel in a free fall through a stationary gas. A region or several regions or zones of the reactor are heated to desired temperatures. This permits rapid heating conditions without the agglomeration and particle sticking problems. However, there are dificulties in the estimation of true residence time for particles going through severe changes in density due to swelling, devolatilization itself, or other reactions. Among the various continuous methods, this seems to be the most successful and is used by many researchers [19–21], including those investigating biomass pyrolysis [22]. After these, several other techniques were introduced. The complexities of experimental determinations are not within the scope of the present book. Good reviews of analytical methods, particularly applied to pyrolysis, can be found elsewhere [23, 24]. Like any other ield, this one continues to evolve, and several others using sophisticated methods have been developed.

REFERENCES 1. Carrasco, F., The evaluation of kinetic parameters from thermogravimetric data: comparison between established methods and the general analytical equation, Thermochimica Acta, 213, 115–134, 1993. 2. Anthony, D.B., and Howard, J.B., Coal devolatilization and hydrogasiication, AIChE J., 22, 4, 625–656, 1976. 3. Stone, H.N., Batchelor, J.D., and Johnstone, H.F., Low temperature carbonisation rates in a luidised-bed, Ind. Eng. Chem., 46, 274, 1954. 4. Zielke, C.W., and Gorin, E., Kinetics of carbon gasiication, Ind. Eng. Chem., 47, 820– 825, 1955. 5. Peters, W., and Bertling, H., Kinetics of the rapid degasiication of coals, Fuel, 44, 317– 331, 1965. 6. Jones, W.I., The thermal decomposition of coal, J. Inst. Fuel, 37, 3–5, 1964. 7. Moseley, F., and Paterson, D., Rapid high-temperature hydrogenation of coal chars. Part 1, J. Inst. Fuel, 38, 13–23, 1965.

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8. Moseley, F., and Paterson, D., Rapid high-temperature hydrogenation of coal chars. Part 2, J. Inst. Fuel, 38, 378–391, 1965. 9. Eddinger, R.T., Friedman, L.D., and Rau, E., Devolatilization of coal in a transport reactor, Fuel, 45, 245–252, 1966. 10. Howard., J.B., and Essenhigh, R.H., Pyrolysis of coal particles in pulverized fuel lames. Ind. Eng. Chem. Process Des. Dev., 6, 74, 1967. 11. Glenn, R.A., Donath, E.E., and Grace, J.R., Gasiication of coal under conditions simulating stage 2 of the BCR two-stage super pressure gasiier, in Fuel Gasiication, Advances in Chemistry Series No. 69, American Chemical Society, Washington, DC, p. 253, 1967. 12. Kimber, G.M., and Gray, M.D., Rapid devolatilization of small coal particles, Combustion and Flame, 11, 360, 1967. 13. Kimber, G.M., and Gray, M.D., Measurements of Thermal Decomposition of Low and High Rank Non-Swelling Coals at M.H.D Temperatures, British Coal Utilization Research Association, Document No. MHD 32, 1967. 14. Badzioch, S., Hawksley, P.G.W., Kinetics of thermal decomposition of pulverized coal particles, Ind. Eng. Chem. Process Des. Dev., 9, 521, 1970. 15. Belt, R.J., Wilson, J.S., and Sebastian, J.J.S., Continuous rapid carbonisation of powdered coal by entrainment and response surface analysis of data, Fuel, 50, 381, 1971. 16. Belt, R.J., and Roder, M.M., Low-sulfur fuel by pressurized entrainment carbonisation of coal, Am. Chem. Soc., Div. of Fuel Chem. Preprints, 17(2), 82, 1972. 17. Coates, R.L., Chen, C.L., and Pope, B.J., Coal devolatilization in a low pressure, low residence time entrained low reactor, in Coal Gasiication, Advances in Chemistry Series, No. 131, American Chemical Society, Washington, DC, 1974, p. 92. 18. Johnson, J.L., Gasiication of Montana lignite in hydrogen and in helium during initial reaction stages, Am. Chem. Soc., Div. of Fuel Chem. Preprints, 20(3), 61, 1975. 19. Shapatina, E.A., Kalyuzhnyi, V.A., and Chukhanov, Z.F., Technological utilization of fuel for energy, 1. Thermal treatments of fuels, 1960 (Paper reviewed by S. Badzoich, British Coal Utilization Research Association Monthly Bulletin, 25, 285, 1961.) 20. Moseley, F., and Paterson, D., Rapid high-temperature hydrogenation of bituminous coal, J. Inst. Fuel, 40, 523, 1967. 21. Feldmann, H.F., Simons, W.H., Mimn, J.A., and Hitshue, R.W., Reaction model of bituminous coal hydrogasiication in a dilute phase, Am. Chem. Soc., Div. of Fuel Chem. Preprints, 14(4), 1, 1970. 22. Yu, Q., Brage, C., Chen, G., Sjöström, K., Temperature impact on the formation of tar from biomass pyrolysis in a free-fall reactor, Journal of Analytical and Applied Pyrolysis, 40–41, 481–489, 1997. 23. Tsuge, S., Analytical pyrolysis—past, present and future, Journal of Analytical and Applied Pyrolysis, 32, 1–6, 1995. 24. Bridgwater, A.V., and Peacocke, G.V.C., Fast pyrolysis processes for biomass, Renewable and Sustainable Energy Reviews, 4, 1–73, 2000.

Remarks on 1 Basic Modeling and Simulation CONTENTS 1.1

Experiment and Simulation ..............................................................................1 1.1.1 The Experimental Method ....................................................................2 1.1.1.1 Example 1.1 ............................................................................2 1.1.1.2 Example 1.2............................................................................3 1.1.1.3 Example 1.3............................................................................4 1.1.1.4 Example 1.4............................................................................6 1.1.1.5 Example 1.5 ...........................................................................7 1.1.2 The Theoretical Method ....................................................................... 8 1.2 A Classiication for Mathematical Models ..................................................... 10 1.2.1 Phenomenological versus Analogical Models .................................... 10 1.2.2 Steady-State Models ........................................................................... 11 1.2.2.1 0D-S Models ........................................................................ 12 1.2.2.2 One-Dimensional Steady Models ........................................ 13 1.2.2.3 Two-Dimensional Steady Models ........................................ 13 1.2.2.4 Three-Dimensional Steady Models ..................................... 14 1.2.3 Dynamic or nD-D Models .................................................................. 16 1.2.4 Which Level to Attack? ...................................................................... 16 1.3 Exercises ......................................................................................................... 17 1.3.1 Problem 1.1 ......................................................................................... 17 1.3.2 Problem 1.2 ......................................................................................... 17 References ................................................................................................................ 17

1.1 EXPERIMENT AND SIMULATION Over the years, I have heard questions such as “If the equipment is working, why all the effort to simulate it?” and “Can’t we just ind the optimum operational point by experimentation?” The basic answer to the irst question is this: We simulate equipment because it is always possible to improve and, given an objective, to optimize an existing operation. Optimization not only increases competitiveness of a company but may also determine its chances of survival. For the second question the answer is that yes, we could ind the optimal point by experimentation; however, experimentation is much more expensive than computation, not to mention carrying the risk of disaster and loss of life due to trial-and-error tests aimed at making improvements. In addition, if any variable of the original process changes, the optimum that was determined through 1

2

Solid Fuels Combustion and Gasification

costly experimentation is no longer valid. For example, if a thermoelectric power unit starts receiving coal with different properties than that previously used, the optimum operational point would not be the same. It is also important to remember that the experimentally found optimum at a given scale is not applicable to other scales, even if several conditions remain the same. In fact, no company today can afford not to use computer simulation to seek optimized design or operation of its industrial processes. One may think that these remarks could hide some sort of prejudice against the experimental method. Nothing could be further from the truth. There is no valid theory without experimentation. A model or theory is not applicable unless it is experimentally veriiable, and this is a fundamental aspect of science. In other words, no matter how sophisticated a mathematical model and consequent simulation are, they would be useless if they cannot reproduce the values of variables measured during real operations within an acceptable degree of deviation. If the simulation does not observe those basic characteristics, it would not be able to predict the behavior of future equipment with some degree of conidence and would therefore be useless as a designing tool. Before going any further, let us make a few comments on experimental and theoretical procedures.

1.1.1 THE EXPERIMENTAL METHOD One may ask about the conditions required for the application of the experimental method in order to obtain or infer valid information concerning the behavior of a process. For this, the following deinitions are useful: • Controlled variables are those whose values can, within a certain range, be imposed freely. An example is the temperature of a bath heated by electrical resistances. • Observed variables are those whose values can be measured, directly or indirectly. An example is the thermal conductivity of the luid in the bath where the temperature has been controlled. Experiments are a valid source of information if, during the course of tests, the observed variables are solely affected by the controlled variables. Any other variable should be maintained at a constant value. 1.1.1.1 Example 1.1 Let us imagine one is interested in determining the dependence on temperature of the thermal conductivity of a liquid with a given composition and under a given pressure. Therefore, the observed variable would be the thermal conductivity of the liquid and the controlled variable would be its average temperature. For that, the thermal conductivity would be measured at various levels of temperature, and during the following must be guaranteed: • The temperature of the liquid bath should be kept as uniform as possible. A small agitation would sufice. • Its chemical composition and pressure must be maintained at constant levels.

Basic Remarks on Modeling and Simulation

3

In doing so, it will be possible to correlate cause and effect: i.e., the values of observed and controlled variables. These correlations may take the form of graphs or mathematical expressions. 1.1.1.2 Example 1.2 The SO2 absorption by limestone can be represented by the following reaction: 1 O 2 ↔ CaSO 4 . 2 The experiments to determine the kinetics of that reaction should be conducted under carefully chosen conditions, and one ought to do the following: CaO + SO 2 +

• Provide a proper barge to hold a small sample of limestone, which would be maintained in a container that could allow continuous injections and withdrawals of gases. • Provide a system to continuously weight the barge with required precision. • See that various mixtures of SO2 and O2, with precisely determined compositions, are injected into the container and pass around the barge. (The concentrations of reactants [SO2 and O2] in the involving atmosphere should be controlled or kept as constant as possible; therefore, a fresh supply of the reacting mixture should be guaranteed.) • Maintain a constant pressure in the capsule containing the barge. • Be sure that the limestone sample has its maximum particle size reduced to a point such that no interference of particle size on the reaction rate can be noticed. (That interference is caused by the resistance to mass transfer of gas components through the porous structure of the particle.) • Maintain the temperature in the container as constant as possible during each value experimental test. (This can be achieved by heating or cooling devices around the barge. Temperature differences between the various regions of the sample should be reduced or kept to a minimum. This can be accomplished by using thin layers of sample and through high heat transfer coeficients between gas and solid sample. One method is to increase the Reynolds number or the relative velocity between gas and particles. Of course, this has limitations because the gas stream should carry no solid.) • Organize the tests with different periods. (At the end of each test, samples of the CaO and CaSO4 mixture would be analyzed. An alternative is to apply algorithms that allow the determination of composition based on the sample weight. If this is accomplished, the weight of the barge could be measured during longer tests.) Following the above guidelines, during the experiments the number of controlled variables could be reduced to the temperature and the concentrations of SO2 and O2 in the incoming gas stream. The degree of conversion of CaO into CaSO4 would be the observed variable against time. In addition, a large number of experimental tests should be carried out to eliminate bias. This is possible to verify by quantitative means.

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Even after all these precautions, the determined kinetics is valid only at the range of temperature and concentration of the experiments. In addition, it is applicable only to the particular kind of limestone tested because other chemical components, different from those involved in the reaction and present in the original limestone, may have catalyst or poisoning activity. The above examples show the correct scientiic experimental procedure. This procedure allows observations that may be applied to understand the phenomenon. Within an established range of conditions, the conclusions do not depend on a particular situation. Therefore, they can be generalized and used in mathematical models combining several other phenomena. It is also important to be aware that experimental optimization of a process presents stringent limitations due to two factors: 1. The number of variables interfering in a given process is usually much bigger than the number of controllable variables. Variation imposed on a single input may represent variations on several process conditions and consequently on several observed variables. 2. Scale effects must be considered. Physical-chemical properties of substances handled by the process do not obey a linear dependence with the geometry of the equipment, such as length, area, and volume. For instance, one may double the volume of a chemical reactor, but densities or viscosities of streams entering or leaving the reactor are not doubled. In some cases, scaling up does not even allow geometric similarity with the experimental or pilot unit. The following example illustrates the false inference from experimental procedures. 1.1.1.3 Example 1.3 Someone is intending to scale up a simple chemical reactor, as shown in Figure 1.1. Solutions of components A and B are continuously injected, and the reaction between them is exothermic, whereas the exiting stream contains product C diluted into the Reactant A Hot water Reactant B Water jacket

Cold water

Agitator Reaction products

FIGURE 1.1

Schematic of a water-jacketed reactor.

5

Basic Remarks on Modeling and Simulation

reacting mixture. The process follows a steady-state regime, and water is used as cooling luid that runs inside a jacket surrounding the reactor. A stirring device maintains the reacting media as homogeneous as possible. An engineer knows the optimum temperature to be maintained in the vessel in order to achieve the maximum output of species C. He or she would try to ind the correct low of cold water through the jacket in order to maintain the optimum temperature. To achieve this, a pilot was built, and through experiments, the best operational condition was set. Now, the engineer intends to build an industrial-scale reactor that delivers ten times the rate of C obtained in the pilot. Someone suggests that the same geometric shape of the pilot be kept, that is, H/D = 2 (where H = level illed with luid and D = diameter). However, to obtain the same concentration of C in the exiting stream, the residence time of reactants in the volume should be maintained. Residence time τ is usually deined by the following equation: τ=

Vρ F

(1.1)

where V is the volume of the reactor (illed with reacting mixture), ρ is the average density of the luid, and F is the mass low of injected luids into the reactor. Assuming a cylindrical reactor, the volume is given by the following equation: V=

πD 2H . 4

(1.2)

The following relation should be set in order to maintain the same residence time: V = 10 V0 .

(1.3)

Here, index 0 indicates the pilot dimensions. To keep the H/D ratio equal to 2, the new diameter would be D = 101/ 3 D 0 .

(1.4)

The area for heat transfer (neglecting the bottom) is given by A = πHD.

(1.5)

A HD D2 = = 2. A 0 H 0 D0 D0

(1.6)

A = 10 2 / 3 A 0 .

(1.7)

Therefore, the area ratio would be

Using 1.4, one gets

6

Solid Fuels Combustion and Gasification

According to this option, the heat exchanging area of the industrial-scale reactor would be only 4.64 times the respective heat exchanging area of the pilot unit. Thus, cooling would not be as eficient as before. Conversely, if one tries to multiply the reactor wall area by 10, the volume would be much larger than 10 times the volume of the pilot. Actually, this would increase the residence time of the reactants by more than 31 times, which is not desirable because (a) the investment in the new reactor would be much larger than intended, and (b) the reactants would sit idle in the reactor because the residence time of reactants would be much larger than the value found through optimization tests. The only possibility is to modify the H/D ratio in order to maintain the same area/volume relationship as found in the pilot. For instance, if one proposes to keep the same diameter as the pilot, the industrial reactor would be 10 times as tall as the experimental unit. Having in mind limitations determined by the factory layout, this solution may not be convenient or practical. This demonstrates that not even similar geometric shape can be kept in a scaling up procedure. The alternative to multiplying the volume by 10 while maintaining the same shape of the pilot would be to install a refrigerating coil inside the reactor. After a proper design, that would provide the required heat transferring rates. 1.1.1.4 Example 1.4 Another example is given for the case of a boiler. Someone wants to optimize the boiler design through experimental tests using a relatively small pilot unit. The unit is composed of a furnace with a tube bank that can be partially retreated from the furnace in order to vary the heat exchange area. Coal is continuously fed into it while water runs into the tube bank. After a few tests in which the fuel feeding rate is kept constant, the engineer inds the fraction of exposed tube area, which provides the maximum rate of steam generation. This can be explained by the following: 1. If the tube area is increased, the heat transfer between the furnace and the tubes is augmented, leading to increases in steam production. However, further increases in the tube area also lead to decreases in the average temperature in the furnace. This reduces the carbon conversion or fuel utilization and therefore diminishes the rate of energy transferred to tubes for steam generation. 2. If the area of the tubes is decreased, the average temperature in the furnace increases and the carbon conversion is enhanced. Therefore, more coal is converted to provide energy, possibly leading to augmentations in the steam generation. However, after a certain point, the lack of suficient tube area for heat transfer would just raise the temperature of the gases leaving the chimney, resulting in a reduction in boiler eficiency. As we have seen, the optimum heat exchanging area to achieve the maximum boiler eficiency should be an intermediate point. However, it is not easy to ind such a solution because of the great number of processes or phenomena inluencing the process. Among them are all sorts of heat transfers among the particles, gases, and

7

Basic Remarks on Modeling and Simulation

tubes; mass transfers among gases and particles; momentum transfers among those phases; and several chemical reactions. To achieve a good solution through experimentations would be a nightmare, if not a virtual impossibility. 1.1.1.5 Example 1.5 Following the previous example, let us now imagine that someone tries to design a bigger boiler, for instance, one to consume twice the previous amount of fuel. Among other possibilities, two alternatives could be imagined: a. To maintain the same ratio between the total area of tube surface and the mass low of coal feeding (or energy input) used at that optimum condition found at the pilot b. To maintain the same ratio between the total area of tube surface and the combustor volume used at that optimum experimental condition A very likely answer to this dilemma is none of the above. To explain, let us simplify the problem by writing the following relations:  = fc Fh c Q

(1.8)

 η = Aα av ∆T n Q

(1.9)

where  is the total rate energy input to the boiler furnace (W) • Q • fc is the fraction of fuel that is consumed in the furnace (dimensionless from 0 to 1); the unreacted fuel leaves the furnace with the stack gas or in the ashes • F is the fuel feeding rate (kg/s) • hC is the combustion enthalpy of the injected fuel (J/kg) • η is the boiler eficiency, i.e., the ratio between the injected energy (through the fuel) into the furnace and the amount used to generate steam • A is the total area of tube surface (m2) • n is a coeficient used to accommodate an approximate equation that accounts for the differences between the laws of radiative and convective heat transfer processes • ΔT is the average difference between the luid that runs inside the tubes and the furnace interior (K) • αav is the equivalent global (or average between inside and outside coeficients of the tubes) heat transfer coeficient (includes convection and radiation) between the luid that runs inside the tubes and the furnace interior (W m–2 K–n) Equations 1.8 and 1.9 are combined to write A=

ηfc Fh c . α av ∆T n

(1.10)

8

Solid Fuels Combustion and Gasification

Let us examine alternative a. According to Equation 1.10, the area of tubes seems to be proportional to the fuel-feeding rate. However, assuming the same tube bank coniguration as in the pilot, its area would not be linearly proportional to the furnace volume. Therefore, the average velocity of gases crossing the tube bank will be different from the values found in the pilot. The heat transfer coeficient will change as well, because it is a strong function of that velocity. Looking at Equation 1.10, it is easy to conclude that alternative a will not work. Now, if one tries alternative b and increases the furnace volume, according to Equation 1.1, the residence time of fuel particles in the furnace will increase as well. That will greatly differ from the optimum found during pilot tests. Therefore, the fuel conversion and the boiler eficiency will not be the same as the pilot, and the second alternative would not work either. As seen, even that simpliied analysis is able to demonstrate the complexity of the question. Hence, one would ask how this type of problem should be faced. Answering that question, among others, will be one of the main goals of this book.

1.1.2 THE THEORETICAL METHOD It is too dificult for the human mind to interpret any single or combination of phenomena in which more than three variables are involved. It is not a coincidence that the graphical representation of inluences is also limited to that number of variables. Some researchers make invalid extrapolations of the experimental method and apply the results to multidimensional problems. For instance, consider the case explained above: trying to infer the kinetics of SO2 absorption using a boiler combustor to which coal and limestone are added. As temperature and concentration (among several other variables) change from point to point in the combustion chamber, no real control over the inluencing variables is possible. The best that can be accomplished is the veriication of some interdependence. The results of such a study might be useful for optimizing a particular application without the pretension of generalizing the results and applying them to other situations, despite any eventual similarities. On the other hand, the theoretical method is universal. If it is based on fundamental equations (mass, momentum, and energy conservations) and correlations obtained from well-conducted experimental procedures, the theoretical approach does not suffer from limitations due to the number of variables involved. Therefore, mathematical modeling is not a matter of sophistication, but the only possible method by which to understand complex processes. In addition, the simple fact that a simulation program is capable of being processed to generate information that describes, within a reasonable degree of deviations, the behavior of a real operation is in itself a strong indication of the coherence of the mathematical structure. The most important properties of mathematical modeling, along with its respective simulation program, can be summarized as follows: • Mathematical modeling requires much less in the way of inancial resources than experimental investigation. • It can be applied to studying conditions in areas that are dificult or impossible to access or where uncertainties in measurements are implicit. Such

Basic Remarks on Modeling and Simulation

• • • •

• •

9

conditions may include very high temperatures, as are usually found in combustion and gasiication processes. In addition, these processes normally involve various phases with moving boundaries, turbulence, etc. It can be extended to infer the behavior of a process far from the tested experimental range. It allows a much better understanding of the experimental data and results and therefore can be used to complement the knowledge acquired from experimental tests. It can be used to optimize the experimental procedure and to avoid tests at uninteresting or even dangerous ranges of operations. It can be used during the scaling-up phase in order to achieve an optimized design of the equipment or process unit. This brings substantial savings of time and money because it eliminates or drastically reduces the need for intermediate pilot scales. The model and its respective program are not static. In other words, they can be improved at any time to expand the range of application, increase reliability, or decrease the time necessary for the computations. The model and the program can also be improved as more and better information become available from experimental investigations. Results published in the literature concerning the basic phenomena involved in the simulated process or system represent a constant source of information. Therefore, the program can be seen as a reservoir of knowledge.

Mathematical modeling is not a task but a process. The development of a mathematical model and its respective computer simulation program is not a linear sequence in which each step follows the former one until the end. The process is composed of a series of forward and backward movements, where each block or task is repeatedly revisited. One should be suspicious of simple answers. Nature is complex, and the process of modeling it is an effort to represent it as closely as possible. The best that can be done is to improve the approximations in order to decrease deviation from reality. A good simulation program should be capable of reproducing measured operational data within an acceptable level of deviation. In most cases, even these deviations have a limit that cannot be surpassed, which is established by several constraints, including the following: • Intrinsic errors—due to limitations in measurement precision—in correlation obtained through experimental procedures • Available knowledge from literature or personal experience • The basic level of modeling adopted (zero, one, or more dimensions) These factors, as well others, are discussed further in Chapter 12. Therefore, only approximations—sometimes crude ones—of the reality are possible. However, despite its simplicity, any model should be mathematically consistent. It is advisable to go from simple to complex, not the other way around. Modeling is an evolutionary movement. Usually, there is a long path from the very irst model

10

Solid Fuels Combustion and Gasification

to a more elaborate version. It is important to include very few effects in the irst trial. Several hypotheses and simpliications should be assumed; otherwise, the risk of not achieving a working model would increase. In addition, evolution requires comparisons between simulation results and reality. If one starts from the very simple, results might be obtained from computations. Then, comparisons against experimental or equipment operational data may provide clues for improvements. During industrial operations, measurements of variables can seldom be made with deviations smaller than 5%. For instance, the average temperature of a stack gas released from an industrial boiler can easily luctuate 30 K around an average of 600 K. Therefore, if the respective model has already produced deviations against measured or published values below that range, one might consider stopping. Improvements in a model should be accomplished mainly by eliminating one (and only one) simplifying assumption at a time, followed by veriication if the new version leads to representations closer to reality by comparison against experimentation. If not, one may eliminate another assumption, and so on. On the other hand, if one starts from complex models, not even computational results may be obtainable. Even if computation results are achieved, there would be little chance of identifying where improvements should be made. If one needs to travel, it is better to have a small and inexpensive car that runs than a luxury sedan without any wheels.

1.2 A CLASSIFICATION FOR MATHEMATICAL MODELS 1.2.1 PHENOMENOLOGICAL VERSUS ANALOGICAL MODELS Industrial equipment or processes operate by receiving physical inputs, processing them, and delivering physical outputs. Among the usual physical inputs and outputs of streams are low rates, compositions, temperatures, and pressures. In addition to those variables, heat and work may be exchanged through the control surface. Phenomenological models intend to reproduce the processes taking place in the equipment as closely as possible. Those models are based on two types of equations: • Fundamental equations,* such as the laws of thermodynamics and mass, energy, and momentum conservation • Auxiliary equations, usually based on empirical or semiempirical correlations The combination of fundamental equations and auxiliary correlations may lead to models that would be valid within the same range as the correlations used. Such a procedure brings some assurance that phenomenological models are relecting reality within that range.

*

The fundamental laws of thermodynamics can be found in any textbook. The main relations are repeated in Chapter 5 and Appendix B. Those related to conservation can be found in any text of transport phenomena [1–4]. In addition, Appendix A summarizes those that are the most important for the purposes of this book.

Basic Remarks on Modeling and Simulation

11

Analogy models, although useful for relatively simple systems or processes, simply mimic the behavior of a process and therefore do not relect reality. Examples of analogy models are those based on mass-and-spring systems or on electric circuits. Since phenomenological models are built on fundamental equations and tested correlations, the respective simulation programs are not just more realistic and reliable than analogy models but also provide much more information than simple descriptions of the properties and characteristics of the output streams. As mentioned before, is extremely important to properly understand the role of variables that are not accessible for direct or even indirect measurements. For instance, such information might be useful in designing new optimized units, as well as visualizing operational controlling methods for the simulated process or operation. In addition, developing a phenomenological model requires setting relationships linking various variables. Consequently, the combination of experimental observations with the theoretical interpretation usually leads to a better understanding of processes taking place in the equipment or industrial process. The phenomenological models can be classiied according to several criteria. The irst branching is set according to the number of space dimensions considered in the model. Therefore, three levels are possible. A second branching considers the inclusion of time as a variable. If time is not included, the model is called steady state; otherwise, it is a dynamic model. Moreover, several other characteristics can be added to model classiications, such as whether laminar or turbulent low conditions are assumed and whether dissipative effects (such as those provided by terms containing viscosity, thermal conductivity, and diffusivity) are included or not. The list could go on. However, in this introductory text, only spatial dimensions and time are considered. During the present text, mathematical models are classiied as follows: 1. Zero-dimensional steady (0D-S): This is the simplest level of modeling and includes no dimension or time as a variable. 2. Three-dimensional-dynamic (3D-D): This includes three space dimensions and time as variables.

1.2.2 STEADY-STATE MODELS Despite being widely used, the term steady-state regime leads to mistakes if not clearly understood. In relation to a given set of space coordinates, a steady-state regime for a control volume* will be established if: 1. The control surface does not deform or move. 2. The mass lows and average properties of each input and output stream remain constant.

*

See Appendix B, section B.1, “Heat and Work.”

12

Solid Fuels Combustion and Gasification

3. The rates of heat and work exchanges between the control volume and surrounds are constant. 4. Despite differences in conditions from point to point inside the control volume, they remain constant at each position. Most industrial equipments work at or near steady-state conditions, at least during the largest portion of their operational life. On the other hand, there is no such a thing as a perfect steady-state regime. Even for supposedly steady operations, some degree of luctuations against time is observed regarding variables such as temperatures, concentrations, pressures, etc. Nevertheless, most of the time, those variables remain within relative narrow ranges during operations of several industrial processes and can thus be treated as constants. 1.2.2.1 0D-S Models Zero-dimensional (0D) models set relations between input and output variables of a control volume without considering the details of the phenomena occurring inside that volume. Therefore, no description or evaluation of temperature, velocity, or concentration proiles inside the equipment being studied is possible. 0D-S models keep those values constant regarding time. In spite that, 0D-S models are very useful, mainly if an overall analysis of an equipment, or system composed of several equipments, is intended. Depending on the complexity of internal phenomena and the information available about a process, this may be the only achievable level of modeling. However, that should not be taken for granted because it may involve serious dificulties, mainly when many subsystems (or equipment) are considered in an industrial unit process. Since 0D models do not consider descriptions regarding internal points inside the equipment, they require assumptions such as chemical and thermodynamic equilibrium conditions at the output streams. Such hypotheses may constitute oversimpliications, leading to false conclusions because of the following points: a) Rigorously, equilibrium at exiting streams would require ininite residence time of the chemical components or substances inside the equipment. The typical residence time of several classes of reactors, combustors, and gasiiers is seconds to minutes. Therefore, outputs with conditions far from equilibrium might occur. For instance (and as will be demonstrated later in the text), gasiiers are among those types of equipment that do not deliver streams at chemical equilibrium. b) Even when equilibrium is assumed, determination of exiting stream compositions requires the value of their temperatures. However, to estimate those, one needs to perform energy balance or balances, which require the compositions and temperatures of the streams to allow computation of enthalpies or internal energies of the exiting streams. Reiterative procedures based on equilibrium assumptions may work if exiting streams are composed of just one or two components. However, reiterative processes with several nested convergence problems might bring awkward computational problems. Most of these situations take signiicant effort and computational time to achieve a solution. Failure to arrive at a solution is not uncommon.

Basic Remarks on Modeling and Simulation

13

c) If the process includes gas–solid reactions—as in combustors or gasiiers— the level of solid fuel (such as coal or biomass) conversion varies signiicantly. This happens even for single pieces of equipment operating at different conditions. Zero-dimensional models should assume the level of fuel conversion because they cannot address values of residence times of different phases inside a real reactor. In addition, they are not able to evaluate internal process, such as heat and mass transfers between gases and solid particles. Moreover, the bulk of solid reacting conversions occur at points of high temperature inside the equipment. That temperature is usually much higher than the temperatures of exiting streams (gas or solid particles). Consequently, to perform the energy balance for the control volume, one needs to guess some sort of average representative temperature at which the reactions and equilibrium should be computed. Apart from being artiicial from the point of view of phenomenology, such guessed values are completely arbitrary. Therefore, 0D (which includes 0D-S and zero-dimensional dynamic [0D-D]) models are very limited and might even lead to wrong conclusions. Besides, they are not capable of predicting a series of possible operational problems common in combustors and gasiiers, such as critical temperatures that might surpass the limits of integrity of wall materials, explosion limits, values that start runaway processes, etc. Apart from all that, 0D models are also extremely deicient in cases of combustion and gasiication of solids because pyrolysis or devolatilization is present. That very complex process introduces gases and complex mixtures of organic and inorganic substances at particular regions of the equipment. The compositions of those streams have a strong inluence on the composition and temperature of the exiting gas, even if equilibrium is assumed. 1.2.2.2 One-Dimensional Steady Models The second level of modeling is to assume that all properties or conditions inside the equipment vary only at one space coordinate. They constitute a considerable improvement in quantity and quality of information provided by the 0D models. Equilibrium hypotheses are no longer necessary, and proiles of the variables, such as temperature, pressures, and compositions, throughout the equipment can be determined. Of course, they might not be enough to properly represent processes where severe variations of temperature, concentration, and other parameters occur in more than one dimension. 1.2.2.3 Two-Dimensional Steady Models Two-dimensional (2D) models may be necessary in cases where the variations in a second dimension can no longer be neglected. For example, let us imagine the difference between a laminar-low reactor and a plug-low reactor. Figure 1.2 illustrates a reactor in which exothermic reactions take place and heat can be exchanged with the environment through the external wall. In Figure 1.2a, the variations of temperature and composition in the axial direction are added to the variations in the radial direction. It is easy to imagine

14

Solid Fuels Combustion and Gasification (a)

Velocity profile

Temperature profile

Inlet of reactants

(b)

Inlet of reactants

Outlet of products

Velocity profile

Temperature profile

Outlet of products

FIGURE 1.2 Schematic views of a laminar low (a) and a plug-low (b) reactor.

that, because of the relatively smaller temperatures at the layer of luid near the walls, modest changes in composition would occur because of chemical reactions. This is a situation that may require 2D modeling to account for the correct exit conditions; otherwise, the average concentration at the exiting section cannot be computed with a reasonable degree of precision. Conversely, a plug-low model would just allow computing temperature, velocity, and concentration averages at each cross-sectional position along the reactor length. However, highly turbulent low may provide the possibility of assuming plug-low regimes. The same would probably be assumed if some sort of packing is present inside the reactor. Of course, drastic variations of temperature would be found at the thin layer near the wall. Nonetheless, that layer is not a representative portion of the lowing mass. Therefore, compared with variations in composition and temperature at the axial direction, it would be fair to neglect variations in the radial coordinate. This is a typical case where a one-dimensional steady (1D-S) model might lead to good results. 1.2.2.4 Three-Dimensional Steady Models Despite the visualizations of processes and phenomena provided by threedimensional (3D) models, they usually involve great mathematical and computational complexity. However, in some situations, they are necessary for a realistic representation. Of course, if the model and simulation procedure are successful, a great deal of information about the process is obtained. For instance, let us imagine the laminar-low reactor (Figure 1.2a), in which a rotation or vortex is imposed on the lowing luid. In some of these cases, a cylindrical symmetry can be assumed, and a two-dimensional model might be enough for a good representation of the process, but for asymmetrical geometries, 3D models are usually necessary. An example of such a process occurs inside commercial boilers burning pulverized fuels. Most of the combustion chambers have rectangular cross-sections, and internal buffers are usually found, not to mention tube banks. As rotational components, as well as

Basic Remarks on Modeling and Simulation

15

strong reversing lows, are present (see Figure 2.10), no symmetry assumption is possible or reasonable. On the other hand, as with everything in life, there is a price to be paid. Let us imagine what would be necessary to set up a complete model. First, the solution of the complete Navier-Stokes or momentum conservation equations should be found. That solution ought to be combined with the equations of energy and mass conservation applied for all chemical species. All these equations must be written for three directions and solved throughout the reactor. Such a system of equations would also require a large number of boundary conditions. Frequently, these conditions involve not just given or known values at interfaces, but also derivatives. Moreover, boundary conditions might require complex geometric descriptions. For example, the injections of reactant streams at the feeding section may be made by such a complex distributing system that even setting the boundary condition would be a very dificult problem. When auxiliary correlations and equations for computations of all parameters are included, the inal set of mathematical equations would be signiicantly large. However, commercially available computational luid dynamics programs have been developed and, in many cases, are capable of solving such problems. Very good results are obtainable, particularly for single-phase systems (gas–gas, for instance). Nonetheless, combustion and gasiication of solid fuels still present considerable dificulties, due to large amounts of chemical reactions and processes combined with directional radiative heat exchanging with interfering media. In most situations, those problems are overwhelming, especially in cases of combustion and gasiication of pulverized fuel suspensions. On the other hand, one must ask whether all that information would really necessary for good design or optimization of a boiler. Is it essential to predict the details of the velocities, concentration, and temperature proiles in all directions inside the equipment? What is the cost-beneit situation in this case? Departing from a previous one- or two-dimensional model, would this three-dimensional model be capable of decreasing the deviations between simulation and real operation to a point where the time and money invested in it would be justiiable? Is it useful to have a model that generates deviations below the measurement errors? Finally, what would be necessary to measure in order to validate such a model against real operations? Along this line, let us ask what sorts of variables are usually possible to measure and what is the degree of precision or certainty of such measurements. Anyone who has worked at an industrial processing plant knows that is not always easy to measure temperature proiles inside equipment. For instance, if combustion of solid fuel is taking place, the composition, temperature, and velocity proiles of gas streams, as well as those of solids, are extremely dificult to determine with a reasonable degree of conidence and reproducibility. On the other hand, average values of temperatures, pressures, mass lows, and compositions of the entering and leaving streams can be measured. Sometimes, average values of variables at a few points inside the equipment are obtainable. This illustrates how important it is for those involved in the mathematical modeling of a particular type of equipment or process to be very well acquainted with its real operation. That provides valuable training that could be very useful when one decides to develop a model and its respective simulation program. If this is not possible, it is advisable to keep in contact with the personnel involved in those operations, as well as reading as many

16

Solid Fuels Combustion and Gasification

papers or reports as possible related to operations of the industrial or pilot units in question. This will be extremely rewarding when simpliication assumptions of a model need to be made.

1.2.3 DYNAMIC OR ND-D MODELS In addition to the above considerations, models might include time as a variable, and they are therefore called dynamic models. Some processes are designed to impose variations of input values of temperature, concentration, velocity, pressure, etc. Others cannot escape from such situations, such as batch operations of reactors and internal combustion engines. In those instances, dynamic models are required. Those approaches are also useful or necessary when control strategies are to be set or for cases where start-up and/or stopping procedures should be monitored or controlled. Safety or economics may demand these approaches. Most of the considerations described above for steady-state models, regarding the necessity of including more dimensions, are also valid for dynamic models.

1.2.4 WHICH LEVEL TO ATTACK? The choice among the various levels of modeling should be based on necessity. Sophistication is not a guarantee of quality. The same is true of extreme simplicity, which could lead to false or naive assumptions about complex phenomena. The following suggestions may serve as a guide: 1. In case of steady-state or near-steady-state processes, it is advisable to start from a 0D-S model (or from 0D-D model if dynamic). Even if this is not the desired level, it is useful just to verify whether one’s understanding of the overall operation of the equipment or process is coherent or not. Overall mass and energy conservation should always hold. 2. Comparisons between simulation results and the measured values should be made. As we have seen, measurements in industrial operations always present relatively high deviations. Unless more details within the equipment are necessary, the present level might be satisfactory if it has already produced relatively low deviations between simulation and measured values. 3. If simulation results and the measured values do not compare well, at least within a reasonable degree of approximation, the model equations must be revisited. The hypothesis and simpliications should also be reevaluated. Then, the process should start again from Step 2. It is also possible that for reasons already explained, the elected level of attack cannot properly simulate the process. The reasons for that have been noted above. In addition, there may be deviations produced by the previous level or the need for more detailed information. In any of those circumstances, it might be necessary to add a dimension and time as a variable. 4. Before starting a higher and more sophisticated level of modeling, it is advisable to verify what can be measured in the equipment pilot or the

Basic Remarks on Modeling and Simulation

17

industrial unit to be simulated. In addition, one should verify whether the measurements and available information would be enough for the comparisons against results from that next simulation level. If they are not suficient, the merit of stepping up the model should receive serious consideration.

1.3 1.3.1

EXERCISES PROBLEM 1.1* Discuss others possibilities for the design of the reactor and solving the scaling-up contradictions, as described in Example 1.3.

1.3.2

PROBLEM 1.2** Based on the considerations raised in Example 1.4, develop some suggestions for a feasible scaling up of the boiler.

REFERENCES 1. Bird, R.B., Stewart, W.E., and Lightfoot, E.N., Transport Phenomena, John Wiley, New York, 1960. 2. Slattery, J.C., Momentum, Energy, and Mass Transfer in Continua, Robert E. Kriefer, Huntington, NY, 1978. 3. Brodkey, R.S., The Phenomena of Fluid Motions, Dover, New York, 1967. 4. Gidaspow, D., Multiphase Flow and Fluidization, Academic Press, San Diego, CA, 1994.

2 Solid Fuels CONTENTS 2.1 2.2

Introduction .................................................................................................... 19 Physical Properties .........................................................................................20 2.2.1 Size Distribution .................................................................................20 2.2.2 Shape...................................................................................................20 2.2.3 Porosity ............................................................................................... 21 2.3 Chemical Properties ....................................................................................... 22 2.4 Thermal Treatment .........................................................................................25 2.4.1 Drying .................................................................................................26 2.4.2 Devolatilization................................................................................... 29 2.4.2.1 Composition and Structure of Original Carbonaceous ....... 31 2.4.2.2 Temperature and Heating Rates ........................................... 31 2.4.2.3 Pressure ................................................................................34 2.4.2.4 Composition of Surrounding Atmosphere ...........................34 2.5 Gasiication and Combustion .......................................................................... 35 2.6 Exercises ......................................................................................................... 36 2.6.1 Problem 2.1 ......................................................................................... 36 References ................................................................................................................ 36

2.1

INTRODUCTION

This chapter is devoted to describing the fundamental properties of the carbonaceous solid fuels most used in commercial combustion and gasiication processes. The objective is to introduce the main characteristics of most common solid fuels and their behavior under heating. No mathematical treatments are presented at this stage; only qualitative aspects of drying and pyrolysis are shown here, leaving quantiications to Chapter 10. From a practical point of view, industrially employed solid carbonaceous fuels can be classiied in three main categories: coals, biomass, and other. Carbonaceous fuels are complex collections of organic polymers consisting mainly of aromatic chains combined by hydrocarbons and other atoms, such as oxygen, nitrogen, sulfur, potassium, sodium, etc. Coals are primarily the results of slow deterioration of biomass, and the degree of that deterioration determines the coal rank. For instance, a lower degree of deterioration is found in lignites, and the maximum degree is found in anthracites. Intermediary stages are the subbituminous and bituminous coals. Therefore, the physical and chemical properties of coal are functions of its age, and the most important are discussed below.

19

20

2.2

Solid Fuels Combustion and Gasification

PHYSICAL PROPERTIES

Among the most important physical properties of solid fuel particles are size distribution, shape, and porosity.

2.2.1

SIZE DISTRIBUTION

Of course, the size of particles plays a fundamental role on combustion and gasiication processes. Prior to being fed into combustors or gasiiers, a solid fuel usually passes through grinding. The degree of particle size reduction depends on the requirements of application. Most combustors and gasiiers operate with grain dimensions in the range from 10−6 to 10−2 m. The grinding method is also important because solid fuels may present special characteristics. If they are not well understood and processed, serious problems may occur during operations of combustor or gasiier feeding devices. For instance, ibrous materials, such as sugarcane bagasse, have broom-like extremities. Often, such particles become entangled, leading to the formation of large agglomerates inside hoppers. This might prevent the bagasse from continuously lowing down to the feeding screws. Sophisticated design and costly systems are necessary to ensure steady feeding operation. Many grinding processes dramatically increase the fraction of particles with broom-like ends, but that can be largely avoided by applying rotary knife cutters. In general, any sample of particles covers a wide range of sizes. The particle size distribution is provided after a laboratory determination where several techniques can be applied. The most commonly used is classiication using vertical stacks of screens with decreasing net apertures from the top to the bottom. A sample of particles is deposited on the top of the pile, usually with 5 to 15 screens, and a gentle rocking movement is applied to the whole system. After a time, if no signiicant change is veriied on the amounts retained by the screens, the mass remaining at each one is measured. The percentage of original mass retained at each screen is provided. Of course, one would expect a distribution approaching normal probabilistic curve, but that is not always the case. The apertures of screens follow standard sizes, and more details are described in Chapter 4. It is easy to imagine that smaller particles tend to be consumed faster and are carried more easily by the gas stream, than bigger ones. Therefore, the particle size distribution inluences not only the rate at which the fuel reacts with oxygen and other gases, but almost all other aspects of combustor and gasiiers operations. Of course, to repeat all computations for each particle size would be rather cumbersome. That is why all mathematical models use some sort of average diameter. There are several deinitions or interpretation for such an average, and those are shown in the next chapter.

2.2.2

SHAPE

In addition to particle sizes, their shape strongly inluences several phenomena found in combustor and gasiiers. The rates of gas–solid reactions—among them the

21

Solid Fuels

oxidation of carbonaceous solids or combustion—depend on the available particle surface area. Thus, for the same volume, the particle with higher surface area should lead to faster consumption. Of course, the minimum would be found for spherical particles. In addition, the particle shape has a strong inluence on the momentum transfers between particles and the gas stream carrying them. Among such parameters is the terminal velocity. The parameter most often used to describe the shape of a particle is the sphericity, which is deined by ϕp =

Surface area of spherical particle . Surfacee area of particle with same volume of the spherical one

(2.1)

Clearly, the sphericity tends to 1 for particles approaching spherical shape, and smaller values are found for particles departing from that form. This parameter is very useful because it allows a simple quantitative description of the average shape of a solid particle sample, and it will be applied in the modeling shown ahead in the text. The sphericity ranges from 0.6 to 0.9 for most ground coals, limestone, and sand particles. The value of 0.7 may be used if no better information is available. Wood chips—usually used to feed the process in paper mills—have a sphericity around 0.2. The sphericity of particles is mainly dictated by the grinding or preparation process, and it is easily determined by laboratory tests. As with several other particle properties, sphericity also suffers strong variation during combustion or gasiication processes.

2.2.3

POROSITY

Fuel solids are usually very porous. Typically, more than half of particle volume is empty because of tunnels that crisscross its interior. A good portion of these tunnels have microscopic diameters. This leads to considerably high values for the total area of their surfaces per mass of a particle, and values around 500 m2/g are common. The area of pore surfaces inside particles strongly inluences the mass transfers of gases into and out of the particle. Obviously, for similar conditions, the larger the available area, the faster the consumption rate of components belonging to the solid matrix by gas–solid reactions. There is some controversy on the methods to determine the internal area of pores, and a lot has been written on the subject, as illustrated by Gadiou et al. [1]. In any case, the above value demonstrates the importance of pores and their structures on the rate of heterogeneous or gas–solid reactions. The porosity can be calculated using two different deinitions of density: • Apparent particle density (ρapp), which is the ratio between the mass of an average particle and its volume, including the void volumes of internal pores. A wide range of values can be found even for a single species of fuel, but for the sake of example, typical igures fall around 1100 kg/m3 for coals and 700 kg/m3 for woods.

22

Solid Fuels Combustion and Gasification

• Real density (ρreal), which is the ratio between the mass of an average particle and its volume, excluding the volumes occupied by internal pores. Again, a wide range of values can be found even for a single species of fuel, but common values fall around 2200 kg/m3 for coals and 1400 kg/m3 for woods. The porosity is deined as the ratio between the volumes occupied by all pores inside a particle and its total volume (including pores). It can be easily demonstrated that Vpores ρ app . ζp = (2.2) = 1− V ρ real Typical values of porosity are around 0.5 (or 50%).

2.3

CHEMICAL PROPERTIES

The composition and molecular structures found in any carbonaceous fuel, such as coal and biomass, are very complex. They involve a substantial variety of inorganic and organic compounds. The largest portion is organic arranged in hydrocarbon chains where, apart from C, H, O, and N, several other atoms are present, such as S, Fe, Ca, Al, Si, Zn, Na, K, Mg, Cl, heavy metals, etc. There is a vast literature on coal structure, and just a few reports are listed here [2–4]. On the other hand, these studies also show that many aspects of coal architecture are still ignored. The ratio between the amount of carbon and the amount of hydrogen in a solid fuel is such a fundamental aspect of its composition that is chosen to classify them. For instance, large differences in that ratio are found between biomasses and coals. Hence, C/H ratio can be seen as a vector of time. The principal reason is the decomposition process, which preferentially releases light gases, which contain a lower C/H ratio than the original biomass. For instance, biomass present C/H ratios around 10, whereas lignite is around 14 and bituminous coals average near 17, leaving anthracites with ratios in the vicinity of 30. Therefore, the inal stage of decomposition is almost pure carbon. On the other hand, one should be careful not to equate the fuel rank with its value for applications. For instance, a high volatile content is very important in pulverized combustion because it facilitates ignition. That aspect is easily understood because once the fuel is thrown into the combustion chamber, fast pyrolysis provides an atmosphere of fuel gases around the particle. Those gases are easily ignited, and the solid particle becomes surrounded by burning gases. Heat is transferred by radiation from the burning layer to the solid particle surface, which in turn increases its temperature and allows its ignition. There are relatively simple analyses to determine the basic fractions and atomic composition in a solid fuel. The simpler one is called proximate analysis, and the fractions determined through such an analysis are as follows: • Moisture, which is obtained by maintaining a sample of solid fuel within an inert atmosphere at 378 K and near ambient pressure until no variation of its mass is detected. The moisture content on a wet basis is given by the ratio between the mass lost by the sample and its original mass.

23

Solid Fuels

• Volatile content, which is found by maintaining the sample in an inert atmosphere at around 1300 K until no variation of its mass is detected. The volatile content at wet basis is given by the ratio between the lost mass and the original mass (before drying) of the sample. • Fixed carbon fraction, which is revealed by reacting the devolatilized sample with oxygen until no mass variation is detected. The ixed carbon content at wet basis is expressed by the ratio between the mass lost during the combustion and the sample original mass. • Ash content at wet basis is given by the ratio between residual mass from combustion and the original mass of the sample. Laboratories usually provide the ultimate analysis, which allows for inding the mass fractions of elements in the solid carbonaceous. The typical chemical species are carbon, hydrogen, nitrogen, oxygen, and sulfur. Ash, which is mostly a mixture of oxides of Fe, Zn, K, Na, Al, Ca, Mg, and Si, is included as a single fraction, and the results are usually expressed on a dry basis. A description of the methods used for such analysis is beyond the scope of the present book, but it can be found in several publications [2–7]. Several works have been dedicated to rank coals on the basis of composition. Among the works referred to most is that of Averit [8]. However, a more recent classiication is presented by Hensel [9] and is summarized below: 1. Anthracite, with typical volatile (wt., daf) content between 1.8% and 10%, carbon from 91% to 94.4%, carbon C/H (elementary carbon and hydrogen ratio) from 23.4 to 46, and combustion enthalpy between 34.4 and 35.7 MJ/kg. 2. Bituminous, with typical volatile content (daf) between 19% and 44.6%, carbon between 77.7% and 89.9%, C/H from 14.2 to 19.2, and combustion enthalpy between 32 and 36.3 MJ/kg. 3. Subbituminous with typical volatile content (daf) between 44.2% and 44.7%, carbon between 73.9% and 76%, C/H from 14.3 to 14.6, and combustion enthalpy between 29 and 30.7 MJ/kg. 4. Lignite with typical volatile content around 47%, carbon around 71%, C/H near 14.5, and combustion enthalpy around 28.3 MJ/kg.

TABLE 2.1 Typical Characteristics of a Subbituminous Coal Component Moisture Volatile Fixed carbon Ash High heating value (MJ/kg) (dry basis)

Mass %, Wet Basis 5.00 38.00 47.60 9.40 30.84

24

Solid Fuels Combustion and Gasification

TABLE 2.2 Typical Ultimate Analysis of a Subbituminous Coal Component C H O N S Ash

Mass %, Dry Basis 73.2 5.1 7.9 0.9 3.0 9.9

For the sake of an example, the following tables present the proximate and ultimate analyses of a typical subbituminous coal. Moisture and mainly ash contents play an important role regarding the quality of fuels. For instance, some coals may reach 40% in ash content, which leads to problems with the ignition of particles. Besides not being an the most utilized among solid fuels, biomass is attractive as an energy source because of the following characteristics: • It is a renewable source, and its application, as fuel, provides near zero overall CO2 emissions to the environment. The overall emissions depend on the methods of harvesting and transportation. • Most biomasses have present low ash content, which decreases problems related to residual disposal, equipment cleaning, and various other operational aspects. However, one inds exceptions, such in cases of rice husks and straws. • It allows lexibility regarding the location of power plants because dedicated forests may be set near the power plant, and not the other way around, as in most coal-based units. The main beneit to this strategy is savings in energy transmission. • Existing industrial units employing biomass for other main objectives can generate electric power for their own consumption or even as a subproduct. This might represent a considerable number of energy sources. For instance, a study [10] showed that signiicant increments in power supply could be obtained through the use of sugarcane bagasse excess* from sugaralcohol mills. On the other hand, compared with coal, biomass presents some disadvantages, such as high moisture, alkali (K, Na), and chlorine contents. Of course, drying demands part of the energy from combustion. If one intends to use the stream produced from gasiication for turbines, careful and costly cleaning is required to greatly reduce particle concentrations and sizes, as well as removing alkali and chlorine, because those contribute to corrosion and erosion of turbine blades. *

Excess here means the bagasse that is not already used for steam or power generation in the mills.

25

Solid Fuels

TABLE 2.3 Typical Characteristics of Sugarcane Bagasse with 20% Moisture Component

Mass %, Wet Basis

Moisture Volatile Fixed carbon Ash High heating value (MJ/kg) (dry basis)

20.00 65.24 12.11 2.65 19.04

TABLE 2.4 Typical Ultimate Analysis of Sugarcane Bagasse Component C H O N S Ash

Mass %, Dry Basis 49.66 5.71 41.08 0.21 0.03 3.31

The composition of biomasses covers a very wide range, and a case-by-case approach is necessary. With this in mind, though, an example is given below as a reference for the case of sugarcane bagasse (after a drying process). An excellent list of compositions and properties of biomass can be found in the literature [11].

2.4

THERMAL TREATMENT

As seen above, various phenomena or processes take place when a sample of solid fuel particles is heated. Of course, the sequence and characteristics of these processes depend on the physical and chemical properties of the fuel, as well as conditions of the atmosphere around the sample, such as temperature, pressure, and composition. In addition, the rate at which heating is imposed on the solid particles plays an important role in the characteristics of the fuel thermal decomposition. The following main steps occur during the sample heating: • In drying, liquid water leaves the particles in the form of steam. • In pyrolysis or devolatilization, gases such as H2, CH4, CO, CO2, H2O, etc., as well as tar—a complex mixture or larger molecules as detailed below—are released to the surroundings. In addition, important reactions and transformations take place inside the particle. • In gasiication, part of the particle solid components reacts with gases in the surrounding atmosphere. If the atmosphere contains oxygen, the gasiication process is called combustion. However, one should be aware that

26

Solid Fuels Combustion and Gasification

during combustion, all other reactions between the solid matrix and many gases take place as well. It is important to notice that the usual nomenclature prefers to not consider combustion and gasiication as part of thermal treatment, mainly because those two involve chemical reactions. However, that nomenclature is not precise, because pyrolysis also involves reactions. Methods to determine the kinetics of pyrolysis, as well as gas–solid reactions in general, are described in the literature [12–35]. Although they are not among the preoccupations of the present book, a few basic notions are presented in Appendix G. The objective is simply to illustrate the quantiication of pyrolysis, as well as combustion and gasiication processes, as presented in Chapters 8–10.

2.4.1

DRYING

Drying is the irst process to take place during the heating of a solid fuel. At atmospheric pressures, it occurs in the temperature range from ambient temperature to around 380 K. Despite its apparent simplicity, drying of a solid particle is a complex combination of events involving three phases: liquid water, vapor, and the porous solid through which the liquid and vapor migrate. In addition, ions of sodium and potassium, among others, are dissolved in the water inside the particle pores, and complex surface tension phenomena occur during the drying process. To better illustrate the various drying characteristics of process, let us refer to Figures 2.1 and 2.2. Throughout this text, numbers are set to specify chemical species. These are properly introduced in Chapter 8. For now, let us just set the number 700 to indicate moisture or liquid water in the midst of a solid phase. ρ700 A B

C

D 0 0

t

FIGURE 2.1 Typical moisture concentration as a function of time during drying of a porous particle.

27

Solid Fuels dρ700 dt

B

C

A

D 0 0

t

FIGURE 2.2 Typical drying rate as a function of time for a porous particle.

Suppose a porous solid particle were suddenly exposed to an ambiance with constant temperature and a concentration of water below the respective saturation condition.* In this situation, Figure 2.1 shows the typical evolution of moisture content in the particle against time, and Figure 2.2 presents the respective drying rates. The concentration of liquid water (or moisture) inside the solid particle is indicated by ρ700,S. The following periods or regions can be recognized: 1. The period from A to B represents the heating of the solid particle. Depending on the temperature of the gas atmosphere around the particle, two situations might occur: i. If that temperature is equal to or above the boiling point of water at the pressure of atmosphere around the particle, the liquid water at the particle surface tends to be at the saturation temperature. ii. If that temperature is below the boiling point of water at the pressure of atmosphere surrounding the particle, the liquid water at the surface tends to be at the wet-bulb temperature computed for the gas mixture near the surface. 2. The period from B to C represents the constant-rate drying region. In a simpliied view, liquid water is stored in the internal pores of the solid particle structure. When the solid is very wet, liquid water migrates to the surface by several mechanisms. This provides a surface continuously covered by a thin layer of liquid water. If the conditions allow, this water evaporates *

As saturation of water in air occurs when the relative humidity reaches 100%, a psychrometric chart can be used to determine the wet-bulb temperature, which would be the temperature at the waterevaporating surface. In general or for any mixture of gases as surrounding atmosphere, the molar fraction corresponding to the water saturation concentration would be given by the ratio between the partial pressure of water vapor (steam) and the ambient pressure. If the temperature equals the steam saturation temperature, boiling occurs, and the molar fraction of water vapor at the liquid surface is equal to 1. Steam tables, as well as good introductions to psychrometry, can be found in classic texts on thermodynamics [36–38].

28

Solid Fuels Combustion and Gasification

to the gas phase around the particle. Therefore, the rate at which water leaves the surface is somewhat independent of the nature of the solid particle. Hence, as long the surface remains wet and the ambient conditions constant, the rate of drying will be constant. This process is also known as irst drying period. 3. The period from C to D represents the region of decreasing drying rate. Here, free water is no longer available at the particle surface, and the wet boundary retracts to the interior of the particle. Thus, phase change from liquid to steam occurs inside the particle. To leave the particle, the steam has to travel through a layer of dried material surrounding the wet core. If external conditions remain constant, the drying rate decreases because of the increase in the thickness of the dried layer. Consequently, the resistance for mass and heat transfers between the wet interface and the particle surface increases. This process is also called second drying period. The rate of mass transfer from the particle surface to the ambiance is affected mainly by the following factors: • The temperature of the particle, especially of the water liquid-vapor interface • The rate of heat transfer between ambiance and particle • The water vapor concentration in the surrounding gas layer If only pure water is present—i.e., no ions are involved—the partial pressure of the water vapor at the liquid-vapor interface is equal to the steam saturation pressure at the temperature of that interface. This is established, no matter whether it is in the irst or second drying period. The gas layer just above the water liquid surface contains water vapor, and in order to be transferred through to the gas mixture, either outside or inside the particle, a concentration gradient of water should exist. Therefore, the lower the concentration of water in the gas mixture, the faster the mass transfer. The process stops when the concentration of water vapor in the gas mixture reaches the saturation value. However, the amount of water needed to provide saturation conditions in a gas mixture is higher for higher temperatures. This shows how important the temperature of the gas phase is for the rate of drying. Another aspect is the time taken by each drying period, and as expected, these times depend on the conditions of the surrounding atmosphere, as well as the properties of the porous particles. Nevertheless, as the second period involves larger masstransfer resistances than the irst period, it is reasonable that former takes longer than the latter. This is especially true for increasing temperatures, because the rate of the diffusion process during the second period is not increased as much as the rate of evaporation of freely available liquid water at the surface. As a wet particle is thrown into a combustor, the following sequence of events usually takes place: a) Fast heating of the particle occurs. Therefore, the region from A to B (Figures 2.1 and 2.2) occurs in a very short time, which is considered instantaneous by most models.

Solid Fuels

29

b) Constant-rate drying is also very fast because of the usually relatively large differences in temperature and water concentration between particle surface and surrounding gas. Hence, very high heat and mass transfers take place. It is very reasonable to assume a very fast irst drying period. c) The rate of drying decreases. This decrease starts almost immediately after the injection of particles into the furnace. Of course, the time taken for the second drying period decreases with the increase of temperature in the gas around the particle.

2.4.2

DEVOLATILIZATION

Devolatilization, or pyrolysis, is a very complex process and involves several reactions, as well as heat and mass transfers, resulting in the release of mixtures of organic and inorganic gases and liquids from the particle into the surrounding atmosphere. This release is set by the increase of particle temperature. Because of that complexity, pyrolysis of coal, biomass, and other carbonaceous fuels is among the most studied subjects [39–124].* Usually, the term pyrolysis is reserved for the devolatilization process when an inert atmosphere surrounds the particle. The term hydropyrolysis is used for devolatilization occurring in a hydrogen atmosphere. Three main fractions are produced during pyrolysis of coal or biomass: • Light gases, including H2, CO, CO2, H2O, CH4, etc. • Tar, composed of relatively heavy organic and inorganic molecules that escape the solid matrix as gases and liquid in form of mist • Char, which is the remaining solid residue As we have seen, for several coal ranks and almost all biomass, volatiles represent a signiicant portion of the carbonaceous fuels. According to the above description, volatiles are essential for the ignition of fuel particles. In the case of gasiiers, devolatilization also supplies fuel gases. Therefore, care should be taken not to assume that fuels with a higher content of carbon or higher C/H ratio are of good quality for applications. Coals with relatively high volatile content—such as subbituminous and bituminous coals—are much more useful for commercial combustion than those related to older deposits, such as anthracite. There are plenty of studies, experimental tests, proposed mechanisms, and applications for the pyrolysis process [13–41, 39–124].† Several aspects of those works will be discussed in Chapter 10. For the moment, it is interesting to note that: • Until the mid-1970s, most works applied or developed pyrolysis models based on single steps or just few steps, with little or no consideration of structures. *

†

This is not an exhaustive list of references the subject; however, they have been organized in chronological order to allow a historical view of developments in pyrolysis modeling. As previously noted, an exhaustive literature review is not among the preoccupations of the present book, and much more information can be found in the literature.

30

Solid Fuels Combustion and Gasification

• Because of the introduction of more sophisticated analysis techniques, such as Thermogravimetric Fourier-Transform-Infrared spectroscopy (TG-FTIR), better insights on the structure of coal and biomass were made possible, leading to more elaborate and more reliable models. A jump in these development occurred in the 1980s, and a mechanism for devolatilization became generally accepted, with the following main steps: 1. Heating leads to expansion and partial release of small amount of gases trapped in the particle pores. 2. Cracking or depolymerization of large organic molecules of the coal or biomass forms smaller structures, called metaplast. 3. Repolymerization by cross-linking of metaplast molecules occurs. 4. Gases and liquid components migrate to the surface. Parts of these are carried by the gas stream described above. 5. During that trip to the surface, part of the liquid component is cracked to gases that escape to the particle, and another part suffers coking and stays within the solid structure. In addition, reactions occur between the escaping gases and the solid and liquid components within the particle pores and surface, leading to more gases. Supplementing the above, the following points became established: • The release of many individual gas species does not occur in a single stage. Some of them present several peaks of releasing rates. • Devolatilization yields a wide variety of components, and hydrogen is required for molecular links in several of them. Therefore, devolatilization is a process that depletes hydrogen from the original carbonaceous matrix, and the remaining char becomes closer to graphite (almost pure carbon) than the original fuel. This is called graphitization. That is why some industrial processes try to enhance the volatile yield by promoting pyrolysis under a hydrogen atmosphere, or hydropyrolysis. One application is retorting of shale within an atmosphere of hydrogen, which is also called hydroretorting. Actually, the devolatilization process can be seen as an artiicial method to accelerate the slow natural decomposition of organic matter, which increases the C/H ratio in the fuel. • The 1990s were marked by the development of sophisticated versions of the former models. Some investigations were also devoted to determining the compositions of tars. For instance, the detailed work at Electric Power Research Institute (EPRI) [96] describes the composition of tar from coals.* Such information was used by several researchers, such as Mahjoub et al. [114]. It was veriied [119] that tar from vitrinites is composed mainly of mono- and diaromatic systems with a preponderance of phenol compounds. In the case of biomass, the composition of tar revealed [33, 102, 103] the *

EPRI [96] reports the following main components: naphthalene, acenaphtylene, acenaphtene, luorene, phenanthrene, anthracene, luoranthrene, pyrene, benzo(a)anthracene, chrysene, benzene, toluene, ethyl-benzene, xylene, phenol, cresols, and 2-4-dimethylphenol.

31

Solid Fuels

presence of a long list of components,* of which the most important are benzene, toluene, phenol, and naphthalene. Other works on this area are noted in Chapter 10. • The recent trend has been on widening the range of models and applications to solid fuels other than coals, especially biomasses. A signiicant portion of the research has also concentrated on emissions of pollutant gases from pyrolysis, mainly NOx. Finally, workers [120] have explored relatively new techniques, such as neural networks, to verify and relate information from several sources. The characteristics, composition, and quantities of chemical species released from devolatilization depend on a large amount of factors. The more important are the following: • Composition and structure of the original carbonaceous • Temperature, pressure, and composition of the atmosphere involving the particles • Heating rate imposed upon the carbonaceous particles The main signiicant aspects of these inluences are discussed below. 2.4.2.1 Composition and Structure of Original Carbonaceous As noted, the composition and structure of a solid fuel are very important in the distribution of components from coal and other solid fuels. Several works [43, 56–58, 63, 65, 66, 69, 70, 74, 76–82] show that the amounts of gases released can be correlated with original fuel compositions. However, on the basis of previous studies, Solomon and Hamblen [63] conirmed that the kinetics of pyrolysis is relatively insensitive regarding coal rank. 2.4.2.2 Temperature and Heating Rates Usually, the devolatilization starts when the carbonaceous solid reaches temperatures just above the drying, or as low as 390 K [125]. The temperatures used for coal or biomass may reach 1300 K. Early research veriied that increases in the temperature or heating rate led to increases in the yield of volatile products [23, 24, 30, 40, 42, 43, 45, 49, 126–128]. For instance, Gregory and Littlejohn [43] studied the inluence of conditions on the total amount of volatile that could be extracted from the coal. They established a simple correlation for the mass low of volatile released given by the following equation: FV = Fp,I ,daf (w V,daf − L '− L '') *

(2.3)

According to Yu et al. [33], the main components of tar at 1073 K are phenol, o-cresol, m-cresol, p-cresol, xylenol, benzene, toluene, p-xylene, o-xylene, indene, naphtalene, 2-methylnaphthalene, 1-methylnaphthalene, biphenyl, acenaphthylene, luorene, phenanthrene, anthracene, luoranthene, and pyrene. Brage et al. [102] list the following as well: o-ethylphenol, pyridine, 2-picoline, 3-picoline, 2-vinylpyridine, quinoline, isoquinoline, and 2-methylquinoline.

32

Solid Fuels Combustion and Gasification

where wV,daf is the mass fraction of volatiles (dry, ash-free basis) in the original coal. This value can be determined by standard proximate analysis. The index I indicates the entering conditions of the particle into the equipment, and L ' = 0.01 exp[26.41 − 3.961 ln(TS − 273.15) + 1.15w V ,daf ]

(2.4)

L '' = 20(w V ,daf − 0.109).

(2.5)

Of course, these correlations do not take into account various factors, such as particle heating rate. In addition, they were limited to a narrow range of coals. Yu et al. [33] veriied that in the range of usual industrial equipment operation (i.e., above 1000 K), the mass fraction of gas from wood devolatilization is greater than amount of tar, and this proportion tends to increase with temperature. This is due to secondary reactions, which are represented mainly by the cracking of tar molecules. In addition, the proportion of components in the tar varies considerably with temperature. As an example of compositions found by Yu et al., tar from wood presented concentrations of benzene from 29.2% (mole basis), for pyrolysis at 973 K, to 62.5% at 1173 K, whereas phenol dropped from 9.3% to 2.0%. It has been shown [33, 102] that higher temperatures favor the production of aromatics against phenols. The rates at which the volatiles are released from the solid carbonaceous fuel are not uniform. For instance, for several biomasses, the temperature of maximum release occurs between 600 and 700 K [103]. However, many components during biomass pyrolysis are released at 100 K below that range [108]. In the case of bituminous coals, the peak temperatures are slightly higher, or between 700 and 800 K, whereas peaks for lignite are around 700 K [97]. These factors are illustrated in Figure 2.3, which presents the typical shape of Derivative Thermo Gravimetry (DTG) or derivative against time for volatile release obtained through thermal gravimetric analysis (see Appendix G) under linear and slow increase of sample temperature. Several pyrolysis mechanisms have been proposed. In the case of coals, Jüntgen [125] presents the following correspondence between temperature and released species: 1. Below approximately 390 K, desorption occurs, releasing H2O, CH4, and N2. 2. Above approximately 520 K, destriation takes place, releasing aromatics and aliphatic as part of tar. 3. Above approximately 670 K, degradation of macromolecules occurs, releasing aromatics (tar) and gases, such as CH4, H2O, and C3H8. 4. Above 870 K, condensation of aromatic structures and decomposition of heterocyclic compounds occur, and more gases, such as H2, CO, and N2, are liberated. In the cases of biomass, Raveendran et al. [103] propose the following temperature ranges regarding devolatilization: 1. 2. 3. 4.

Zone I: 773 K, mainly lignin decomposition

33

Solid Fuels dρ1000 dt

0 0

t T (K) 450

500

600

700

900

FIGURE 2.3 Typical DTG or derivative of volatile release against time obtained from thermogravimetric (TG) analysis.

A heating rate is imposed during any thermal analysis, and that inluences the pyrolysis process. Indeed, it is common to classify the pyrolysis as slow, moderate, or fast. Slow pyrolysis assumes a heating rate of the sample below 10 K/s, whereas fast pyrolysis refers to rates above 103 K/s. This main reason for this classiication is related to the process of carbonaceous solid utilization. Fast pyrolysis occurs in almost all combustions of pulverized solid, where values reaching 105 or even 106 K/s can be found. Fluidized beds (bubbling and circulating) impose lower values, or around 102 to 104 K/s. Moderate to slow pyrolysis may happen in sections of movingor ixed-bed combustion or gasiications. Solomon and Hamblen [63] present comparisons of peak evolution for several gases from coal pyrolysis conducted at heating rates varying from 2.5 × 10 –3 K/s to 30 K/s. However, on average, it is possible to verify the following points: a) The irst peak for H2O release occurs around 730 K and the second peak around 800 K. b) The irst peak for CO2 takes place at temperatures from 660 K and 800 K, and the second peak happens in the range from 890 K to 1030 K. c) Tar production peaks from 690 K to 790 K. d) The irst peak for CH4 takes place from 770 K and 880 K, and the second peak occurs from around 880 K to 980 K. e) The irst peak for CO occurs from 750 K to 850 K, second peak occurs from 930 K to 1110 K, and a third peak is veriied around 1180 K. f) Hydrogen production happens at higher temperatures than most components, with a single peak from 970 K to 1040 K. g) Nitrogen also shows a single peak around 1180 K.

34

Solid Fuels Combustion and Gasification

After that, a model for species release considering the kinetics for each individual component with various linking strengths to the main coal structure was introduced [63]. With contribution of several researchers, this model evolved over the years, leading to the functional group/depolymerization-vaporization cross-linking, or FG-DVC model [81, 82]. Further details are discussed in Chapter 10. Afterward, it was veriied that release peaks also vary with the coal origin [91]. For instance, the temperatures of maximum release of various species during heating of various coals at rate of 30 K/s are summarized below: • • • • • • • • • •

SO2: 636 ± 20 K CO2: 712 ± 19 K CO: 743 ± 30 K Tar/aliphatic: 758 ± 22 K H2O: 798 ± 32 K CH4: 833 ± 7 K SO2 (second peak): 873 ± 16 K CO2 (second peak): 973 ± 45 K CO (second peak): 1046 ± 30 K H2O (second peak): 1080 ±28 K

Obviously, the above descriptions are approximations of pyrolysis processes because the compositions and amounts of products, as well as their releasing temperatures, are dictated by interlaced and complex factors. In addition to intrinsic kinetics, heat and mass transfers take place during pyrolysis. The time for the completion of devolatilization depends on all these processes combined. However, for very fast heating and maintaining the sample at constant temperature above 900 K, pyrolysis is completed before 7 minutes [54]. The behavior of biomass is not too different. 2.4.2.3 Pressure It has been veriied that increases in pressure decrease volatile yield [13]. Apart from inluences in secondary reactions, it seems that increases in pressure enhance the residence time of volatile matter in the char structure. This allows further polymerization, which augments the portion of blocked pores, therefore explaining the decrease in volatile yield. In case of high-volatile coals, such a process also enhances particle swelling [1]. In addition, Gadiou et al. [1] veriied that higher pressures enhance the H/C ratio of released volatile, leading to further graphitization of the remaining char. The work developed by Gadiou and collaborators conirmed earlier indings that the remaining char porosity rises with pressure, but its reactivity decreases. 2.4.2.4 Composition of Surrounding Atmosphere Pyrolysis is conducted in an inert atmosphere because the volatile yield can be affected when a reacting atmosphere is present. For instance, devolatilization conducted in a hydrogen atmosphere (or hydropyrolysis) leads to increases in volatile yield. Apart from what has been briely discussed above, it is important to notice that the lack of hydrogen results in products with longer chains with lower mobility

Solid Fuels

35

toward the particle surface. These heavier polymers tend to block the fuel particle pores, preventing further escape of lighter molecules. Of course, all combustion or gasiication processes involve several reacting gases. Therefore, one should be careful during the application of pyrolysis models for real industrial processes. Most of the models neglect the cross-inluence and assume that pyrolysis kinetics are not affected by the surrounding atmosphere. In some cases, this approximation seems acceptable. Nonetheless, it is important to be aware that this may be a source of deviations between simulations and real operations.

2.5

GASIFICATION AND COMBUSTION

The modeling of processes and phenomena related to gasiication and combustion of solids is the main preoccupation of the present book. For now, the objective is just to introduce a few fundamental deinitions to allow better understanding of equipment operations, as shown in the following chapters. It is also important to understand that, in general terms, gasiication is the transformation of solid fuel components into gases. This is usually accomplished by thermal treatments or chemical reactions, or by a combination of both. Therefore, devolatilization is part of gasiication process, as well as combustion or the reaction of carbonaceous fuel and oxygen. However, in the usual context of thermal sciences, gasiication reactions are the ones taking place between char and gases other than oxygen. In addition, one should be aware that in a real process, devolatilization, gasiication, and combustion reactions might occur simultaneously. Actually, it is almost impossible to have combustion without other reactions typical of gasiication because gases formed during combustion also react with the solid frame. For instance, hydrogen is oxidized to water during combustion, which in turn reacts with carbon. There are an immense number of gasiication reactions. The most important are the following: • Carbonaceous solid and water (in a simpliied notation: C + H2O = CO + H2); this is one of the most important reactions for the production of fuel gases • Carbonaceous solid and carbon dioxide, or simply C + CO2 = 2CO • Carbonaceous solid and hydrogen, or simply C + 2H2 = CH4 Of course, these are simpliications, and other components would be involved as reactant and products since any usual carbonaceous fuel contains H, N, O, S, etc. A more detailed representation of those reactions is presented in Chapter 8. As has been seen, many species resulting from gasiication reactions are at their reduced or less oxidized forms, such as CO instead CO2, H2 instead H2O, H2S instead SO2, and NH3 (or even HCN) instead of NO or other oxides. As a rough rule, gasiication reactions are endothermic. That is why in the great majority of processes, the energy to promote the gasiication reactions comes from the partial combustion of the solid fuel. The contact between the solid fuel particles and gases can be obtained through several means, the most important of which are moving and luidized beds, as well as suspensions.

36

Solid Fuels Combustion and Gasification

Combustion is a gasiication process in which oxygen is present among the gases in contact with the fuel. The main combustion reaction takes place between carbon and oxygen, leading to carbon monoxide and dioxide. Actually, as will be shown in Chapter 8, for temperatures typically found in combustion processes, practically speaking, only carbon monoxide is produced from that reaction. Carbon dioxide would be the typical product only at relatively low temperatures. Carbon monoxide oxidizes to dioxide if enough oxygen is still available. Several other gases are produced by the reaction of oxygen with the components of solid fuel. Usually, the gases from combustion are in their most stable form among the possible oxides. For instance, the oxidation of nitrogen in the fuel usually leads to NO, instead N2O and NO2. Finally, a general note on the usual notation for nitrogen oxides could be useful at this point. Nitrogen, as a component of the solid fuel, can be attacked by reacting gases such as O2 much more easily than the nitrogen molecule (N2) in the air. The literature distinguishes the NO from these two sources by calling the NO produced from fuelnitrogen oxidation thermal-NO, and the NO from air-nitrogen oxidation air-NO. As mentioned before, the formation of NOx is a complex subject, and possible reactions to predict its formation are shown in Chapter 8 and 9. This topic is important since nitrogen oxides are strong pollutants and responsible for part of the process that results in what is called acid rain, which is caused mainly by sulfuric and nitric acids formed in the atmosphere from sulfur and nitrogen oxides (SOx and NOx). These acids fall back to the earth’s surface, causing health and environmental damage.

2.6 2.6.1

EXERCISES PROBLEM 2.1* Demonstrate Equation 2.2.

REFERENCES 1. Gadiou, R., Bouzidi, Y., and Prado, G., The devolatilization of milimetre sized coal particles at high heating rate: the inluence of pressure on the structure and reactivity of the char, Fuel, 81, 2121–2130, 2002. 2. Smith, K.L., Smoot, L.D., Fletcher T.H., and Pugmire, R.J., The Structure and Reaction Processes of Coal, Plenum Press, New York, 1994. 3. Smoot, L.D., General characteristics of coal, in Pulverized-Coal Combustion and Gasiication, Smoot, L.D., and Pratt, D.T., Eds., Plenum Press, New York, 1979, pp. 123–132. 4. Ergun S., Coal classiication and characterization, in Coal Conversion Technology, Wen, C.Y., and Lee, E.S., Eds., Addison-Wesley, Reading, MA, 1979, pp. 1–53. 5. Smith, K.L., and Smoot, L.D., Characteristics of commonly-used U.S. coals—towards a set of standard research coals, Prog. Energy Combust. Sci., 16, 1–53, 1990. 6. Miles, T.R., Miles, T.R., Jr., Baxter, L.L., Bryers, R.W., Jenkins, B.M., and Oden, L.L., Alkali Depositions Found in Biomass Power Plants: A Preliminary Investigation of Their Extend and Nature, National Renewable Energy Laboratory, Golden, CO, 1995. 7. Jenkins, B.M., Bakker, R.R., and Wei, J.B., On the properties of washed straw, Biomass and Bioenergy, 10(4), 177–200, 1996.

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8. Averit, P., Coal Resources of the United States, U.S. Geological Survey Bulletin No. 1412, 1975. 9. Hensel, R.P., Coal: Classiication, Chemistry, and Combustion, Coal-Fired Industrial Boilers Workshop, Raleigh, NC, Fossil Power Systems, Combustion Engineering, Inc., Windsor, CT, 1981. 10. Hollanda, J.B., Frydman, I., Trinkenreich, J., Bertelli, L.G., Macedo, I., Rodrigues, L., Lorenz, K., Leal, M.R.L.V., Campos, R.M., Asti, A., Serfaty, M., Assad, L.S., Dastre, L.D., Spinelli, M.A.S., Ortiz, J.N.G., Pracchia, L.C., Lewis, A., Benitez, S., Silverstrin, C.R., Pereira, V., Simões Neto, J., and Cunha, M.L.M., A Study on Optimization of Electric Power Production from Sugar-Cane Mills (Estudo em Otimização da Produção de Energia Elétrica em Usinas Alcooleiras), Final Report (Eletrobras, Copersucar, BNDES, CESP, CPFL, Eletropaulo), November, Sao Paulo, Brazil, 1991. 11. Jenkins, B.M., Baxter, L.L., Miles, T.R., Jr., and Miles, T.R., Combustion properties of biomass, Fuel Processing Technology, 54, 17–46, 1998. 12. Carrasco, F., The evaluation of kinetic parameters from thermogravimetric data: comparison between established methods and the general analytical equation, Thermochimica Acta, 213, 115–134, 1993. 13. Anthony, D.B., and Howard, J.B., Coal devolatilization and hydrogasiication, AIChE J., 22(4), 625–656, 1976. 14. Stone, H.N., Batchelor, J.D., and Johnstone, H.F., Low temperature carbonisation rates in a luidised-bed, Ind. Eng. Chem., 46, 274, 1954. 15. Zielke, C.W., and Gorin, E., Kinetics of carbon gasiication, Ind. Eng. Chem., 47, 820, 1955. 16. Peters, W., and Bertling, H., Kinetics of the rapid degasiication of coals, Fuel, 44, 317, 1965. 17. Jones, W.I., The thermal decomposition of coal, J. Inst. Fuel, 37, 3, 1964. 18. Moseley, F., and Paterson, D., Rapid high-temperature hydrogenation of coal chars: part 1, J. Inst. Fuel, 38, 13–23, 1965. 19. Moseley, F., and Paterson, D., Rapid high-temperature hydrogenation of coal chars: part 2, J. Inst. Fuel, 38, 378–391, 1965. 20. Eddinger, R.T., Friedman, L.D., and Rau, E., Devolatilization of coal in a transport reactor, Fuel, 45, 245, 1966. 21. Howard., J.B., and Essenhigh, R.H., Pyrolysis of coal particles in pulverized fuel lames, Ind. Eng. Chem. Process Des. Dev., 6, 74, 1967. 22. Glenn, R.A., Donath, E.E., and Grace, J.R., Gasiication of coal under conditions simulating stage 2 of the BCR two-stage super pressure gasiier, in Fuel Gasiication, Advances in Chemistry Series, No. 69, American Chemical Society, Washington, DC, p. 253, 1967. 23. Kimber, G.M., and Gray, M.D., Rapid devolatilization of small coal particles, Combustion and Flame, 11, 360, 1967. 24. Kimber, G.M., and Gray, M.D., Measurements of Thermal Decomposition of Low and High Rank Non-Swelling Coals at M.H.D. Temperatures, British Coal Utilization Research Association, Document No. MHD 32, 1967. 25. Badzioch, S., and Hawksley, P.G.W., Kinetics of thermal decomposition of pulverized coal particles, Ind. Eng. Chem. Process Des. Dev., 9, 521, 1970. 26. Belt, R.J., Wilson, J.S., and Sebastian, J.J.S., Continuous rapid carbonisation of powdered coal by entrainment and response surface analysis of data, Fuel, 50, 381, 1971. 27. Belt, R.J., and Roder, M.M., Low-sulfur fuel by pressurized entrainment carbonisation of coal, American Chemical Society, Div. of Fuel Chem. Preprints, 17(2), 82, 1972. 28. Coates, R.L., Chen, C.L., and Pope, B.J., Coal devolatilization in a low pressure, low residence time entrained low reactor, in Coal Gasiication, Advances in Chemistry Series, No. 131, American Chemical Society, Washington, DC, 1974, p. 92.

38

Solid Fuels Combustion and Gasification

29. Johnson, J.L., Gasiication of Montana lignite in hydrogen and in helium during initial reaction stages, American Chemical Society, Div. of Fuel Chem. Preprints, 20(3), 61, 1975. 30. Shapatina, E.A., Kalyuzhnyi, V.A., and Chukhanov, Z.F., Technological utilization of fuel for energy, 1: Thermal treatments of fuels, 1960. (Paper reviewed by Badzoich, S., British Coal Utilization Research Association Monthly Bulletin, 25, 285, 1961.) 31. Moseley, F., and Paterson, D., Rapid high-temperature hydrogenation of bituminous coal, J. Inst. Fuel, 40, 523, 1967. 32. Feldmann, H.F., Simons, W.H., Mimn, J.A., and Hitshue, R.W., Reaction model of bituminous coal hydrogasiication in a dilute phase, American Chemical Society, Div. of Fuel Chem. Preprints, 14(4), 1, 1970. 33. Yu, Q., Brage, C., Chen, G., and Sjöström, K., Temperature impact on the formation of tar from biomass pyrolysis in a free-fall reactor, J. Analytical and Applied Pyrolysis, 40–41, 481–489, 1997. 34. Tsuge, S., Analytical pyrolysis—past, present and future, J. Analytical and Applied Pyrolysis, 32, 1–6, 1995. 35. Bridgwater, A.V., and Peacocke, G.V.C., Fast pyrolysis processes for biomass, Renewable and Sustainable Energy Reviews, 4, 1–73, 2000. 36. Moran, M.J., and Shapiro, H.N., Fundamentals of Thermodynamics, 3rd Ed., John Wiley, New York, 1996. 37. van Wylen, G.J., and Sonntag R.E., Fundamentals of Classical Thermodynamics, John Wiley, New York, 1973. 38. Kestin J., A Course in Thermodynamics, Vols. I and II, Hemisphere, New York, 1979. 39. van Krevelen, D.W., and Schuyer, J., Coal Sciences, Elsevier, Amsterdam, 1957. 40. Badzioch, S., Rapid and controlled decomposition of coal, British Coal Utilization Research Association Monthly Bulletin, 25(8), 285, 1961. 41. Pitt, G.J., The kinetics of the evolution of volatile products from coal, Fuel, 41, 267, 1962. 42. Loison, R., and Chauvin, R., Pyrolyse rapide du charbon, Chimie et Industrie, 91, 3, 269–274, 1964. 43. Gregory, D.R., and Littlejohn, R.F., Survey of numerical data on the thermal decomposition of coal, Coal Utilization Res. Assoc. Monthly Bull., 29(6), 173, 1965. 44. Eddinger, R.T., Friedman, L.D., and Rau E., Devolatilization of coal in a transport reactor, Fuel, 45, 245, 1966. 45. Wiser, W.H., Hill G.R., and Kertamus, N.J., Kinetic study of the pyrolysis of a highvolatile bituminous coal, Ind. Eng. Chem. Process Des. Dev., 6, 133, 1967. 46. Roberts, A.F., A review of kinetics data for the pyrolysis of wood and related substances, Combustion and Flame, 14, 261–272, 1970. 47. Badzioch, S., and Hawksley, P.G.W., Kinetics of thermal decomposition of pulverized coal particles, Ind. Eng. Chem. Process Des. Dev., 9, 521, 1970. 48. Fine, D.H., Slater, S.M., Saroim, A.F., and Williams, G.C., Nitrogen in coal as a source of nitrogen oxide emission from furnaces, Fuel, 53, 120–125, 1974. 49. Anthony, D.B., Howard, J.B., Hottel, H.C., and Meissner, H.P., Rapid devolatilization of pulverized coal, Fifteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, PA, 1975, 1303. 50. Cheong, P.H., PhD thesis, California Institute of Technology, Pasadena, CA, 1976. 51. Shaizadeh, F., and Chin, P.P.S., Thermal Deterioration of Wood, American Chemical Society Symposium Series, 43, 57, 1977. 52. Suuberg, E.M., Peters, W.A., and Howard, J.B., Product Compositions and Formation Kinetics in Rapid Pyrolysis of Pulverized Coal, Seventeenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, PA, 1979, 117–120.

Solid Fuels

39

53. Solomon, P.R., and Colket, M.B., An entrained low reactor with in situ FTIR analysis, Seventeenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, PA, 1979, 131. 54. Horton, M.D., Fast pyrolysis, in Pulverized-Coal Combustion and Gasiication, Smoot, L.D., and Pratt, D.T., Eds., Plenum Press, New York, 1979, 133–147. 55. Bradbury, A.G.W., Sakai, Y., and Shaizadeh, F., A kinetic model of pyrolysis of cellulose, Journal of Applied Polymer Science, 23, 3271–3280, 1979. 56. Gavalas, G.R., Coal Pyrolysis, Elsevier, Amsterdam, 1981. 57. Gavalas, G.R., Cheong, P.H., and Jain, R., Model of coal pyrolysis. 1. Qualitative development, Ind. Eng. Chem. Fundam., 20, 113–122, 1981. 58. Gavalas, G.R., Cheong, P.H., and Jain, R., Model of coal pyrolysis. 2. Quantitative formulation and results, Ind. Eng. Chem. Fundam., 20, 122–132, 1981. 59. Pillai, K.K., The inluence of coal type on devolatilization and combustion in luidized beds, J. Institute of Energy, Sept., 142–150, 1981. 60. Howard, D.F., Williams, A.P., and Howard, J.B., Mineral matter effects on the rapid pyrolysis and hydropyrolysis of a bituminous coal. 1. Effects on yields of char, tar and light gaseous volatiles, Fuel, 61, 155–160, 1982. 61. Fuller, E.L., Jr., Coal and Coal Products: Analytical Characterization Techniques, American Chemical Society, Washington DC, 1982. 62. Pillai, K.K., A schematic for coal devolatilization in luidized bed combustors, J. Institute of Energy, Sept., 132–133, 1982. 63. Solomon, P.R., and Hamblen, D.G., Finding order in coal pyrolysis kinetics, Prog. Energy Combustion Sci., 9, 323–361, 1983. 64. Antal, M.J., Jr., Effects of reactor severity on the gas-phase pyrolysis of cellulose- and kraft lignin-derived volatile matter, Ind. Eng. Chem. Prod. Res. Dev., 22(2), 366–375, 1983. 65. Solomon, P.R., and King, H.H., Tar evolution from coal and model polymers: Theory and experiments, Fuel, 63, 1302–1311, 1984. 66. Jüntgen, H., Review of the kinetics of pyrolysis and hydropyrolysis in relation to the chemical constitution of coal, Fuel, 63, 731–737, 1984. 67. Chan, W.R., Kelbon, M., and Krieger, B., Modelling and experimental veriication of physical and chemical processes during pyrolysis of a large biomass particle, Fuel, 64, 1505–1531, 1985. 68. Suuberg, E.M., Unger, P.E., and Lilly, W.D., Experimental study on mass transfer from pyrolysing coal particles, Fuel, 64, 956–962, 1985. 69. Nunn, T.R., Howard, J.B., Longwell, J.P., and Peters, W.A., Product compositions and kinetics in the rapid pyrolysis of sweet gum hardwood, Ind. Eng. Chem. Process Des. Dev., 24(3), 836–844, 1985. 70. Nunn, T.R., Howard, J.B., Longwell, J.P., and Peters, W.A., Product compositions and kinetics in the rapid pyrolysis of milled wood lignin, Ind. Eng. Chem. Process Des. Dev., 24(3), 844–852, 1985. 71. Suuberg, E.M., Lee, D., and Larsen, J.W., Temperature dependence of crosslinking processes in pyrolysis of coal, Fuel, 64, 1668–1671, 1985. 72. Simmons, G.M., and Gentry, M., Kinetic formation of CO, CO2, H2, and light hydrocarbon gases from cellulose pyrolysis, J. Analytical and Applied Pyrolysis, 10, 129–138, 1986. 73. Squir, K.R., Solomon, P.R., Carangelo, R.M., and DiTaranto, M.B., Tar evolution from coal and model polymers: 2. The effects of aromatic ring sizes and donatable hydrogens, Fuel, 65, 833–843, 1986. 74. Solomon, P.R., Serio, M.A., Carangelo, R.M., and Markham, J.R., Very rapid coal pyrolysis, Fuel, 65, 182–193, 1986. 75. Fong, W.S., Peters, W.A., and Howard, J.B., Kinetics of generation and destruction of pyridine extractables in a rapidly pyrolysing bituminous coal, Fuel, 65, 251–254, 1986.

40

Solid Fuels Combustion and Gasification

76. Solomon, P.R., Beér, J.M., and Longwell, J.P., Fundamentals of coal conversion and relation to coal pyrolysis, Energy, 12(8–9), 837–862, 1987. 77. Serio, M.A., Hamblen, D.G., Markham, J.R., and Solomon, P.R., Kinetics of volatile evolution in coal pyrolysis: experiment and theory, Energy & Fuels, 1, 138–152, 1987. 78. Xu, W.C., and Tomita, A., Effect of coal type on the lash pyrolysis of various coals, Fuel, 66, 627–631, 1987. 79. Niksa, S., and Kerstein, A.R., On the role of macromolecular coniguration in rapid coal devolatilization, Fuel, 66, 1389–1399, 1987. 80. Niksa, S., Rapid coal devolatilization as an equilibrium lash distillation, AIChE J., 34, 790–802, 1988. 81. Solomon, P.R., Hamblen, D.G., Carangelo, R.M., Serio, M.A., and Deshpande, G.V., General model of coal devolatilization, Energy & Fuels, 2, 405–422, 1988. 82. Solomon, P.R., Hamblen, D.G., Carangelo, R.M., Serio, M.A., and Deshpande, G.V., Models of tar formation during coal devolatilization, Combustion and Flame, 71, 137– 146, 1988. 83. Alves, S.S., and Figueiredo, J.L., Pyrolysis kinetics of lignocellulosic materials by multistage isothermal thermogravimetry, J. Analytical and Applied Pyrolysis, 13, 123–134, 1988. 84. Bruinsma, O.S.L., Geerstsma, R.S., Bank, P., and Moulijin, J.A., Gas phase pyrolysis of coal-related aromatic compounds in a coiled tube low reactor: 1. Benzene and derivatives, Fuel, 67, 327–333, 1988. 85. Uden, A.G., Berruti, F., and Scott, D.S., A kinetic model for the production of liquids from the lash pyrolysis of biomass, Chem. Eng. Comm., 65, 207–221, 1988. 86. Boroson, M.L., Howard, J.B., Longwell, J.P., and Peters, W.A., Product yields and kinetics from the vapor phase cracking of wood pyrolysis tars, AIChE J., 35(1), 120–128, 1989. 87. Yun, Y., Maswadeh, W., Meuzellaar, H.L.C., Simmleit, N., and Schulten, H.R., Estimation of coal devolatilization modelling parameters from thermogravimetric and time-resolved soft ionisation mass spectrometric data, American Chemical Society, Div. Fuel Chem., 34(4), 1308–1316, 1989. 88. Solomon, P.R., Best, P.E., Yu, Z.Z., and Deshpande, G.V., A macromolecular network model for coal luidity, Preprint, American Chemical Society, Div. Fuel Chem., 34(3), 895–906, 1989. 89. Grant, D.M., Pugmire, R.J., Fletcher, T.H., and Kerstein, A.R., Chemical model of coal devolatilization using percolation lattice statistics, Energy & Fuels, 3, 175–186, 1989. 90. Solomon, P.R., Hamblen, D.G., Yu, Z.Z., and Serio, M.A., Network models of coal thermal decomposition, Fuel, 69, 754–763, 1990. 91. Solomon, P.R., Hamblen, D.G., Serio, M.A., Smoot, L.D., and Brewster, B.S., Measurement and Modeling of Advanced Coal Conversion Process, Fourteenth Quaterly Report for U.S. Department of Energy, Contract No. DE-AC21-86MC23075, Morgantown Energy Technology Center, Morgantown, WV, Advanced Fuel Research, Inc., East Hartford, CT, Brigham Young University, Provo, UT, 1990. 92. Fletcher, T.H., Kerstein, A.R., Pugmire, R.J., and Grant, D.M., Chemical percolation model for devolatilization. II: Temperature and heating rate effects on product yields, Energy & Fuels, 3, 54–60, 1990. 93. Niksa, S., Flashchain theory for rapid coal devolatilization kinetics. 1: Formulation, Energy & Fuels, 5, 647–665, 1991. 94. Niksa, S., Flashchain theory for rapid coal devolatilization kinetics. 2: Impact of operating behavior of various coals; 3: Modeling the behavior of various coals, Energy & Fuels, 5, 665–683, 1991. 95. Fletcher, T.H., Kerstein, A.R., Pugmire, R.J., and Grant, D.M., A chemical percolation model for devolatilization. III: Direct use of 13C NMR data to predict effects of coal type, Energy & Fuels, 6, 414–431, 1992.

Solid Fuels

41

96. EPRI, Chemical and Physical Characteristics of Tar Samples from Selected Manufactured Gas Plant Sites. Project No. 2879-12. Final report. Atlantic Environmental Services Inc., Colchester, CT, 1993. 97. Garcia, A.N., Estudio Termoquimico y Cinetico de la Pirolisis de Residuos Solidos Urbanos, PhD thesis, University of Alicante, Alicante, Spain, 1993. 98. Solomon, P.R., Fletcher, T.H., and Pugmire, R.J., Progress in coal pyrolysis, Fuel, 72, 587–597, 1993. 99. Font, R., Marcilla, A., Devesa, J., and Verdu, E., Kinetic study of the lash pyrolysis of almond shells in a luidized bed reactor at high temperatures, J. Analytical and Applied Pyrolysis, 27(2), 245–273, 1993. 100. Kocaefe, D., Charette, A., and Catonguay, L., Green coke pyrolysis; investigation of simultaneous changes in gas and solid phases, Fuel, 74(6), 791–799, 1995. 101. Font, R., and Williams, P.T., Pyrolysis of biomass with constant heating rate: inluence of the operating conditions, Thermochimica Acta, 205, 109–123, 1995. 102. Brage, C., Yu, Q., and Sjöström, K., Characteristics of evolution of tar from wood pyrolysis in a ixed-bed reactor, Fuel, 75(2), 213–219, 1996. 103. Raveendran, K., Ganesh, A., and Khilar, K.C., Pyrolysis characteristics of biomass and biomass components, Fuel, 75(8), 987–998, 1996. 104. Caballero, J.A., Font, R., and Marcilla, A., Kinetic study of the secondary thermal decomposition of kraft lignin, J. Analytical and Applied Pyrolysis, 38, 131–152, 1996. 105. Ghetti, P., Ricca, L., and Angelini, L., Thermal analysis of biomass and corresponding pyrolysis products, Fuel, 75(5), 565–573, 1996. 106. Pindoria, R.V., Megaritis, A., Messenböck, R.C., Dugwell, D.R., and Kandiyoti, R., Comparison of the pyrolysis and gasiication of biomass: effect of reacting gas atmosphere and pressure on Eucalyptus wood, Fuel, 77(11), 1247–1251, 1998. 107. Della Rocca, P.A., Cerella, E.G., Bonelli, P.R., and Cukierman, A.L., Pyrolysis of hardwoods residues: on kinetics and chars characterization, Biomass and Bioenergy, 16, 79–88, 1999. 108. Órfão, J.J.M., Antunes, F.J.A., and Figueiredo, J.L., Pyrolysis kinetics of lignocellolosic materials—three independent reaction model, Fuel, 78, 349–358, 1999. 109. Jones, J.M., Pourkashanian, M., Rena, C.D., and Williams, A., Modelling the relationship of coal structure to char porosity, Fuel, 78, 1737–1744, 1999. 110. Heidenreich, C.A., Yan, H.M., and Zhang, D.K., Mathematical modeling of pyrolysis of large coal particles—estimation of kinetics parameters for methane evolution, Fuel, 78, 557–566, 1999. 111. Bridgwater, A.V., Meier, D., Radlein, D., An overview of fat pyrolysis of biomass, Organic Geochemistry, 30, 1479–1493, 1999. 112. Donskoi, E., and McElwain, D.L.S., Optimization of coal pyrolysis modeling, Combustion and Flame, 122, 359–367, 2000. 113. Wiktorsson, L.P., and Wanzl, W., Kinetics parameters for coal pyrolysis at low and high heating rates—a comparison of data from different laboratory equipment, Fuel, 79, 701–716, 2000. 114. Mahjoub, B., Jayr, E., Bayard, R., and Gourdon, R., Phase partition of organic pollutants between coal tar and water under variable experimental conditions, Water Research, 34(14), 3551–3560, 2000. 115. Ross, D.P., Heidenreich, C.A., and Zhang, D.K., Devolatilization times of coal particles in a luidized-bed, Fuel, 79, 873–883, 2000. 116. Alonso, M.J.G., Borrego, A.G., Alvarez, D., Parra, J.B., and Menendez, E., Inluence of pyrolysis temperature on char optical texture and reactivity, J. Analytical and Applied Pyrolysis, 58–59, 887–909, 2001. 117. Bonelli, P.R., Della Rocca, P.A., Cerella, E.G., and Cukierman, A.L., Effect of pyrolysis temperature on composition, surface properties and thermal degradation rates of Brazil Nut shells, Bioresource Technology, 76, 15–22, 2001.

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118. van Dyk, J.C., Development of an alternative laboratory method to determine thermal fragmentation of coal sources during pyrolysis in gasiication process, Fuel, 80, 245– 249, 2001. 119. Iglesias, M.J., Cuesta, M.J., and Suárez-Ruiz, I., Structure of tars derived from lowtemperature pyrolysis of pure vitrinites: inluence of rank and composition of vitrinites, J. Analytical and Applied Pyrolysis, 58–59, 255–284, 2001. 120. Carsky, M., and Kuwornoo, D.K., Neural network modeling of coal pyrolysis, Fuel, 80, 1021–1027, 2001. 121. Stenseng, M., Jensen, A., and Dam-Johansen, K., Investigation of biomass pyrolysis by thermogravimetric analysis and differential scanning calorimetry, J. Analytical and Applied Pyrolysis, 58–59, 765–780, 2001. 122. Das, T.K., Evolution characteristics of gases during pyrolysis of maceral concentrates of Russian coking coals, Fuel, 80, 489–500, 2001. 123. Arenillas, A., Rubiera, F., Pevida, C., and Pis, J.J., A comparison of different methods for predicting coal devolatilization kinetics, J. Analytical and Applied Pyrolysis, 58–59, 685–701, 2001. 124. Demirbas, A., Gaseous products from biomass by pyrolysis and gasiication: effects of catalyst on hydrogen yield, Energy Conservation and Management, 43, 897–909, 2002. 125. Jüntgen, H., Coal characterization in relation to coal combustion, in Fundamentals of the Physical-Chemistry of Pulverized Coal Combustion, Lahaye, J., and Prado, G., Eds., Martinus Nijhoff, Dordrecht, Netherlands, 1987, pp. 4–59. 126. Dryden, I.G.C., Chemistry of coal and its relation to coal carbonization, J. Inst. Fuel, 30, 193, 1957. 126. Jones, W.I., The thermal decomposition of coal, J. Inst. Fuel, 37, 3, 1964. 128. Peters, W., and Bertling, H., Kinetics of the rapid degasiication of coals, Fuel, 44, 317, 1965.

3 Equipment and Processes CONTENTS 3.1 3.2 3.3

Introduction .................................................................................................... 43 Elements of Gas–Solid Systems ..................................................................... 43 Moving Bed .................................................................................................... 48 3.3.1 Applications of Moving Beds ............................................................. 52 3.4 Fluidized Bed.................................................................................................. 53 3.4.1 Bubbling Fluidized Beds .................................................................... 53 3.4.2 Circulating Fluidized Beds ................................................................. 59 3.4.2.1 A Few Aspects of Circulating Bed Designing ..................... 61 3.4.3 Applications of Fluidized Beds .......................................................... 63 3.4.4 Comparisons between Fluidized-Bed and Moving-Bed Processes .....66 3.4.5 Atmospheric and Pressurized Fluidized-Bed Gasiiers......................66 3.4.6 Gas Cleaning ...................................................................................... 70 3.5 Suspension or Pneumatic Transport ............................................................... 71 3.5.1 Applications of Suspensions ............................................................... 73 3.6 A Few Basic Aspects of Fuels ........................................................................ 74 References ................................................................................................................ 76

3.1

INTRODUCTION

This chapter presents the basic characteristics of equipment usually applied for solid fuel combustion and gasiication. Although no mathematical models are shown at this point, the text below should facilitate discussion related to modeling and simulation of that equipment. The following fuel combustion and gasiication techniques will be described: • Moving bed • Fluidized bed • Pneumatic transport Other units, such as cyclonic and rotary reactors (or rotary furnaces), are also used industrially [1, 2] and may be classiied under one of the above methods. For instance, the irst can be seen as pneumatic transport with swirl and the second as moving bed with angular movement.

3.2

ELEMENTS OF GAS–SOLID SYSTEMS

The subject of gas–solid systems is very broad and includes a large variety of processes in which gas and solid particles are in contact and reacting. Examples of such processes 43

44

Solid Fuels Combustion and Gasification

are combustion of carbonaceous solids, such as coal or biomass; gasiication of these solids; drying; carbonization; sulfur dioxide absorption using limestone or dolomite* in stack gas cleaning; oil shale retorting; and catalytic cracking. The equipment in which gas–solid contact takes place needs to be speciied. For instance, the pulverized solid particles can be injected into a chamber illed with reacting gases. If they have relatively small density or sizes, the particles may be kept in suspension while reacting with the ascending gas low. Hence, suspensions can be thought of as pneumatic transport processes, but pulverized combustion and gasiication is the term commonly used for that class of processes [3–5]. Conventional combustion chambers or furnaces of boilers found in industry operate by such a technique. Among the most commonly used methods of promoting gas–solid reactions is to force gas streams through beds of particles. If the particles remain ixed in their positions, the equipment is called a ixed-bed reactor. If the particles are allowed to move without detaching from each other, the process is classiied as moving bed. Fluidized-bed and pneumatic transport can be established when the momentum transfer with the gas stream is such that particles detach from each other. Figures 3.1 and 3.2 demonstrate a simple and precise method of classifying beds of particles in contact with lowing gas streams. The basic parameters are the luid supericial velocity and its pressure drop in the bed. The supericial velocity U is Gases & particles

Gases & particles

Gases & particles

Gases & particles

Gases & particles

Gases & particles

Gases in

Gases in

Gases in

Gases in

Gases in

Gases in

Fixed bed (a)

Minimum fluidization (b)

Bubbling fluid. bed (c)

Turbulent fluid. bed (d)

Fast fluid. bed (e)

Pneumatic transport (f )

z

FIGURE 3.1 Various situations for beds of particles.

*

From now on, sulfur solid absorbent, such as limestone or dolomite, will simply be called absorbent.

45

Equipment and Processes Pressure loss Due to accommodation

Fixed bed

Bubbling fluidized bed Superficial velocity

0 Minimum fluidization

Minimum bubbling

FIGURE 3.2 Supericial velocity against pressure drop.

deined as the average velocity of the luid (gas or liquid) in the axial or vertical direction, measured as if no particle or packing is present in the equipment. Of course, if the temperature and pressure of the injected luid remain constant, the supericial velocity is proportional to its low rate. Figure 3.1a illustrates a region where the intensity of momentum exchange between luid and solids is not enough to provoke noticeable movement of the particles. This situation corresponds to region for velocities below the minimum luidization one, or Umf. Figure 3.2 shows that before reaching Umf, the pressure drop in the luid—measured across the bed—increases almost linearly with its supericial velocity [6–11]. Fixed- or moving-bed equipment operates within this region of supericial velocity. Actually, most of the so-called ixed beds operate as moving beds because particles are continuously fed into the bed and move downward, as illustrated in Figures 3.3 and 3.4. If the supericial velocity of the gas is gradually increased, there will be a situation when the bed starts expanding. Neighboring particles detach from each other, and a seemingly random movement starts. Actually, the movement is a combination of circular paths (see Figures 13.2 and 14.1). The supericial velocity at this point is called minimum luidization velocity, or Umf (Figure 3.1b). However, minimum luidization condition is often achieved when the supericial velocity is slightly higher than the minimum luidization velocity. As detailed in Chapter 4, this is so because some classes of powders present resistance to movement due to accommodation among particles or to electrostatic charges. Of course, such resistance does not occur when the process is reversed, i.e., departing from higher velocities than Umf toward a minimum luidization condition. Therefore, a sort of hysteretic effect is observed in the process. Any additional increase in the mass low of injected luid through the bottom will provoke the appearance of bubbles that will rise through the bed, as illustrated in Figure 3.1c. Actually, there is a difference between the minimum luidization velocity and the velocity where bubbling starts. This usually small difference occurs only in the case of ine powders, and the velocity at which the bubbling process starts is called minimum bubbling velocity [6–7, 10–11]. A classiication of solid

46

Solid Fuels Combustion and Gasification Carbonaceous particles

Exiting top stream = gas + carried particles

Drying Devolatilization

Insulation Gasification

Combustion Ash layer

Ash

Grate Gasification agents

FIGURE 3.3 Schematic view of updraft moving-bed gasiiers.

Carbonaceous particles

Insulation Drying Devolatilization Gasification agents

Combustion

Gasification Exiting gas stream

Ash layer

Grate

Ash

FIGURE 3.4 Schematic view of downdraft moving-bed gasiiers.

Equipment and Processes

47

particle types based on their quality, as well as size and density, has been developed by Geldart [7] and allows mapping of luidization regimes. Quantiication of several situations is presented in Chapter 4. Within the luidization region, the pressure loss per unit of bed length remains almost constant. This situation covers a relatively wide range of supericial velocity. Additional increases in the supericial velocity lead to further expansion of the bed and increases in the bubble sizes. Depending on the cross-sectional area of the bed, a situation in which the bubbles takes most of the bed cross-section may be achieved. That is called a slugging-low regime, and in it, large lumps of solids are pushed upward by raising bubbles. When such a bubble breaks at the bed surface, the particles in the respective lump drop back into the bed. Therefore, large masses of solid are thrown up and down, causing undesirable and dangerous vibrations in the equipment. Such a condition might lead to fatigue of the structure supporting the luidized bed. During slugging-low regime (not represented in Figure 3.2), the pressure loss in the bed usually increases [7]. Above the bubbling bed there is a region known as freeboard, the main purpose of which is to allow space for disengaging the particles carried by the gas low. Therefore, the bed is much denser then the freeboard, in mass of solids per volume of equipment. Within bubbling regime, the bubbles are almost free of particles. However, some particles pass through the bubbles. The rate of particles that short-circuit the bubbles increases with the increase of supericial velocity (Figure 3.1d). Actually, when crossing of particles through bubbles is negligible, the process is called a slow luidization regime. If that crossing rate becomes sizable, a turbulent luidization regime is achieved. Further increases of supericial velocity eventually lead to the disappearance of the bubbles as clearly visible entities. This stage is called fast luidization, in which the bed surface is not easily recognizable and clusters of particles dart to and from the freeboard. Such a situation is found in most circulating luidized-bed operations, where a region of dense particles replaces the bed found in bubbling regimes, whereas a leaner region replaces the freeboard. During fast luidization, most particles are carried by the gas stream (Figure 3.1e), and the pressure losses in such multiphase low situations involve a large variety of possibilities with no standard behavior. Therefore, calculations should be developed in a case-by-case basis. More precise deinitions and methodology to verify the actual regime at which a bed is operating will be described in Chapter 4. When computing values for velocities, one should remember that the composition, temperature, and pressure of the gas vary signiicantly throughout the bed. Hence, the terminal velocity (of free-fall terminal velocity) should change from point to point in the bed. During circulating luidized-bed operations, most of the particles are blown upward, and the bigger ones remain luidized in the lower part of the equipment or dense region. Bigger particles might also be carried away after size reduction by chemical consumption, thermal shocks, and grinding. This last process occurs mainly because of attrition between the particles. That is why circulating beds are often used solid grinding equipment.

48

Solid Fuels Combustion and Gasification Exiting gas and particles not captured by cyclone Piping Cyclone and hopper

Raiser

Returning leg or stand piping

Lean region

Fuel feeding and other solids

Aeration Valve

Dense region Air and/or other gases

FIGURE 3.5 Scheme of circulating luidized-bed equipment.

The name circulating is used because of the fact that most of the entrained particles are recycled to the dense region after their capture by a cyclone system (see Figure 3.5). Beyond the fast luidization, there is the pneumatic-transport region (Figure 3.1f). It is achieved when the supericial velocity surpasses the terminal velocities of all particles present in the bed. Thus, all particles are carried by the gas stream. From the point of view of basic phenomena, this situation characterizes particle suspensions in gas. It is also important to note that bubbles are not the only way by which the gas stream can pass through a luidized bed. More detailed descriptions of channeling, jetting, and spouting beds can be found in the literature [7]. For now, the above material is enough to explain the operation of most equipment dealing with gas–solid mixtures.

3.3

MOVING BED

Moving-bed combustors and gasiiers have been used for centuries, and therefore descriptions are found in a large number of references. An excellent review is presented by Hobbs et al. [12]. Figures 3.3 and 3.4 show simpliied schemes of moving-bed gasiier conigurations. The particles of carbonaceous particles—for instance coal, biomass, and various residuals—are fed through the top of the reactor and slowly low to the bottom,

49

Equipment and Processes

where the solid residual is withdrawn. In the updraft version (Figure 3.3), combustion and gasiication agents—normally air and steam—may be injected through the grade or distributor at the bottom, while in the downdraft version (Figure 3.4), those gases are introduced near the top. In their downward movement, the carbonaceous particles undergo the following main processes: drying, devolatilization, gasiication, and combustion. However, there are no sharp delimitations between drying, devolatilization, gasiication, and combustion regions. For instance, a descending particle may be going through devolatilization in its outer layers while it is drying in the inner layers. In addition, a particle may be undergoing devolatilization and simultaneous gasiication and combustion processes. The percolating stream provides reactant gases, and therefore, substantial variation of its composition and temperature occurs during its travel through the bed. For example, in the case of updraft version, the oxygen content of the injected stream is consumed at the lower layers of the bed (combustion zone), leading to increases in its carbon dioxide concentration. Moreover, in the oxygen-rich zone and beyond, CO2 and H2O react with carbon in the solid phase to produce CO and H2, as shown by the following main reactions: 2C + O 2 ⇔ 2CO

(3.1)*

2CO + O 2 ⇔ 2CO 2

(3.2)

C + CO 2 ⇔ 2CO

(3.3)

C + H 2O ⇔ H 2 + CO

(3.4)

At the same time, the gases produced react among themselves, leading to further changes in the gas phase composition. One of the most important of these is known as the shift reaction: CO + H 2 O ⇔ H 2 + CO 2

(3.5)

Of course, the above system of reactions is just a simpliication, and many other reactions are involved. A more complete set of reactions that better represents the gasiication process is presented in Chapter 8 and elsewhere [13–17]. Following its upward movement, the gas stream also receives the products from devolatilization of solid phase. That process is a very important combination of reactions and phenomena occurring within the particles. A simpliied representation of devolatilization is shown below: Volatiles → Tar *

(3.6)

As noted in Chapter 2, at high temperatures, the reaction between carbon and oxygen produces mainly carbon monoxide. Then, CO is oxidized to produce carbon dioxide. However, if oxygen is present in excess of what is necessary for complete combustion of fuel, very small concentrations of CO will remain.

50

Solid Fuels Combustion and Gasification

Volatiles → Gases

(3.7)

 b  b b b Tar +  514 + 531 + 551 + 546 + b 563  O 2 → b 514 CO 4 2 2  2  +

b 531 H 2 O + b 546 NO + b 563SO 2 2

(3.8)

Tar → Gases

(3.9)

Tar → Coke

(3.10)

It is important to stress again that the above scheme is a simpliication, used here to illustrate basic aspects of the process. A detailed discussion of devolatilization stoichiometry and kinetics is presented in Chapter 10. Finally, the gas stream receives water vapor from the process of drying the solid particles. The gas that is produced is withdrawn at the top of the updraft gasiier (Figure 3.3). In the case of a downdraft process, the gases from drying and devolatilization are forced through the gasiication and combustion regions (Figure 3.4). From the point of view of energy generation and consumption, the system composed of reactions represented by Equations 3.1 and 3.2 is exothermic, involving 394 MJ per kmol of carbon (assuming complete and irreversible reactions at 298 K). Those reactions are the main source of energy required to promote and sustain the mostly endothermic gasiication reactions. For instance, the enthalpy of Equation 3.3 (298 K) is –174 MJ/kmol, and that of Equation 3.4 is –131 MJ/kmol. Figure 3.6 illustrates this by showing a typical temperature proile found in updraft moving-bed gasiication processes. Since the gas stream is injected through the grid at the base of the bed, it exchanges heat and mass with the descending solid particles. Therefore, the temperatures of particles and gases tend to approach. After leaving the combustion region, the particles are at relatively high temperatures; hence, the gas crossing this region experiences a sharp increase in temperature. A surge in the gas temperature of both phases happens when the oxygen of gasifying agents starts to be consumed by the combustion of the descending solid. The derivative of the gas phase temperature remains positive until the oxygen is exhausted. The gas stream, now rich in CO2 and H2O and at a high temperature, continues reacting with the descending solid. These basic reactions are endothermic, thus forcing the temperature to decrease. Many of those reactions continue to occur until the gas exits the gasiier. Before leaving the reactor, the gas stream receives the components from solid devolatilization and drying, and the exiting gas usually contains a signiicant amount of tar. As seen in Chapter 2, tar is a complex mixture of inorganic and organic components with relatively long chains. Despite temperatures usually above 600 K, most of the tar is still liquid and leaves the gasiier as droplets suspended in the gas stream or mist. That stream also carries ine particles, and this combination is easily deposited at the interior of equipment ahead of the gasiier, as for instance heat-exchangers. With time, those deposits accumulate while tar goes through cracking, leading to

51

Equipment and Processes Temperature

Ash Produced gas Gas in

Feeding solid Height Grid

Top of bed

FIGURE 3.6 Typical temperature proiles in an updraft moving-bed gasiier.

hard solid layers. Removing such dregs is very dificult and often requires replacing entire tube sections or the whole heat-exchanger. In an effort to avoid such problems, the downdraft or concurrent gasiier was developed. As shown in Figure 3.4, the gas and solid phases low in the same downward direction. The gas stream leaving the devolatilization region—and therefore rich in tar—is forced through the combustion region. In the process, most of that tar is cracked and burned, resulting in cleaner gas than the equivalent provided by updraft model. On the other hand, compared with the updraft technique, the downdraft technique presents limitations on controllability when applied to large-diameter or high-power output equipment [18]. This is mostly due to formation of preferential channels, which prevent the tar-rich stream from meeting the hot core of the combustion region. Some manufactures have tried to mitigate the problem by inserting slowly rotating paddles. Those may distribute the solid particles more uniformly across the combustion region and diminish the formation of preferential channels. Nevertheless, such devices also bring technical problems, and there are severe limitations on the application of moving-bed technology in large-scale production. A typical composition of gas from atmospheric updraft, as well as downdraft, moving-bed gasiiers, when air and steam are used as gasiication agents, is as follows (mol %, dry basis): CO = 20%, CO2 = 10%, H2 = 15%, CH4 = 2%. This is a low-heating value gas in the range of 5 MJ/kg and is therefore limited to processes where high lame temperatures are not needed. Much higher gas combustion enthalpies can be obtained if the correct mixtures of pure oxygen and steam are used as gasiication agents. Methane is an important and desirable component in the gas produced. Most of it is liberated during devolatilization, whereas the remainder is produced from the following reaction: C + 2H 2 ⇔ CH 4

(3.11)

52

Solid Fuels Combustion and Gasification

Since the volume tends to contract during the above reaction, methane production is favored by increasing the operational pressure of the gasiier. Moving-bed gasiiers and combustors have been used largely because of their simplicity and degree of controllability. That is why, despite some disadvantages compared with more modern processes, several commercial units of moving-bed gasiiers are still in use throughout the world. Some of them have been used for coal gasiication in order to generate synthesis gas.

3.3.1

APPLICATIONS OF MOVING BEDS

Besides the examples given above for gasiiers, moving-bed or ixed-bed techniques are widely used for several other purposes, such as drying, extraction, retorting, absorption, and calcination. In the speciic ield of combustion, the process occurring on the grates of boilers can be understood as a ixed-bed one. As illustrated in Figure 3.7, the gas stream with oxidant (air) is blown at the bottom of the bed or through the grate, and it percolates through the bed of reacting carbonaceous particles. The grate moves or pushes the solid material at such a rate that particles leave almost completely converted into ash. In other words, enough residence time is usually provided to the particles to ensure a high level of conversion of carbon from solid to gas phase. There are several versions of moving-bed combustors and gasiiers [1–2, 12]. Most of them operate with air and mixtures, or air and steam in the case of gasiiers, and ash or residual is withdrawn from the bottom as solid particles. One variety applies oxygen or preheated air in order to achieve temperatures above the ashfusion temperature. In this case, the residual leaves the reactor as luid or slag. More Coal

Coal Coal

Air Grate

Air Air Ash Grate Grate

Ash

Air (a)

(b)

Ash Air (c)

FIGURE 3.7 Schematic view of common ixed-bed combustor and grates: (a) grate with bars (oscillating); (b) grate with mobile bars in steps; (c) rolling (chain) grate.

53

Equipment and Processes

details regarding ash-softening temperatures can be found in Chapter 16, as well as throughout the literature [19].

3.4

FLUIDIZED BED

As introduced above, there are two basic types of luidized-bed equipment: bubbling and circulating. Many modern boilers and gasiiers operate by those techniques. Their basic characteristics and comparisons of these processes with others and among themselves are described below.

3.4.1

BUBBLING FLUIDIZED BEDS

Figure 3.8 illustrates the basic characteristics of bubbling luidized-bed equipment. Gas or a mixture of gases—such as air, steam, or others—can be injected together or separately into the plenum at the bottom of the equipment. From there, such a stream passes through the gas distributor, which promotes a uniform lux of gas needed to ensure homogeneous luidization conditions throughout the cross-section of the equipment. Simple distributor designs, such as porous or perforated plates, are used in bench and pilot scale units. More elaborate ones with bubble caps or perforated lutes are used in large-scale industrial units and may also include sophisticated systems with features that provide automatic removal of lumps. These lumps of solid particles may form because of the eventual agglomerations that are usually started because Gas exit Freeboard Cyclone

Steam out

Water in Solid fuel, inert, absorbent hoppers Steam out Particle recycling

Water in Bed

Withdrawal by overflow

Distributor Air and/or steam

FIGURE 3.8 A basic scheme of luidized-bed equipment.

54

Solid Fuels Combustion and Gasification

of localized surpassing of the ash-softening temperature.* In addition, distributors of industrial-size equipment reserve a few injection points for the introduction of ignition gas (natural gas, liqueied petroleum gas [LPG], etc.). Those gases are burned during the start-up of the boiler or gasiier to heat the bed composed of inert particulate solids, such as sand or alumina. The inert bed is luidized with air that is maintained under bubbling conditions. Heating is provided by the injection of a combustible gas (natural gas, LPG, etc.) above or inside the bed, and the operation is kept until the average bed temperature is reached at around 800 or 900 K. From that point, the ignition gas injection is turned off and feeding of solid fuel starts. Above the distributor, the bed is composed of two main phases: emulsion and bubbles. The emulsion is a combination of solids and gases. It retains almost all particles of the bed; hence, gas percolates through the particles. Therefore, most of the gas–solid reactions occur within the emulsion. The circulation of particles† in the emulsion is very high, and it provides a certain degree of homogeneity of temperature throughout the bed. The intimate contact between gases and particles in the emulsion promotes closeness among their temperatures throughout most of the bed. The high rates of heat and mass transfer between phases, as well as other characteristics, lead to lower and more uniform average temperature in a luidized bed compared with values found at speciic regions of other techniques, such as moving-bed and suspension combustion equipment. As noted below, those characteristics of bubbling beds provide several advantages over many more conventional processes. Bubbles grow and accelerate while traveling from the distributor to the top of the bed. This growth is due to the decrease in the static pressure with the bed height and to coalescence of bubbles. The acceleration process creates a pressure difference between the top and bottom parts of each bubble. Because of the relatively lower pressure behind each bubble, small particles in the emulsion are drawn into those regions, creating what is known as the bubble wake. Therefore, these small particles share the momentum with their respective bubbles and are thrown into the freeboard once the bubble bursts at the top of the bed. Although the bubbles are almost free of particles, there are situations in which particles could short-circuit them. However, those cases do not represent a signiicant fraction of the particles in the system. Above the bed, one inds the freeboard, which provides space for inertial separation between particles and the carrying gas. This separation is a complex process in which, generally but not necessarily, most particles with terminal velocities above the average gas velocity return to the bed while those with terminal velocities below that average tend to be carried away and may reach the top of the freeboard. Of the latter, those that do leave the equipment with the exiting gas stream. The rate of gas–solid separation depends on a series of factors, such as the following: 1. The upward momentum of particles leaving the bed, which is provided mainly by the bubble burst 2. The momentum of the upward-lowing gas stream * †

Processes leading to particle agglomeration are detailed below in the present chapter. The circulation phenomena are described in detail in Chapter 14.

55

Equipment and Processes Fine particles leaving the freeboard

Height

Freeboard TDH

Bed

Bubbles

Gas injection

Mass flux of particles

FIGURE 3.9 Scheme illustrating the entrainment process.

3. The particle momentum abating because of collisions with descending particles and gravity A particle may reach the top of the freeboard and leave the system if its upward momentum is enough to overcome the countereffects. In addition, depending on the difference between the terminal velocity and the supericial velocity, the gas drag could favor the upward movement of particles, or not. The combination of all factors leads to the separation or disengagement of particles from gas in the freeboard, and the mass low of particles traveling in the upward direction decreases for higher positions in the freeboard. That upward low (kg/s) at each position is called entrainment rate, or simply entrainment.* Actually, for typical situations, the entrainment decreases according to an exponential function in the vertical direction [6–7, 20], as illustrated in Figure 3.9. The mass low particles reaching the top of freeboard and removed from the equipment is called elutriation rate or just elutriation. After reaching a certain height in the freeboard, the upward low of particles or entrainment no longer decreases (or decreases very slowly). Therefore, no matter how high the freeboard, the elutriation remains practically constant. This is called TDH or Transport Disengaging Height and is deined as the height at which the derivative of entrainment low against height is just 1% of its value at the top of the bed [20]. TDH is a fundamental parameter during the design of luidized beds.

*

Some authors prefer to use lux, or mass low per unit of freeboard cross-sectional area (kg s –1 m–2), to designate entrainment of particles.

56

Solid Fuels Combustion and Gasification

It is also interesting to comment that during the entrainment process, the low of particles is not uniform throughout the horizontal cross-section of the freeboard. Actually, particles tend to low upward at the central part of the freeboard and downward at regions near the reactor walls. This effect is mainly caused by smaller gas velocities near the walls. Relatively large particles carried upward by the fast gas in central regions eventually reach positions near the wall, where the gas velocity is lower than the particle terminal velocity. In other words, particles are lifted at freeboard central regions, whereas peripheral regions work as downcomers. Since the fraction of freeboard cross-section affected by the presence of the wall is relatively small, most of the cross-section has a uniform velocity. In addition, the average velocity of particles traveling upward is larger than those going in the opposite direction. This is why the center is leaner in particles than regions near the vertical walls. This behavior is observed in turbulent and fast luidization regimes as well. Hence, circulating luidized beds operate with strong differences in volumetric concentration of solids between central and peripheral regions of the reactor’s horizontal cross-section. This justiies two-dimensional approaches for modeling of such equipment. However, one-dimensional attacks have led also to reasonable approximations, as shown in Chapter 18. The process of separation or removal of ines from the mixture of particles and gases has several aspects, and quantitative treatment of these variables is shown in the Chapter 14. Unlike the situation found in most moving beds, the carbonaceous particles fed into bubbling or circulating fluidized furnaces go through very fast drying and devolatilization. Thus, in most cases, these processes are almost completed near the feeding point of the particles, and that is the only region affected by the devolatilization. The extent of this region is a strong function of the devolatilization reaction rates and the local velocity of carbonaceous particles. For instance, in the case of bubbling beds, the devolatization region depends on the circulation rate of particles in the bed and the rate of pyrolysis reactions. As those reactions are strongly influenced by temperature, that and circulation rate of particles are the most important parameters determining the size of region where devolatilization process can be observed. Figure 3.10 exemplifies the devolatilization-affected region in the case of a bubbling fluidized-bed gasifier, with the feeding position located at 0.3 m above the gas distributor or base of the bed (zero height). That figure illustrates how devolatilization leads to a surge in the tar concentration near the feeding point. After that, tar decomposes mainly by cracking and cooking. Details about these processes are described in Chapter 10. Figure 3.11 illustrates typical temperature proiles found in the bed section of a bubbling luidized boiler. It should be noted that the carbonaceous solid temperature reaches its maximum value near the bed base (zero height). This fact could seem to be a paradox because the particles are in contact with the cold gas injected through the distributor. Nonetheless, because of a relatively high local oxygen concentration in the emulsion gas, carbonaceous particles have a much higher combustion rate near the distributor. This can also be explained by the following:

57

Equipment and Processes 9.E-2 H2S NH3 TAR/OIL

8.E-2

Molar fraction

7.E-2 6.E-2 5.E-2 4.E-2 3.E-2 2.E-2 1.E-2 0.E1

0

1

2

3 4 Height (m)

5

6

7

FIGURE 3.10 Tar concentration throughout a bubbling luidized-bed gasiier.

1. Particles move very fast inside the bubbling bed, following a circular vertical path, as shown in Figure 3.12. 2. Those paths can be subdivided into smaller ones. However, for now it is enough to know that this mainly vertical movement of particles, including that of fuel, is fast. 3. Thus, in a brief space of time, a particle could be taken from the middle of the bed to regions near the distributor. 1800 1600

Temperature (K)

1400 1200 1000 800 600 400 Emuls. gas 200

0

0.1

Bubble 0.2

Carbonac. 0.3

0.4

Absorbent 0.5

Average 0.6

0.7

Height (m)

FIGURE 3.11 Typical temperature (K) proiles of various phases as a function of the height in a bubbling bed.

58

Solid Fuels Combustion and Gasification

Freeboard

Bed

FIGURE 3.12 Typical circulatory movements of particles found in bubbling luidized beds.

4. The regions near the distributor are rich in oxygen because of the fresh air coming from below. 5. The main combustion process (represented by Equations 3.1 and 3.2) experiences a rapid increase in its rate. 6. As the rate of energy release due to the combustion process is much higher than the rate of energy consumed by gasiication endothermic reactions and transferred to the incoming gas, the temperature of fuel particles nearing the distributor increases. Figure 3.12 also shows the sharp increase in the temperature of gas lowing through the emulsion near the distributor. This is due to the intense mass and heat transfers between the emulsion gas and the fast-burning carbonaceous particles. The fraction of gas diverted to the bubble phase experiences much lower rates of heat and mass transfer with the gas and particles in the emulsion. Therefore, the gradient of bubble temperature in the vertical direction is lower than that observed for the gas in the emulsion. Nonetheless, in some situations, the temperature of the bubbles may even surpass the average temperature of the bed. This can be understood by the following sequence of events [13, 14, 21–25]: 1. As the concentration of oxygen in the emulsion drops to almost zero at regions not very far from the distributor, fuel gases (H2, CO, etc.)—produced by gasiication reactions—may concentrate in that phase. 2. These gases diffuse to the relatively cold bubbles. 3. Despite initially being low, the temperature of the bubbles keeps increasing and eventually reaches the point of ignition of the gas mixture inside the bubbles.

Equipment and Processes

59

4. The fast combustion might lead local bubble temperature to surpass the average in the bed. 5. Once the fuel gases are consumed, the bubble temperature tends to approach the average in the bed. For short or shallow combustor beds, stored fuel gases in the bubble may not have the opportunity to burn inside the bed. They do so at the top of the bed when the bursting bubbles exposes fuel gases to oxidant atmosphere of the freeboard. This phenomenon may lead to lames at the bed surface. This occurrence is usual in cases of combustors or boilers but not easy to ind in gasiiers, for two reasons. The irst reason has to do with the rate of oxygen injection in gasiiers, which is below that necessary for complete combustion of the solid fuel. Actually, the ratio of oxygen actually injected over to the one necessary for complete combustion is called the oxygen ratio. Usually, the best values for the oxygen ratio are around 0.2 or 0.3 (20%–30%) [21–24], but exceptions may be found. The second reason is that gasiiers usually use deeper beds than combustors to allow enough residence time for relatively slow gasiication reactions, as well as total mass transfer of oxygen from bubbles to emulsion. It is important to note that lames at the surface of the bubbling beds of gasiiers is a strong indication of bad design or bad operational conditions, because all fuel gas accumulated in the emulsion would be burned. The other consequence is relatively low carbon conversions and therefore low gasiication eficiency. The temperatures of any inert or sulfur-absorbent particles tend to follow the bed average. This is because no highly exothermic or endothermic reactions take place in those phases.

3.4.2

CIRCULATING FLUIDIZED BEDS

Like the bubbling luidized-bed (BFB) technique, the circulating luidized-bed (CFB) technique has found numerous applications, such as boilers, gasiiers, reformers, and dryers with different designs in all sizes. Actually, it is likely that one would ind more units operating using the CFB technique than the BFB technique. The text below shows the main reasons for that. Despite many similarities, there are several important differences between bubbling and circulating beds in their dynamics and other aspects. Among them the most important are: 1. The basic characteristics of the regions inside the equipment are different. In circulating luidization, the dense region is the equivalent to the bed of bubbling equipment, while the lean region is the equivalent to the freeboard of bubbling equipment. However, unlike in bubbling beds, the dense region does not retain the largest portion of particles held in the equipment, and a signiicant portion is found in the lean region. This is so because much higher supericial velocities are found in circulating than in equivalent bubbling versions; thus, most particles are dragged to the lean region. 2. Since CFBs operate with higher supericial velocities than BFBs, the circulation rates in the dense region are much higher, and particles have enough

60

Solid Fuels Combustion and Gasification

momentum to overcome the resistance imposed at the boundary between a bubble and the neighboring emulsion. Despite the differences in nature, that resistance may be understood as similar to that imposed by surface tension at the interface between a liquid and gas bubble in it. 3. As seen before, CFBs operate above the turbulent luidization, where bubbles formed retain much more particles than those in bubbling versions. Further increases in gas velocity lead to complete fast luidization, where bubble interfaces are no longer visible. Such an effect leads to closer contact between particles and gases in circulating units compared with bubbling ones. This is probably one of the main reasons for the good carbon conversions observed in CFB operations. Because of the bubble breakage, the slugging regime found at the limits of bubbling bed operations also disappears. The transition from bubbling to turbulent regime is gradual [7]. Actually, the denomination of a dense region is solely due to the higher average concentrations of particles found in the emulsion compared with the lean region of CFBs. 4. Temperature peaks are different. Because of the relatively low fractions of particles held in dense region, most of the chemical transformations occur in the lean region of circulating beds. Unlike BFB equipment, CFB combustors or gasiiers present peaks of fuel oxidation rates in the lean region, as illustrated in Figure 3.13. Moreover, fast and localized temperature surges lead to great variations of void fraction and therefore particle concentration per volume throughout the lean region compared with freeboards of BFBs. It is interesting to note that despite localized high temperatures, the high void fractions found in lead regions decrease the probability of particle agglutinations. This occurs, irst, because the high temperatures occur in the gas phase because 3000

Temperature (K)

2500

2000

1500

1000 Gas 500

0

1

2

Carbonac.

Inert

4 3 Height (m)

Average 5

6

7

FIGURE 3.13 Typical temperature proiles in the lean region of circulating luidized beds.

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61

of gas–gas or homogeneous reactions, whereas the carbonaceous particles remain at relatively low temperatures (Figure 3.13). Second, even if temperatures surpass the ash-softening limit, the particles are too apart for agglutinations to take place. Having that in mind, circulating units allow operations with a wider range of oxygen ratios. For instance, boilers can function with smaller air excesses, and gasiiers might use larger oxygen ratios, promoting high temperatures at points in the lean region. Such possibilities usually provide higher performances of combustors and gasiiers compared with the equivalent bubbling units. 5. Internal arrangements are different. Unlike in BFBs, no tube banks are usually found inside circulating boilers. The high momentum of particles found in circulating units would incite extremely fast erosion of tube surfaces. Steam is generated in external heat exchangers, where the gas leaving the cyclone system enters the boiler tube banks. In this respect, the circulating technique is not as advantageous as the bubbling technique, where high heat transfer coeficients can be obtained between tubes in direct contact with gases and particles inside the bed. 6. Contact between gases and particles is different. In addition to the effects described above, CFBs usually present greater relative differences of velocities between gas and solid phases compared with similar BFBs. This also leads to greater heat and mass transfer rates between those phases in circulating versions. The consequences are more eficient combustion and gasiication processes found in CFB units than in the equivalent BFB units. There are qualitative methods to help decide whether a luidization regime is characteristic of bubbling or circulating beds. Among the methods are the following: a) Graphs delimiting regions according to luidization parameters (for instance, Grace [26] shows a chart correlating the luidization regime against the Archimedes number and a dimensionless velocity) b) Surpassing the transport velocity or at least the second turbulent limiting velocity, as described in Chapter 4 c) Comparisons between the transport and supericial velocity d) Comparisons between the supericial velocity and the average terminal velocity Among the extensive literature on transitions from bubbling to turbulent and fast luidization regimes, the text edited by Geldart [7] presents detailed discussions on the subject. The luidization parameters mentioned above are deined precisely in Chapter 4. 3.4.2.1 A Few Aspects of Circulating Bed Designing As shown above, circulating luidized beds demand reinjection of solids collected by the cyclone system into the main column or raiser. This is not a trivial problem, and a badly designed returning system may prevent the proper operation of the equipment. Figure 3.5 shows the main components of circulating luidized-bed equipment. The solid particles in the returning leg should be injected into the raiser, and for proper operation the following should be provided:

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Solid Fuels Combustion and Gasification

1. Enough pressure at the entrance into the raiser in order to overcome the pressure losses imposed by the dense region (bed), lean region, piping, cyclone, and valve 2. A valve that ensures suficient and controlled injection of particles back into the bed 3. Aeration injection to ensure the valve proper operation There are various types of valves: mechanical (or forced) and nonmechanical. The mechanical ones use screws or any device to force the particles into the equipment. The nonmechanical uses only the aeration gas, but its shape should be well designed to control the low rate of particles. The most common types of nonmechanical valves are shown in Figure 3.14. Among the most used are the L- and V-valves. However, for cohesive, solid particles,* reinjection is possible only through mechanical methods. The positioning and low rate of the aeration stream is critical. Injections near the lower part of the valve may lead to insuficient low of solids into the equipment and consequent stoppage due to accumulation of solids in the returning leg. For instance, in cases of L-valves, Knowlton and Hirsan [27] suggest that the position of aeration injection should be at 2 times the recycling tube diameter above the center line on the valve’s horizontal branch. Additional details on the design of returning legs and valves, including dimensioning, can be found in the literature [7, 27, 28].

Aeration

Aeration

J-valve

L-valve

Aeration Loop-seal valve or V-valve

Reverse seal Aeration

FIGURE 3.14 Commonly found valves applied in circulating-bed equipment. *

For C and sometimes A types of particulate solids, as shown in Chapter 4.

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Equipment and Processes

3.4.3

APPLICATIONS OF FLUIDIZED BEDS

The luidized-bed technique can be applied to a wide range of industrial equipment, such as boilers, gasiiers, oil shale retorting, reactors using solid catalysts, etc. Since the 1960s, great interest has emerged in the application of the luidized-bed technique for boilers. Compared with conventional systems, such as suspensions or grate burning, the luidized-bed technique shows the following advantages: 1. High degree of controllability 2. Low SOx and NOx emissions 3. Homogeneity and relatively low average temperature throughout the equipment 4. High turn-down ratios 5. In the case of BFBs, high heat transfers coeficients to tubes immersed in the bed 6. More compact equipment for the same power output 7. In the case of gasiication, very low concentration of tar or oil in the produced gas 8. Allowing the consumption of a wide range of feedstock with low heating value that could hardly be used in more conventional processes Most of the above advantages are easily understood. For instance, the low emissions of sulfur dioxides are made possible by adding limestone to the bed. Calcination of its particles takes place as soon as the limestone enters the bed, as follows: CaCO3 → CaO + CO 2 Then, the sulfur oxide might react with the calcium oxide through the following reaction: CaO + SO 2 + 1/2O 2 → CaSO 4 As seen, the product is a stable solid, which can be discharged from the bed without environmental problems. Similar reactions can be written by using magnesium instead of calcium. Therefore, the same effects can be obtained by adding dolomite (which contains MgCO3) to the bed. In general, limestone and dolomite are called sulfur absorbents. Compared with the conditions in conventional combustion chambers, the average temperatures in bubbling or circulating luidized beds are much lower and more uniform, thus leading to lower NOx emissions. In moving or pneumatic transport combustors, overall heat transfer coeficients between immersed tubes and the surrounding gas-particle mixture are usually below 100 W m–2 K–1 [3, 4]. On the other hand, values around three times as large can be found in bubbling luidized beds [6, 29]. This is basically due to the high circulation rates of particles in the bed. However, particles impacting on the tube surfaces provoke considerable erosion rates at particular points. This problem has been solved by a special design of the tube banks [30].

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Solid Fuels Combustion and Gasification

Fluidized-bed gasiiers provide much lower tar concentrations in the exiting gas stream than similar units operating under the updraft moving-bed technique. Usually, the generated gas is used in furnaces or injected in turbines for power generation. Frequently, the feasibility of such applications is very sensitive to the gas cleaning system. This is why luidized-bed gasiiers have found an increasing number of applications in advanced turbine power generation units. These processes are called coal integrated gasiication (CIG)/gas turbine and biomass integrated gasiication/gas turbine (BIG/GT), and some basic characteristics of those are described below. Although it is not mandatory or necessary, several luidized-bed combustors or gasiiers operate with inert solid particles mixed with the fuel because they usually provide the following advantages: • Easier temperature control. Since inert particles are not involved in reactions, they work as an energy inertial mass. That avoids eventual sudden variations in operational conditions, which may provide conditions for steep changes in localized values of bed temperature. In other words, the presence of inert particles might avoid agglomeration of particles and collapse of the bed due to local surpassing of the ash-softening temperature. • Fluidization stability. During the transition from ixed to luidized bed, small particles slip into the gaps between large ones and luidize before them. Smooth luidization can be achieved when the gas velocity is increased, and hysteresis (Figure 3.2) might be avoided [6]. Hence, it is easy to conclude that the presence of a wide range of particle sizes helps to absorb sudden variations of gas velocity during operations. Despite those advantages, care should be taken to prevent segregation. For instance, sand particles usually present densities two to three times those of wood or charcoal. Even within similar sizes, the lighter particles luidize before the heavier ones. This occurs because a bubbling bed behaves like a liquid, and particles with too different density or size might separate into layers. The lighter ones tend to loat or concentrate near the bed surface and the denser ones in lower layers. Actually, luidized-bed behavior is a bit more complex than this simple analogy, and segregation occurs because of differences in minimum luidization velocity among particles with differences in average size and density. Particles with lower minimum luidization velocity tend to loat, thus segregating them from other solids in the bed. There are several consequences of segregation; among them, the most important occurs when carbonaceous fuel particles are the loating ones. Then pyrolysis would occur near the top of the bed, and gas with tar would pass straight through the freeboard. On many occasions, that would not provide the right conditions and enough residence time for complete cracking or cooking of tar components. If no proper cleaning system is present, tar and ine particles carried by the exiting gas would deposit inside cyclone walls, as well as in other equipment downstream. Those deposits accumulate, which may seriously impair normal operation of boilers, gasiiers, or any other unit. In most cases, such auxiliary equipment would be damaged beyond repair. Another similarly grave problem takes place when tar impregnates refractory walls,

Equipment and Processes

65

which in time would need to be replaced, resulting in high maintenance costs. In addition, eficiency losses will result if tar is not consumed or converted into fuel gases by cracking. Those problems can be overcome by the following steps: 1. Avoiding great differences in luidization dynamics of different species, which is accomplished by decreasing the average size of the denser particles or increasing the average size of the lighter particles 2. Increasing the mass low of gases through the distributor, which leads to higher circulation of particles in the bed and forces them to mix A method to predict the occurrence of segregation in luidized beds is provided in Chapter 14. Another point to consider before adding inert particles is “diluting” the bed. The rate of solid conversions depends on the residence time of particles in the equipment. As the volume available inside an operating luidized-bed unit is ixed, too-high feeding rates of inert particles would compromise the residence time of solid fuel. Such an effect may also be provoked by high rates of recirculation of particles collected by cyclones to the bed. Those particles usually present higher carbon conversions than the average in the bed. Therefore, high recirculation tends to decrease the average amount of carbon per unit of volume in the bed, with a consequent decrease in the overall rate of carbon consumption or conversion. In addition, due to either endothermic reactions (in cases of gasiiers) or heat exchanges with tubes (in cases of boilers), particles reaching the cyclones tend to be at much lower temperature than those in the bed. Reinjecting those particles into the lower bed tends to decrease its average temperature and therefore the rates of reactions. Recirculation may be interesting for units operating with relatively low carbon conversion; however, it might be detrimental to processes and should be carefully evaluated. An important aspect of luidized beds is related to the quality of feeding fuel. Experience shows that even fuels containing high concentrations of ash present no major dificulties to consumption in BFBs or CFBs. Actually, small ash particles detached from the original fuel play the part of inert matter, with the advantages already discussed. The same cannot be said for other processes, especially for combustion or gasiication using suspensions of pulverized fuel. In addition, luidized beds, especially BFBs, can operate with a relatively wide range of particle sizes. This is mainly because luidization provides a good exchange of momentum transfers between particles. This allows larger particles to stay luidized in the midst of the bubbling bed. It is no coincidence that bubbling luidized beds have been used as combustor chambers of large boilers where relatively rough coal is fed mixed with other fuels or residues. The same is not valid for circulating luidized beds, which require a narrower range of particle size distribution. In addition to the applications shown here, luidized beds can be used in combination to achieve several objectives. Those interested in more details will ind very good descriptions in the literature [6]. The complexity of all these and other factors does not allow simple decisions on which technique to adopt in a given situation. Such decisions can be reached only thorough mathematical simulations combined with experimental veriication.

66

3.4.4

Solid Fuels Combustion and Gasification

COMPARISONS BETWEEN FLUIDIZED-BED AND MOVING-BED PROCESSES

It is interesting to note the basic points of comparison between moving- and luidized-bed gasiiers. These are presented in Tables 3.1 through 3.3. The reasons for these and several other characteristics will be better understood through the presentations of models (Chapters 7 and 13), as well as discussions of computations (Chapters 12 and 16).

3.4.5

ATMOSPHERIC AND PRESSURIZED FLUIDIZED-BED GASIFIERS

A powerful instrument by which to evaluate a given process or equipment is its eficiency. Of course, it is possible to set several descriptions for eficiency, and those will be precisely shown in Chapter 5. However, the most commonly used deinition for gasiication eficiency is η=

FG h G FS h S

(3.12)

Here, FG is the mass low (kg/s) of exiting product gas, FS is the mass low (kg/s) of feeding solid fuel, hG is its combustion enthalpy (J/kg), and hS is the combustion enthalpy of the feeding solid fuel (J/kg). If the enthalpy of the produced gas is computed at the temperature found as it leaves the gasiier, Equation 3.12 would provide the “hot eficiency”, whereas if it is computed for the tar-free gas and at 298 K and 101.325 kPa (or standard atmospheric conditions), one obtains the “cold eficiency.” Common values of gasiication cold eficiencies are around 50%.

TABLE 3.1 Comparison between Moving-Bed and Fluidized-Bed Gasifiers (Part 1) Characteristic or Feature

Moving-Bed Gasifier

Fluidized-Bed Gasifier

Usual number of phases involved

One gas plus 1 for each solid involved. For instance, combustors and gasiiers for coal involve just one solid; therefore, there are two phases.

Supericial gas velocity

Relatively low or below the minimum luidization. Increases with the increase of the supericial velocity. Varies signiicantly and passes through extreme peaks.

Two gas (emulsion gas and bubble) plus 1 for each solid involved. For instance, a coal combustor in which limestone is added and ash segregates from the burned layers of coal particles involves 5 phases. Some authors include an additional phase—the cloud—that surrounds the bubbles. Above the minimum luidization and all the way to the limit with pneumatic transport. Remains constant within the luidization range. Almost constant, and the average is much lower than the peaks observed in moving beds where an equivalent process is taking place. Could be large between bubbles and emulsion.

Pressure loss in the bed Average temperature in equipment involving exothermic or endothermic reactions Difference of temperature between phases

Relatively small between gas and solid phases.

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Equipment and Processes

TABLE 3.2 Comparison between Moving-Bed and Fluidized-Bed Gasifiers (Part 2) Characteristic or Feature Moving-Bed Gasifier

Fluidized-Bed Gasifier

Concentration of tar in the gas exit stream

Very low because devolatilization occurs inside the bed. This ensures almost total destruction of the tar by cracking and coking.

Flexibility concerning the quality of feed stock that can be consumed

Controllability

Operational risks

Level of pollutant emissions

High for countercurrent equipment because devolatilization occurs near the gas exit point. The concentration of tar in the gas exit stream is lower downstream because the devolatilization occurs before the gas stream passes through the combustion region. On the other hand, this last class of equipment presents limitations on the size due to controllability problems. Moderate to low. Limitations are found due to the minimum carbon concentrations necessary in the combustion region. High temperatures should be provided to the gasiication region by the gases leaving the combustion zone. This is more critical for atmospheric equipment operating with air. Moderate to low. Countercurrent provides a good level of controllability but high response time (around 1 hour to achieve a new steady-state regime in pilot units to 8 hours in industrial-size units). Downstream ones are limited to the range of 1 MW power output because of the great dificulty of maintaining a uniform temperature proile around the combustion region in large equipment. Moderate because of possible interruptions caused by blockages caused by excessively high temperatures in the combustion region, which may surpass the ash-softening level and to tar acting as bonding between particles in the devolatilization region. The problem is more critical when small particles or ibrous ones are fed into the equipment. High and dificult to avoid.

High. The bed operates at relatively low temperatures, which can be achieved using low carbon concentration feed stocks. In fact, the average carbon concentration in the bed are around 2–5%. High. Beds operate at low and uniform temperatures. Also, the response time is around few minutes within a wide range of equipment sizes. The basic reason for this is the high mixing ratio of gases and solids in the bed.

Moderate to low. Problems of bed collapsing may occur because of localized high temperatures that surpass the ash-softening temperature.

Low and easy to control. The basic reason is the possibility of adding pollutant-absorbing solids to the bed. These mix very well with other solid and maintain a good contact with the gas phases. Relatively low temperatures minimizes the thermal-NOx production.

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Solid Fuels Combustion and Gasification

TABLE 3.3 Comparison between Moving-Bed and Fluidized-Bed Gasifiers (Part 3) Characteristic or Feature Moving-Bed Gasifier Fluidized-Bed Gasifier Carbon conversion

High carbon conversions can be achieved with relatively small efforts.

Cold eficiency

Relatively high. If tar is not included as an useful fuel, the eficiency drops. Values around 70% are common for atmospheric gasiiers using air. High (around 80%). Moderate (60%) to high (85%). This last range is achievable for pressurized gasiiers. Moderate, or around High, or around 4/1. Even higher values can be 2/1. achieved for large equipment where the bed can be divided into sections.

Hot eficiency Turn-down ratio (ratio between the maximum and the minimum power output for the same equipment maintaining in a single operation and maintaining the same level of eficiency) Capital cost

Moderate to low. Carbon conversions above 0.95 are dificult to achieve because of compromises between gas quality and luidization regime. This is more critical for gasiiers using air instead oxygen. The increase in the air low to reach high carbon conversions leads to large bubbles that require deep beds to complete the oxygen transfer to the emulsion. The other alternative is to use supericial gas velocities near the minimum luidization value, but this will require beds with larger diameters. This last possibility can be combined with the preheating of the gas stream, which is to be injected into the bed. Therefore, the expansion of the gas will be smaller than the cold injection, which leads to smaller distances from the minimum luidization condition for points far from the distributor. In both cases (gasiication using air or oxygen), the residence time of particle in the system needs to be relatively high compared with moving-bed gasiiers or with the burning in a luidized-bed combustor. Pressurization decreases the problem. Moderate for atmospheric equipment using air as an oxidant agent. Values around 60% for atmospheric equipment using air are common, and 70% for pressurized equipment using oxygen can be obtained with relatively little effort.

Moderate. Some care Low because of the relatively low and uniform should be taken on the temperature in the bed. ash removal device and material near the combustion region.

Equipment and Processes

69

The main factor that might, in some situations, prevent a bubbling luidized-bed gasiier to operate under relatively high eficiency is the role played by the mass transfers between the bubbles and the emulsion. That point has been partially discussed above. Nonetheless, because of its importance to understanding bubbling gasiication, a more detailed description is given here: a) The gasiication process requires a good region of the bed to be kept under reducing conditions, i.e., without oxygen. b) The oxygen, mixed with other gases or not, is injected into the bed through the distributor. Part of this stream lows through the emulsion. The remaining portion lows through the bed in the form of bubbles. c) The oxygen lowing through the emulsion is consumed in the irst layers of the bed, near the distributor. d) However, as the bubbles remain relatively cool through a large portion of the bed, the oxygen consumption in that phase is slow, and its concentration stays relatively high. e) Compared with the rate of oxygen chemical consumption due to combustion in the emulsion, the transfer of oxygen from the bubbles to the emulsion is a very slow process. f) Therefore, for a good portion of the bed, bubbles work as a source of oxygen that slowly diffuses to the emulsion. g) Consequently, oxidant conditions might remain for a large portion or height in the bed, and the fuel gases produced are consumed in the emulsion phase by the oxygen diffusing from bubbles. There are several possible solutions to increase the gasiication eficiency. Usually, they also improve the carbon conversion and therefore the gasiication eficiency. The most commonly used alternatives are the following: 1. Increasing the bed height. This would provide extra residence time for gas– solid contact at reducing conditions. 2. Increasing the size of particles in the bed. This would lead to a lowing regime nearer the minimum luidization regime, therefore reducing the size of bubbles. The effect is complete exhaustion of oxygen occurring in shorter distances from the bed base. 3. Increasing the pressure in the bed. Apart from the beneicial effect of pressure regarding increases in the methane production (as already explained), pressurized systems lead to relatively smaller bubbles and therefore higher performance of pressurized against atmospheric luidized gasiication processes. This increase of eficiency has been shown in several works [25, 26, 31–40]. 4. If economics allow, using oxygen instead of air as gasifying agent. Without nitrogen, the size of bubbles decreases (with the effect explained in point 2, above). In addition, nitrogen is a diluting gas, and if it is not present, the rate of oxygen diffusion from bubbles to emulsion increases. Moreover, the inal concentration of fuel components in the produced gas is considerably

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Solid Fuels Combustion and Gasification

enhanced, leading to higher combustion enthalpy compared with processes using air as the gasifying agent. 5. Recycling particles collected in the cyclone (or ilters) to the bed. This process increases the residence time of particles in the bed. However, recycling brings a very important side effect that might lead to poorer eficiencies. Apart from those effects already shown, recycling tends to decrease the average particle size in the bed, providing low regimes farther from minimum luidization. This also increases the average size of bubbles in the bed. A compromise should be sought by applying partial recycling.

3.4.6

GAS CLEANING

As mentioned before, luidized-bed techniques have been successfully applied in power generation systems such as CIG and BIG/GT. Other systems are based on BFB boilers, in which the steam is injected into the respective turbines and the lue gas is cleaned and injected into gas turbines. Despite the relatively tar-free conditions, the gas from luidized-bed combustors or gasiiers contains alkaline and particles. Before the gas is injected into turbines, stringent conditions must be met to avoid or minimize blade erosion and corrosion, as well as deposition of particles on their surfaces. For instance, erosion is caused mainly by particles, and the reported [41] tolerances prescribe maximum concentrations between 2 and 200 mg/(standard m3). In fact, there is a compromise between the maximum particle size and the concentration of particles in the turbine-entering stream [42]. Meadowcroft and Stringer [43] explain that damage by erosion on the turbine blades is caused only by particles above 3 µm. At velocities between 200 and 300 m/s, usually found in the gas between the turbine blades, only particles smaller than that can follow the streamlines, thereby avoiding collision with the blades. Corrosion of blades is provoked mainly by the presence of alkaline. Some have reported [44, 45] that concentrations below 200 parts per billion (ppb) should be achieved to ensure longer operations, whereas others [46] recommend maximum values as low as 20 ppb. In addition, corrosion and erosion may combine synergistically. Compared with isolated erosion and corrosion effects, their combination could lead to a threefold increase in the rate of material loss from the turbine blades [43]. Moreover, it has been shown [43] that deposition of particles on the blades might accelerate the corrosion process. The main cause of these depositions is the presence of ash particles, which stick to the blade surface. This process occurs when the temperature of the traveling particles surpasses the ash-softening limit. Even if softening is not reached, deposits of condensing alkaline also carrying very small ( 10 4.

(4.16c)

and

The Archimedes number is given by Equation 4.8.

88

4.2.5

Solid Fuels Combustion and Gasification

TRANSITION FROM TURBULENT TO FAST FLUIDIZATION

As noted in Chapter 3, the transition from turbulent to fast luidization is characterized by the complete blur of bubble interface. Of course, there is not a sharp jump from any condition to another, simply because nature does not work that way. The literature present graphs in which those transitions are represented by a region or gas velocity range. Many authors have tried to present more quantitative approaches. Among them, Perales et al. [13] introduce a method to estimate the transport velocity, which is the limiting value for supericial velocity below which fast luidization cannot take place. As fast luidization is a main characteristic of circulating luidization regimes, the transport velocity provides a convenient method to determine the transition between bubbling and circulating processes, which is given by the following equation [13]: U tr = 1.45

µG N 0Ar.484 . ρG d Pav

(4.16)

The above is valid for 20 < NAr < 5 × 104.

4.2.6

AIR RATIO AND AIR EXCESS

Air ratio is among the basic parameters used for almost every technical decision on combustors and gasiiers, and it is given by ϖ=

Factua1-air Fstoichiometric-air

(4.17)

where Factual-air is the mass low of air actually injected into the combustion chamber, and Fstoichiometric-air is the theoretical minimum mass low that would be necessary for the complete (or stoichiometric) combustion of the fuel. The air excess is usually expressed as percentage and is deined as follows: fair = 100

Factual-air − Fstoichiometric-air = 100( ϖ − 1). Fstoichiometric-air

(4.18)

In simpliied calculations, nitrogen is commonly assumed as the inert or nonreacting component. In these cases, the air ratio and the oxygen ratio are equivalent. As an example, take the case of methane combustion. The basic overall combustion reaction would be CH 4 + 2O 2 → CO 2 + 2H 2 O.

(4.19)

In case of combustion in air, some authors like to include the nitrogen in the reaction: CH 4 + 2(O 2 + 3.76 N 2 ) → CO 2 + 2H 2 O + 7.52 N 2 .

(4.20)

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Basic Calculations

Of course, the factor 4.76 is the molar ratio between nitrogen and oxygen in common air. However, this approach may lead to misunderstandings because if nitrogen is not taken as a reactant, it should not be included in equilibrium computations. On the other hand, nitrogen does take part as a reactant in combustion and gasiication processes. Therefore, Equation 4.20 should be used only as an approximation. Following the example, if 1 mol of methane were injected into a combustor with 3 mols of oxygen (diluted in air), the air or oxygen ratio would be ϖ=

Factual-air Fstoichiometric-air

3 F =  actual-air = Fstoichiometric-air 2

(4.21)

or 1.5. Therefore, the air or oxygen excess would be 50%. As has been shown, the value of the air ratio in a combustor is equal to or greater than 1, and for gasiiers, it is below 1. Actually, most combustors work with an air ratio in the range between 1.2 and 1.5, or air excess from 20% to 50%. For gasiiers, air ratios between 0.2 and 0.4 are within the usual range. All combustors and gasiiers tend to operate at higher temperatures if the air ratio approaches unit or stoichiometric conditions. In combustors, if the air excess increases, the surplus amount of air would consume part of the energy provided by the fuel to be heated at the same temperature as the exiting stream. Therefore, the average and also the peak temperatures tend to decrease in relation to operations with lower air excesses. For gasiiers, lower air ratios provide less oxygen, and therefore less fuel can be burned or oxidized. This leads to lower average and peak temperatures. Figure 4.1 illustrates by a simple graph the possible situations for combustors and gasiiers. Actually, the real situations are not so simple. For instance, large combustors do not behave as well-mixed reactors, and part of the injected fuel may not contact oxygen or air. Hence, air excesses are usually required to improve the contact with Average temperature

0.0

1.0

Air ratio

FIGURE 4.1 Typical temperature behavior for combustors and gasiiers in relation to air ratio.

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Solid Fuels Combustion and Gasification

the fuel and approach complete combustion. The temperature peak is not as high as indicated assuming complete combustion, and it occurs at an air ratio a bit above the stoichiometric one.

4.2.7

DENSITIES FOR SOLID PARTICLES

Because of their nature and the way they should be used in calculations, several deinitions for density can be found. The most common deinitions are the following: • Apparent density of a particle, given by ρp,app =

mp V

(4.22)

where mp is the mass of the particle, and V is the total volume (or volume delimited by an envelop surface). • Real or skeletal density of a particle, given by ρp,real =

mp Vp − Vpores

(4.23)

where Vpores is the volume taken by the pores inside the particle. • Global or bulk density of a particle bed, deined as ρp,bulk =

m V

(4.24)

where m is the total mass of the bed, and V is its volume. Therefore, this volume is the sum of the volume of individual particles and the void space between them. This deinition leads to another very important one. This is the void fraction of particle bed, given by ε=

VG V = 1− S V V

(4.25)

where VG is the volume occupied by the gas phase in the interstices of the particles. It does not include the volume of pores inside the particles themselves. VS is the volume occupied by the solid particles in the bed. This volume is the total volume occupied by the particles in the bed.

4.3

TYPES OF PARTICULATE SOLIDS

The characteristics of solid particles should also include aspects complementary to those indicated above, among them chemical and physical characteristics. On the side of chemical aspects, apart from the composition and structure of solid fuel, its reactivity should be taken into account. Reactivity represents the inluence of species in the fuel matrix on the rate of heterogeneous reactions. The characteristics of chemical reactivity will be discussed further in Chapter 9.

Basic Calculations

91

On the physical side, characteristics of particles, such as size, form, and porosity, as well as type of pores, exert great inluence not only on reaction rates but also on luidization dynamics. For instance, there are some powders that tend to stick together because of static electricity or accommodation due to too-irregular shapes. Those have been called by Geldart [14, 15] cohesive or class C. His classiication depends on buoyancy-related parameter (difference between particle apparent density and luid density) and average particle diameter. Other types are A (aeratable), B (sand-like), and D (spoutable). As an example, for a buoyancy parameter around 1000 kg/m3, those classes are found within the following ranges of average diameters: • • • •

Below 20 µm for class C 20–200 µm for class A 200–1000 µm for class B Above 1000 µm for class D

Above 1000 kg/m3, the range for class C remains constant or always below 20 µm, but other ranges follow logarithmic linear variations according to the buoyancy parameter. For instance, at around 4000 kg/m3, various classes are found according to the following ranges of average diameters: • 20–60 µm for class A • 60–500 µm for class B • Above 500 µm for class D Powders smaller than class A may still present cohesiveness among particles or interparticle forces, but those can be overcome during luidization. Such forces are negligible for solids of classes B and D. Fortunately, most of fuels and solids added to luidized beds fall into those last categories, and no problems related to cohesiveness happen during luidizations. Almost all correlations developed for the luidization dynamics are useful for particulate solids within those last classes. Of course, Geldart proposes an approximate qualitative classiication, which is useful as a general guide for applications of luidization to various particulate materials. More details on his classiication can be found in the literature [14, 15]. Apart from Geldart’s classiication, there are other solid particulates that, despite not leading to luidization problems, might bring dificulties during storage, transporting, and feeding into reactors, especially into pressurized ones. Among them are the ibrous materials, such as sugarcane bagasse. Those particles easily entangle to form cages in the hopper, which prevent the particles from lowing downward to feeding screws. Proper design and even special systems using vibrators or internal hopper paddles are commonly applied to solve those problems. Feeding those ibrous particles into pressurized reactors is not a trivial problem because compression produces hard lumps that block the feeding system. Gradual increases of pressure between hoppers in cascade might solve the problem, but those systems are expensive. Another alternative is feeding those materials as slurries. Nonetheless, this may result in a loss of eficiency in boilers and gasiiers.

92

Solid Fuels Combustion and Gasification

4.4

TIPS ON CALCULATIONS

In this section, basic guidelines are suggested for the irst steps in preliminary estimations of the main dimensions of combustors or gasiiers. The application of those rules is best illustrated through an example. Suppose a case in which it is desirable to compute a few basic equipment operational parameters and geometry. The particle size distribution, apparent density, composition, and mass low of the solid fuel to be used are known. In addition, the type of combustor has already been chosen. The procedure is as follows: 1. Set a desirable air ratio or excess. For most combustors, values around 1.2, or 20%, are common. Fluidized bed combustors can operate with air excess as low as 10%. In these calculations, use the ultimate analysis of the carbonaceous solid that provides the mass fractions of carbon, hydrogen, oxygen, nitrogen, sulfur, and ash. Use the form for its total oxidation reaction: a a a   CH a H Oa O N a N Sa S +  1 + H − O + N + a S  O 2 → CO 2   4 2 2 +

(4.26)*

aH H 2 O + a N NO + a SSO 2 2

Most gasiication processes use air ratios from 20% to 35%. Take the value of 25% for a irst calculation. The coeficient aj in Equation 4.26 should be computed on dry solid basis. Therefore, the proximate and ultimate analysis of the feeding carbonaceous solid must be known. It should also be remembered that those coeficients are molar ratios, or aj =

w j MC wC M j

(4.27)

where wj is the mass fraction (dry basis) of component j in the feeding solid, as given by the ultimate analysis.† 2. Compute the value for the minimum luidization velocity assuming the gas at the entering conditions (Umf,0). For updraft moving beds, it may be interesting to avoid excessive entrainment of small and lighter particles, which will be trapped between larger particles. In this case, use the apparent *

†

Despite the low importance of nitrogen oxidation in the calculations below, the form chosen is just to maintain coherence in relation to typical carbonaceous fuel composition, which includes nitrogen. Equation 4.27 allows us to obtain a representative formula of carbonaceous solid that includes the volatile, ixed carbon, and ash fractions. Therefore, it is useful only for overall mass balances of processes in which a carbonaceous solid is being fed. As the released volatile usually has a composition different from that of the char (Chapter 10), that equation may not be useful when one is interested in the composition of the solid at a particular point in the equipment interior. This is the case with most combustion and gasiication process, where the solid fuel is devolatilized just after feeding.

93

Basic Calculations

particle density of the ash for calculations. This is so because the particles near the bottom of the equipment are almost devoid of all fuel and moisture. An approximate value for that density is given by ρA = ρS,I wash,I

3.

4.

5. 6. 7.

(4.28)

Table 4.3 lists the apparent densities of most common particles. It is safe to assume the average size of particles equals that of the feeding ones. However, for luidized beds, where inert or absorbent (limestone or dolomite) materials are also fed into luidized beds, the average density and particle size of the added solid should be used in the computations. This is because inert or absorbent commonly represents 80%–90% of the mass of solids in the bed. This observation is valid for all following steps. Compute the minimum luidization velocity assuming the gas to be at its expected maximum temperature (Umf,1). That temperature may be assumed to be around 1500 K for moving beds and suspensions and 1200 K for luidized beds. To be on the safe side, in cases of pneumatic transport, use the particle apparent density and the size of the feeding particles. Compute the value for the terminal velocity of the average particle diameter assuming the gas to be at the entering conditions (UT,0) and at maximum gas temperature (U T,1). Do the same for the terminal velocity for the biggest particle level at those two temperatures (Utall,0 and Utall,1, respectively). Apply the size distribution and average density of the feeding particles. Compute the irst and second limiting turbulent velocities at entering (Utur1,0, Utur2,0) and maximum (Utur1,1, Utur2,1) gas temperatures. Use the average particle size and average density of the feeding particles. Compute the transport velocity at entering and maximum gas temperatures (Utr,0, Utr,1) for the average size and density of the feeding particles. Use the average particle size and average density of the feeding particles. Set a supericial velocity (U0) for the entering gas in order to fall into the range of the desired kind of equipment. The following ranges, which include a requirement for the maximum supericial velocity (U1), may be used: • Moving bed: U0 < Umf,0 and U1 < Umf,1.

TABLE 4.3 Usual Values for Apparent Densities of Particles Solid Species Coal Wood Charcoal Limestone Sand

Apparent Particle Density (kg m–3) 700–2000 200–1000 100–800 2000–3200 2500–3500

94

Solid Fuels Combustion and Gasification

• Bubbling luidized bed (see Section 4.4): Umf,0 < U0 < Uturb1,0 and Umf,1 < U1 < Utur1,1; in particular: • 5Umf,0 ≅ U0 < Uturb1,0 for combustors. • 3Umf,0 ≅ U0 < Uturb1,0 for gasiiers. Usually, bubbling beds operate below the irst turbulent limit; however, this is not an ironclad rule, and conditions in the bed may fall into the turbulent luidization region. • Circulating luidized beds (see Section 4.4): Utr,0 < U0 < Utall,0 and Utr,1 < U1 < Utall,1. Usually, circulating beds operate above the transport velocity, as well as above the second turbulent limit; however, this is not an ironclad rule, and conditions may be found below transport conditions but always above the second turbulent luidization region. • Pneumatic transport or suspension: U0 > Utall,0. The above rules allow selection of a possible range of supericial velocity for the injected gas stream. Take, for instance, a simple average of maximum and minimum values of that range. With the computed mass low of gas (step 1), it is possible to estimate the cross-sectional diameter of the equipment. In addition to the above procedure, historic values related to bubbling luidized bed gasiiers (atmospheric or pressurized) show that the following approximate relationship can be used as a irst and approximate relationship by which to estimate the rate of fuel consumption or the bed cross-sectional area: Fcar ,I = 6.0 × 10 −7 SP.

(4.29)

Here, Fcar,I is the mass low of feeding carbonaceous solid in kg/s, S is the crosssectional area of the gasiier in m2, and P is the operational absolute pressure in Pa. Therefore, that correlation can be applied to check the conditions found after following the routine detailed above. Of course, there should be careful veriication of whether the bubbling conditions are satisied, and that requires the computation of characteristics of the average particle in the bed during steady-state conditions. For this, only a comprehensive simulation of the operation at hand is required. Details of a possible modeling leading to that are within the scope of the present book and are shown in Chapters 13 through 15.

4.5

OBSERVATIONS

For luidization of very ine particles, the difference between the minimum luidization and the bubbling velocity can be observed. That last value can be computed by the following equation [14]: U mb = 2.07

d p,avρG0.06 exp(0.716w 45 ) µ G0.347

(4.30)

Here, w45 is the mass fraction of the sample of the powder with diameters below 45 µm. In this case, to avoid segregation or agglomeration problems, it is advisable

95

Basic Calculations

to always operate combustors, as well as gasiiers, at conditions that surpass the minimum bubbling velocities. To date [16], there is no precise deinition that ensures where the fast luidization regime starts. The transport velocity is a good indication, but it is not valid outside the range indicated by Equation 4.16. One graph that indicates an approximate region can be found in the literature [14]. However, that also presents a considerable level of uncertainty. The criteria introduced here at least ensure that a luidization regime is achieved with a high rate of entrainment, which requires operating under the circulating luidization concept. As a suggestion, the above methodology might be applied to check the results using the chart presented in the literature [14, 17, 18].

4.6

EXERCISES

4.6.1

PROBLEM 4.1* Devise a reiterative procedure and write a simple computer program (in any computer language) to compute the terminal velocity of particles. Use Equations 4.12 through 4.15. Assume particle density, particle diameter, and gas properties as input data.

4.6.2

PROBLEM 4.2*

Given the following conditions for a bed of inert particles: • • • • • •

Bed internal diameter: 1.118 m Particle apparent density: 3000 kg/m3 Temperature of the air injected through the distributor: 370 K Pressure of the air injected through the distributor: 100 kPa Mass low of injected air: 0.7 kg/s Particle size distribution according to the table below

Determine the operational regime regarding the luid dynamics, i.e., whether it will be a moving bed, a luidized bed, a circulating bed, or a pneumatic transport. Use a value for acceleration due to gravity equal to 9.81 m/s2. Compare your computation, using the equations described in Sections 4.2.4 and 4.2.5

TABLE 4.4 A Particle Size Distribution Tyler Number (mesh) #04 #10 #16 #20 #48 #80 #100 Pan or smaller than #100

Mass Percentage Retained 10.20 9.85 36.55 20.40 8.80 4.95 2.85 6.40

96

Solid Fuels Combustion and Gasification related to transitions from bubbling to turbulent and from turbulent to fast luidization and the criteria shown in Section 4.4.

4.6.3

PROBLEM 4.3*

A solid residue is produced at the rate of 1 kg/s, and a process should be chosen for its incineration. The proximate and ultimate analyses of the residue are given below: Table 4.4 gives the particle size distribution of feeding coal. Its apparent particle density at feeding is 1400 kg/m3. If use of the updraft moving bed technique is intended, estimate as follows: a) The mass low of air necessary to operate the equipment with 20% oxygen excess. Assume injected air at 298 K and a maximum temperature of 1500 K, as suggested in Section 4.3. b) The minimum diameter of the equipment to avoid luidization at any point.

4.6.4

PROBLEM 4.4**

Using the data given for Problem 4.3, compute the equipment diameter if incineration in a circulating luidized bed is intended.

4.6.5

PROBLEM 4.5**

Repeat Problem 4.3 if gasiication in bubbling luidized bed is intended. Sand with the same particle size distribution as the feeding coal and apparent density of 3000 kg/m3 is to be used.

TABLE 4.5 Proximate Analysis of a Bituminous Coal Fraction Moisture Volatile Fixed carbon Ash

Percentage (wet basis) 5.0 38.0 47.6 9.4

TABLE 4.6 Ultimate Analysis of a Bituminous Coal Component C H N O S Ash

Percentage (dry basis) 73.2 5.1 0.9 7.9 3.0 9.9

Basic Calculations

97

REFERENCES 1. Kunii, D., and Levenspiel, O., Fluidization Engineering, 2nd Ed., John Wiley, New York, 1991. 2. Wypych, P.W., Design considerations of long-distance pneumatic transport and pipe branching, in Fluidization Solids Handling and Processing, Yang, W.-C., Ed., Noyes, Westwood, NJ, 1999, pp. 712–772. 3. Allen, T., Particle Size Measurement, 2nd Ed., Chapman and Hall, London, 1975. 4. Geldart, D., and Abrahamsen, A.R., Fluidization of ine porous powders, recent advances in luidization and luid-particle systems, AIChE Symp. Series, 77(205), 160–165, 1981. 5. Perry, J.H., Green, D.W., and Maloney, J.O., in Perry’s Chemical Engineers Handbook, 7th Ed., McGraw-Hill, New York, 1997. 6. Grace, J.R., in Handbook of Multiphase Systems, Hetsroni, G., Ed., Hemisphere, Washington, DC, 1982. 7. Babu, S.P., Shah, B., and Talwalkar, A., Fluidization correlations for coal gasiication materials—minimum luidization velocity and luidization bed expansion ratio, AIChE Symp. Series, 176(74), 176–186, 1978. 8. Ergun, S., Fluid low through packed columns, Chemical Engineering Progress, 48, 91–94, 1952. 9. Wen, C.Y., and Yu., Y.H., A generalized method for predicting the minimum luidization velocity, AIChE J., 12, 610–612, 1966. 10. Kestin J., A Course in Thermodynamics, Vols. I and II, Hemisphere, New York, 1979. 11. Poling, B.E., Prausnitz, J.M., and O’Connell, J.P., The Properties of Gases and Liquids, 5th Ed., McGraw-Hill, New York, 2000. 12. Reid, C.R., Prausnitz, J.M., and Poling, B.E., The Properties of Gases and Liquids, 4th Ed., McGraw-Hill, New York, 1987. 13. Perales, J.F., Coll, T., Llop, M.F., Puigjaner, L., Arnaldos, J., and Casal, J., On the transition from bubbling to fast luidization regimes, in Circulating Fluidized Bed Technology III, Basu, P., and Horio, M., Eds., Pergamon Press, Oxford, United Kingdom, 73–78, 1991. 14. Geldart, D., Gas Fluidization Technology, John Wiley, Chichester, United Kingdom, 1986. 15. Geldart, D., Behavior of gas-luidised beds, Powder Technology, 6, 201–215, 1973. 16. Kwauk, M., and Li, J., Fluidization regimes, Powder Technology, 87, 193–202, 1996. 17. Reh, L., Verbrennung in der. wirbelschicht, Chem-Ing-Techn., 40, 509–514, 1968. 18. Grace, J.R., Gas Fluidization course, Center for Professional Advancement, East Brunswick, NJ, 1984. 19. Basu, P., Combustion and Gasiication in Fluidized Beds, CRC Press, New York, 2006. 20. Horio, M., and Morishita, K., Flow regimes of high velocity luidization, Jpn. J. Multiphase Flow, 2(2), 117–136, 1988. 21. Lanneau, K.P., Gas–solid contacting in luidized beds, Trans. Inst. Chem. Eng., 38, 125, 1960. 22. Kehoe, P.W.K., and Davidson, J.F., Inst. Chem. Eng. (London) Symp. Series, 33, 97, 1971. 23. Massimila, L., AIChE Symp. Series, 69(128), 11, 1973.

5 Zero-Dimensional Models CONTENTS 5.1 5.2

5.3

5.4 5.5 5.6 5.7

Introduction .................................................................................................. 100 Basic Equations............................................................................................. 100 5.2.1 Mass Balance .................................................................................... 100 5.2.2 Energy Balance ................................................................................. 101 5.2.2.1 Isenthalpic .......................................................................... 102 5.2.3 Eficiencies........................................................................................ 103 5.2.3.1 Based on Enthalpy ............................................................. 103 5.2.3.2 Special Eficiencies ............................................................ 104 Species Balance and Leaving Composition.................................................. 105 5.3.1 Balances Involving Chemical Species .............................................. 106 5.3.2 Chemical Reaction ............................................................................ 106 5.3.3 Chemical Equilibrium ...................................................................... 107 5.3.3.1 Computation of Equilibrium Coeficients.......................... 108 5.3.4 The Composition of Leaving Streams .............................................. 108 5.3.4.1 Algebraic Approach ........................................................... 108 5.3.4.2 Differential Approach ........................................................ 110 Useful Relations ............................................................................................ 111 5.4.1 Example 5.1 ...................................................................................... 113 5.4.2 Example 5.2 ...................................................................................... 116 Summary of the 0D-S Model ....................................................................... 117 Flame Temperature ....................................................................................... 118 5.6.1 Example 5.3 ...................................................................................... 119 5.6.2 A More Realistic Approach .............................................................. 121 Exercises ....................................................................................................... 121 5.7.1 Problem 5.1 ....................................................................................... 121 5.7.2 Problem 5.2 ....................................................................................... 121 5.7.3 Problem 5.3 ....................................................................................... 121 5.7.4 Problem 5.4 ....................................................................................... 121 5.7.5 Problem 5.5 ....................................................................................... 121 5.7.6 Problem 5.6 ....................................................................................... 121 5.7.7 Problem 5.7 ....................................................................................... 122 5.7.8 Problem 5.8 ....................................................................................... 122 5.7.9 Problem 5.9 ....................................................................................... 122 5.7.10 Problem 5.10 ................................................................................... 122 5.7.11 Problem 5.11 ................................................................................... 122 99

100

Solid Fuels Combustion and Gasification

5.7.12 Problem 5.12 ................................................................................... 123 5.7.13 Problem 5.13 ................................................................................... 123 5.7.14 Problem 5.14 ................................................................................... 123 References .............................................................................................................. 124

5.1

INTRODUCTION

As already introduced, zero-dimensional modeling is the simpler level of modeling and simulation for equipment or processes. Even if one is intending to develop multidimensional models, it is strongly advisable to apply this approach, at least for a irst veriication of the basic characteristics of the equipment or system operation. From now on, equipment or any part of it would be called control volume (CV). To each CV it is possible to associate a control surface (CS) enveloping it. Zero-dimensional models are based on the fundamental equations of mass and energy conservation, which can be applied to the whole CV or identiiable parts of that volume. These computations should be as rigorous as possible, because any conclusion about the operation of a piece or set of equipment should be airtight concerning mass and energy conservation. All professionals with a degree in an exact science are familiar with those conservation equations. However, they are presented here to • Introduce the notation that is used at several other points in the book • Refresh concepts and generalizations useful to build zero-dimensional models, as well to evaluate their results • Illustrate some calculations for cases of combustors and gasiiers The treatment below is given with regard to combustors and gasiiers in general. A generalization of a zero-dimensional (0D) model that may be used for any equipment or industrial process is shown in Appendix B. That method would allow simulation of systems with many units, such as those in power generation plants.

5.2

BASIC EQUATIONS

A CV is an open system since mass can be transported through the CS. Energy can be conveyed from and to a CV, just as heat or work can. These transferences can be observed at the only CS that surrounds the CV. The two basic equations are derived from the equations for conservation of mass and energy.

5.2.1

MASS BALANCE

The general mass balance for each CV of a system composed of nCV pieces of equipment or parts is found by nSR

∑F

I

i SR i SR , i CV

i SR =1

=

dM i CV , dt

1CV ≤ i ≤ n CV

(5.1)

101

Zero-Dimensional Models

6, I6= 1

1, I1=1 Equipment (iCV = 1)

8, I8= 1

4, I4=1

nSR = 9

FIGURE 5.1 Example to illustrate the notation used.

where Micv is the mass found in the CV iCV at a given time t. The associated parameter I iSR , i CV indicates the direction (in or out) of low F for each stream iSR crossing the CS of the CV iCV. If the stream iSR is entering the CV, it should be assigned a value of +1, and if it is leaving, it should be assigned –1. The value 0 is left for streams that neither enter nor leave that CV. Figure 5.1 illustrates the notation and a simple case where streams 1 and 4 enter and streams 6 and 8 leave the CV, which has been named equipment number 1. The total number of streams (nSR) is 9; therefore, I2,1, I3,1, I5,1, I7,1, and I9,1 are 0. This notation is convenient because it simpliies the application to computations where any number of linked CVs (nCV) is involved. Of course, the parameters I iSR ,i CV will be represented by a matrix with nSR × nCV elements. Of course, for 0D-S models, the mass balances become nSR

∑F

i SR =1

5.2.2

I

i SR i SR , i CV

= 0 , 1 ≤ i CV ≤ n CV.

(5.2)

ENERGY BALANCE

The energy balance—also known as irst law of thermodynamics—can be written as nSR

∑F

i SR

i SR=1

  u i2SR i −W  i CV = dE i CV , 1 ≤ i CV ≤ n CV + + gz iSR  I iSR,i CV + Q h i CV  SR  2 dt

(5.3)

and, for a steady-state operation, nSR

∑F

i SR =1

i SR

  u i2SR i −W  i = 0, + + gz iSR  I iSR,i CV + Q h i  SR CV CV 2  

1 ≤ i CV ≤ n CV.

(5.4)

Unless otherwise indicated, the steady-state operation will be assumed from this point on. This is so because most industrial units operate at or near a steady-state regime, or they might to some degree be approximated by that class of regime. The

102

Solid Fuels Combustion and Gasification

deduction of similar forms for the above equations can be found in any classical text on thermodynamics [1–3]. Equation 5.4 also uses index iSR to indicate the conditions of parameters or properties related to a stream, such as the following: • Enthalpies (h); these should include the formation and sensible, as deined below (Section 5.4) • Velocities (u) • Vertical positions (z) Each of those values is an average computed over the cross-section of the stream iSR at the point it crosses the CS of a CV iCV. To use a more relaxed language,* the notation for signals can be summarized as follows:  i , is assigned a positive value if it is made from • The rate of heat transfer, Q CV another CV or environment to the CV (iCV) and negative if in the opposite direction. In the absence of adiabatic walls between the CV and the neighboring CV or environment, heat is transferred from the body with a higher temperature to that with a lower temperature.  i is assigned a positive value if it is • The rate of work (or power) transfer W CV made from the CV (iCV) to another CV or environment and negative if in the opposite direction. Power is performed by the CV if the action could be translated by the lifting of a weight in the environment, and consumption if the iteration could be translated by the resisted descent of a weight in the environment (see Appendix B). An example is a steam turbine. It delivers power and usually loses heat to the ambi i would be positive, and Q  i would be negative. ent environment. In this case, W CV CV 5.2.2.1 Isenthalpic As we have seen, modeling allows veriication of the limiting operational conditions or maximum possible performances of equipment and processes. Once such analyses are completed, the application of proper coeficients permits determination of the performance of real operations. Limiting or ideal conditions require few assumptions; the usual ones are the following: a) Operation is adiabatic. Therefore, heat transfer through the CS is zero, or  = 0. Q

(5.5)

b) No work is involved or consumed, or  = 0. W

*

For a more rigorous approach, consult Appendix B.

(5.6)

103

Zero-Dimensional Models

c) Kinetic and potential energy are negligible compared with the variations of enthalpy among inputs and outputs, and nSR

∑F

i SR

i SR =1

 u i2SR  + gz iSR  I iSR,i CV = 0.   2 

(5.7)

In the present text, a model is called isenthalpic when it operates at steady-state conditions and follows requirements given by Equations 5.5, 5.6, and 5.7. Therefore, Equation 5.4 becomes nSR

∑F

i SR

h iSR I iSR,i CV = 0.

(5.8)

i SR =1

It is important to stress that isenthalpic here means no overall variation of enthalpy. This differs from the usual concept that classiies isenthalpic processes as those involving just one input and one output stream with same enthalpy. An example is the throttling of a luid through a valve, which also follows Equation 5.8. Of course, for most equipment, only approximations of that limit are possible. For instance, complete adiabatic conditions are never met and only the following can, sometimes, be assumed. nSR

∑ FiSR h iSR I iSR ,iCV >>

i SR =1

nSR

∑F

i SR

i SR =1

 u i2SR  + gz iSR  I iSR,i CV   2 

(5.9)

Equipment that approaches the isenthalpic model is here called near-isenthalpic. Examples of near-isenthalpic equipment are combustors, gasiiers,* heat exchangers, mixers, valves, and distillation columns. How well they approach that limit is provided below by the application of eficiencies.

5.2.3

EFFICIENCIES

The literature presents various concepts of eficiency, and the most important among those are discussed below. 5.2.3.1 Based on Enthalpy The eficiency for near-isenthalpic equipment may be computed from nSR

∑F

i SR

h iSR I iSR,i CV − (1 − ηental, i CV) H ref = 0.

(5.10)

i SR =1

*

Some gasiiers include mixing devices and therefore involve power transfer through their CS. However, in most situations, those amounts are negligible compared with the variations of enthalpy among input to output streams.

104

Solid Fuels Combustion and Gasification

Alternatively, this equation can be written in a molar basis. The eficiency ηental is related to the particular deinition of reference enthalpy Href. For instance, for a heat exchanger that reference is deined as the enthalpy variation in the hot stream or in the cold stream. In other cases, the deinition should be precise because it depends on the adopted standard or reference. However, it should be stressed that according to Equation 5.10, decreases in eficiency for near-isenthalpic CVs are mainly due to loss of heat to the environment. Therefore, an adiabatic process would present 100% eficiency since no power is involved and variations in kinetic and potential energy are considered negligible. Some examples of commonly assumed eficiencies follow: • Combustor or furnaces, around 0.97. Here Href is understood as the low heating value (LHV) (wet basis)* of the fuel times its mass low. Note that eficiency would decrease only if there is heat exchanged with the environment. Standard recommendations [4] set that eficiency at 97%. This is completely different from the eficiency of the combustion process itself, or combustion eficiency, which is deined as the fraction of fuel converted to gas oxidized products. The combustion eficiency varies too much among processes; nonetheless, values around 99% are easily found. Note that losses by unconverted fuel would be accounted for by the enthalpies in the exit stream or by terms inside the sum of Equation 10. It is important, once again, to call attention to the term (1 − ηental,i CV) H ref

• • • •

which represents only the heat transfer of energy to environment. No other losses—such as unconverted fuel—are involved. Gasiiers, around 0.97. The same observations as for combustors are valid. Heat exchangers, between 0.96 and 0.98. Here, Href is the variation of enthalpy in a luid stream (hot or cold side) through the exchanger times its mass low. Mixers and splitters, around 0.99. Href is the mass low of the entering streams multiplied by their respective enthalpies. Boilers, between 0.95 and 0.97. Now, H ref should be computed in a way similar to that for combustors.

5.2.3.2 Special Efficiencies Of course, the above equations, especially those related to isenthalpic and near-isenthalpic equipment, are useful for a zero-dimensional model of combustor or gasiier. However, it is important to stress again that the usual deinitions for combustion and gasiication eficiencies are completely different from the above. They cannot be related to the deinition used in Equation 5.10, just because those include losses due to, for instance, unconverted fuel. The main preoccupation of such deinitions is to allow comparisons among processes and how well the fuel is consumed to *

See Appendix B.

105

Zero-Dimensional Models

provide a leaving stream with either high temperature or high heating value (HHV) as fuel. In the case of combustors, it is common to compute the fraction of fuel converted into lue gases. Typical values are as follows: • For combustors consuming solid fuels, such as coal and biomass: • 98%–99% in pulverized suspension, moving beds, or circulating luidized beds • 95%–99% in bubbling luidized beds • 99% or more in combustors consuming gas or liquid fuels In case of gasiiers, the eficiency based on fraction of fuel consumption is normally 5%–10% below those found for combustors. However, in the case of gasiiers, the eficiency is usually deined by Gasifier Efficiency =

Fout h out Ffue1h fue1

(5.11)

where the index out refers to the leaving gas product. LHV is normally used for the enthalpy of solid fuel.* Some authors used to add the values of mass low multiplied by the enthalpy of all entering streams to the denominator of Equation 5.11. Actually, this is a more realistic and representative approach. Gasiier producers also provide the eficiency according to the following basis: • Hot basis. For that, the combustion enthalpy of exiting gas mixture computed at 298 K should be added to the sensible enthalpy of that stream. The composition should consider all components, including water and tar. Typical values in moving-bed and luidized-bed gasiiers are around 70%–90%. • Cold basis. Only the combustion enthalpy at 298 K of the leaving gas stream is used, and its composition should be taken on a dry and tar-free (or dry and clean) basis. Typical values in moving-bed and luidized-bed gasiiers vary from 50% to 70%. In addition to those deinitions, one based on exergy is presented in Appendix B. That may provide a very useful method for analysis of combustor and gasiier performance.

5.3

SPECIES BALANCE AND LEAVING COMPOSITION

Of course, if no chemical reaction is involved inside a CV, the chemical species do not suffer modiication. The composition obtained by mixing all leaving streams is the same as that obtained by mixing all entering streams. However, if chemical reactions occur inside the CV, the composition of each leaving stream should be known. *

Other authors use the HHV of the fuel. Therefore, care should be taken when comparing different processes. Details on the enthalpy of fuels can be found in Appendix B.

106

5.3.1

Solid Fuels Combustion and Gasification

BALANCES INVOLVING CHEMICAL SPECIES

The total low of a stream iSR in or out of the CV is given by n CP

∑F

FiSR =

i CP ,i SR

(5.12)

i CP =1

where the index iCP refers to the chemical component or species in the stream iSR. Combination of Equations 5.2 and 5.12 provides nSR

∑F

I

i CP ,i SR i SR ,i CV

= 0, 1 ≤ i CP ≤ n CP .

(5.13)

i SR =1

In Equation 5.3, the total enthalpy hSR in the stream iSR is given by the contributions from each chemical species present in the stream. These enthalpies include the formation enthalpy of each species plus its sensible enthalpy. Therefore, the internal energy increase or decrease due to chemical reactions is automatically accounted for. Obviously, combustors and gasiiers involve a great number of reactions, and methods to estimate (at least approximately) the chemical composition of the exit streams of a CV should be established.

5.3.2

CHEMICAL REACTION

A representation of chemical reaction i can be given by n CP

∑ν

j, i

Ξj = 0

j =1

(5.14)

where the species j is symbolized by formula Ξj. In the present book, the stoichiometric coeficients ν are positive for the products and negative for the reactants. A general description sets the stoichiometric coeficient as 0 for species j not participating in reaction i. As will be described in Chapter 8, the reactions between two chemical species are in fact a complex combination of several elementary reactions. The sequence of these reactions is called the chain mechanism and involves many stable and unstable chemical species (ions). Most works in the area of combustion and gasiication just show the overall reaction, which is the inal result of the combination of elementary reactions. In general terms, the rate of a reaction i can be written as ri =

1 dρ j ⇒ m = nCP − νm ,i ⇐ n = nCP νn ,i = k i ∏ ρ m − k i ∏ ρ n ν j,i dt m =1 n =1

(5.15)

where the superscript ⇒ indicates the forward reaction and ⇐ the reverse reaction. Equation 5.15 can be presented under other forms, such as when partial pressures replace the molar concentrations ρ .

107

Zero-Dimensional Models

Since chain mechanisms are the real processes, most reaction rates are not correctly described by Equation 5.15, and the concentration exponents are not equal to the respective stoichiometric coeficient ν. In addition, many reactions involve components such as catalysts or poisoning agents that, despite taking part in the chain mechanism, do not appear in the overall form. One example is carbon monoxide combustion, in which water acts as a catalyst, and its concentration should be involved in the calculations of the reaction rate.* Usually, the reaction rate coeficients can be described as a function of temperature in the following form: k i = T ai b ie

−

i E  RT

(5.16)

.

The parameter ai is 0, in the classical Arrhenius form for such coeficients. Although for some situations, the activation energy can be deduced using the classical thermodynamics, its value, as well as that of other parameters involved in the above relationship, is determined through experimentation. This is also due to possible effects of catalysts or poisons. In the case of combustion or gasiication, reactivity is another interfering factor in reaction kinetics. This point, as well as the values of the various parameters in several reactions, is discussed in some detail in Chapter 11.

5.3.3

CHEMICAL EQUILIBRIUM

If the residence time of reacting species in the reactor is long enough, chemical equilibrium will be approached. At that, the direct and inverse rates are equal, and the equilibrium coeficient can be deined as follows: ⇒

n CP

∑  ν ν   P  j =1 K i = ⇐ = ∏ ρ j j,i =  ∏ x j j,i      P0   j=1 j=1 ki ki

n CP

n CP

ν j,i

.

(5.17)

Despite the observation made before concerning possible exceptions to the form of Equation 5.15, the equilibrium relationship in Equation 5.17 is always valid. The reason for that lies in the fact that chain reactions—composing the overall reaction— involve unstable ions. Nonetheless, at equilibrium, the concentration of those unstable species is negligible, and only stable molecules remain. Hence, at equilibrium, the overall form does show the true relationship between chemical species. Near equilibrium conditions and once the forward reaction rate is known, Equation 5.17 can be used to determine the rate of reverse reaction. It also important to note that chemical equilibrium is possible only if physical equilibrium has been achieved as well. In other words, if the system has achieved chemical equilibrium and contains various phases, the equilibrium concerning pressure, as well mass and energy transfers between the phases, is also established. *

Details are described in Chapter 8.

108

Solid Fuels Combustion and Gasification

5.3.3.1 Computation of Equilibrium Coefficients As shown in Appendix B, equilibrium conditions for a reaction lead to the following equation: 0  ∆G K i = exp  −   RT

  . 

(5.18)

The above equation provides a method by which to compute the relationship between the chemical species when a reaction reaches equilibrium. It is valid for conditions where the gaseous components might be assumed to be ideal gases. For conditions that depart from the ideal, the pressures should be replaced by fugacities or activity coeficients to give [1] the following:  0  nCP ν j,i  ∆G aˆ j . K i = exp  − =   ∏  RT j=1

(5.19)

In the case of solid-gas reactions, a reasonable approximation is to assume the activities of solids as 1 [5]. Accordingly, Equation 5.18 should include only the partial pressure of gaseous components. For instance, the following reaction C + H 2O ⇔ CO + H 2 would lead to the following relationship between partial pressures at equilibrium K=

5.3.4

PCO PH2 . PH2 O

THE COMPOSITION OF LEAVING STREAMS

As noted above, a zero-dimensional model cannot determine the point-by-point conditions inside equipment. In order to compute the conditions of leaving streams, most such models are forced to assume chemical equilibrium at those streams, which is often a crude approximation. However, if equilibrium is assumed, the composition of a CV exiting stream can be estimated. Such calculations should be seen as a irst approach, providing the basis for further improvements. In addition, these results might allow interesting conclusions regarding limits of eficiency and performance of a given process. Two basic approaches can be used for chemical equilibrium calculations: algebraic and differential. 5.3.4.1 Algebraic Approach Assume methane combustion, or CH 4 + 2O2 ⇔ CO2 + 2H 2O.

(5.20)

109

Zero-Dimensional Models

Imagine that a mixture of methane and oxygen is injected into a combustion chamber (with or without some carbon dioxide and water). Despite the variations of temperature and pressure throughout the chamber, let us assume that the leaving conditions are known. If the initial concentrations (kmol/m3) of methane (ρ CH4,0), oxygen (ρ O2,0), carbon dioxide (ρ CO2 ,0), and water (ρ H2O,0) are known and assuming that after the reaction, the concentration of methane is decreased by a certain amount (a) (kmol/m3), the equilibrium condition at the end of the process leads to

(ρ K= (ρ

CO2,0 CH 4 ,0

=

)( − a ) ( ρ

) − 2a )

)( − a ) ( ρ

) − 2a )

+ a ρ H2O,0 + 2a

( (ρ

O 2 ,0

2

ρ CO2,0 + a ρ H2O,0 + 2a CH 4 ,0

O 2 ,0

2

 P  P  0

(5.21)

2

2

2+1−1− 2

.

The inal concentrations would be ρ j = ρ j,0 + ν ja

(5.22)

where νj is the stoichiometric coeficient of component j in the reaction. Therefore, once the temperature of the exiting stream is known, Equation 5.18 allows the equilibrium constant to be determined for concentrations in that exit steam computed by Equations 5.21 and 5.22. Now let us assume a process involving several reactions i (1 < i < nRE). It is possible to show that for a system in which several reactions occur simultaneously, the equilibrium will require that all of them achieve the equilibrium as if isolated [1]. Moreover, if ideal gases are involved, the equilibrium parameters K i are functions of the temperature. If ideal gases are not involved, those parameters are functions of temperature and pressure. In any case, if the thermodynamic state is established, K i can be determined, and the above equations may be solved to provide the concentrations of all nCP components at the equilibrium concerning reactions i. It is also important to note that computations of equilibrium involving many reactions would be coherent only if those reactions are independent.∗ In other words, no reaction among those involved can be written as a combination of the others selected to model the process. For a CV operating in steady-state mode, the application of a zero-dimensional model would require the following conditions before the chemical equilibrium of leaving stream could be assumed and computed: • Perfect plug-low or perfect mixing of all components • Long residence time in the CV • Knowledge of the exiting stream temperature at which the equilibrium is assumed ∗

See Chapter 8.

110

Solid Fuels Combustion and Gasification

The perfect mixing condition means that no difference (or only a negligible difference) in composition can be detected among all points inside the CV. The perfect plug-low regime means that no diffusion among the various species is allowed. This ensures that only chemical reactions can modify the composition of the reacting front, and no other effects, such as diffusion among the present species, disturb it. 5.3.4.1.1 Presence of Other Nonparticipating Components Very often, a chemical reaction occurs in the presence of other nonparticipating components, such as an inert gas. For instance, for simpliied models of fuel combustion in air, nitrogen is taken as inert. However, the inert gas “dilutes” the mixture and therefore the concentration of all gases in the mixture. As an example, imagine the combustion of propane in air, which may be written as follows. C3H8 + 5O2 → 3CO2 + 4H 2O

(5.23)

If the concentration of propane decreases by a, the inal total concentration (moles per volume unit) at equilibrium would be ρ = (ρ C3H8,0 − a ) + (ρ O2 ,0 − 5a ) + (ρ CO2,0 + 3a ) + (ρ H2O,0 + 4a ) + (ρ N2,0 ) = ρ C3H8,0 + ρ O2,0 + ρ CO2,0 + 3.762ρ O2,0 + a + ρ N2,0 .

(5.24)

Note that here, the total number of moles in the CV does not remain constant, as in the case of methane combustion, and the initial number of moles of nitrogen is just 3.762 (the usual molar N2/O2 proportion in air) times the number of moles of oxygen. The molar fraction can now be written, and the method of solution would follow as explained above. It should be noted that this form is general. For instance, it does not require the initial number of moles of oxygen to be stoichiometric, i.e., 5 times that of propane. This leaves the method useful for cases where excess oxygen or air is used in the combustion process. 5.3.4.2 Differential Approach Observing Equation 5.24, it is easy to imagine that the solutions should present some restrictions. Among them, it is required that concentrations of all chemical species be positive or that the molar or mass fractions of all involved species should add up to 1. In the solution of a complex system, several such restrictions may lead to cumbersome computations. Besides, the previous approach is limited to zero-dimensional models only. An alternative to that approach is to use differential equations. Let there be a system or equipment in which nRE reactions are occurring. In this case, the following system of differential equations can be written as follows.  m = nCP dρ j i = nRE ⇒ 1 = ∑ k i ν j,i  ∏ ρ m− νm ,i − Ki dt i =1  m =1

n = n CP

∏ ρ n =1

νn , i n

  

(5.25)

111

Zero-Dimensional Models

This provides an excellent method by which to determine the chemical equilibrium for a system because the derivatives of the concentration of chemical species j approach zero when the equilibrium is approached as well. Thus, if the system formed by the above differential equations is solved and the solution is taken for a long residence time, the compositions would be very near those at the equilibrium condition. The value of residence time that provides a near-equilibrium condition can be obtained by comparing concentrations at increasing values of residence time. If no meaningful variations in composition are seen, the respective residence time could be used for other integrations involving the same system at similar temperature, pressure, and initial concentration conditions. In addition, the independent variable time t can be replaced by space (for instance, z). This may be useful for future higher dimensional models in which one or more spatial coordinates would be considered. As an example, for a irst-dimensional model, it is possible to write dρ j 1 dρ j = dz u dt

(5.26)

where u is, for instance, the average velocity in the direction of the main stream inside a plug-low combustor. The boundary condition for each differential Equation 5.25 is the average concentration of component j after mixing all streams entering the CV. This is called mixing-cup or bulk entering composition. Various methods can be applied to solve the nonlinear differential system in Equation 5.25. Among them are the variable-pace methods, such as Adams and GEAR. Those are commonly available in commercial mathematical libraries. The details of numerical solutions are not in the scope of the present text, but an extensive body of literature on the subject is available [6–9].

5.4

USEFUL RELATIONS

The following relations might help during calculations, as well as helping to express the results in more convenient forms: • The mass fraction of component iCP in the exit stream iSR may be computed by Fi CP,iSR = FiSR w i CP,iSR.

(5.27)

• The relationship between mass and molar fractions is given by w i CP,iSR =

x i CP,iSR M i CP M iSR

(5.28a)

or x i CP,iSR =

w i CP,iSR M i CP n CP

∑w j=1

j,i SR

M j,iSR

.

(5.28b)

112

Solid Fuels Combustion and Gasification

• The average molecular mass of the stream is given by n CP

M iSR = ∑ x j,iSR M j,iSR.

(5.29)

j=1

• The molar fractions of components in the stream are given by x i CP,iSR =

ρ i CP,iSR . ρ iSR

(5.30)

• The molar density of a stream is related to the average density of the stream by the simple relationship ρ iSR =

ρiSR . M iSR

(5.31)

For an initial approach, the computations for average enthalpy and entropy of each stream can be approximated by h iSR =

n CP

∑w

i CP ,i SR

h i CP,iSR

(5.32)

s

(5.33)

i CP =1

and s iSR =

n CP

∑w

i CP ,i SR i CP ,i SR

.

i CP =1

More elaborate studies on these averages can be found elsewhere [10, 11]. The enthalpies and entropies of an ideal gas can be computed by the following equations: T

h = h f + ∫ c dT

(5.34a)

T0

or in mass basis T

h = h f + ∫ c dT

(5.34b)

T0

and T

s =

c

P

∫ T dT − R ln P

(5.35a)

0

T0

or T

s=

c

R

P

∫ T dT − M ln P

T0

0

.

(5.35b)

113

Zero-Dimensional Models

In the case of an ideal (incompressible) liquid, the above equations become T

h = h f + ∫ c dT − (P − P0 )v

(5.36)

T0

and T

s =

c

∫ T dT.

(5.37)

T0

Convenient tables for properties, including formation enthalpies and entropies at the reference temperature T0 (normally taken as 298 K), as well as approximations of speciic heat as a function of temperature, are available in the literature [1–3, 5, 10]. One table for selected substances is presented at the end of Appendix B (Table B.1). The corrections for departure from the ideal behavior can be computed using the Redlich-Kwong equations or even more elaborate relations. This subject can be found in the literature [10–11], but it is beyond the scope of the present text. Finally, for fuels, the deinition of representative formation enthalpy (see Appendix B, Section B.7), based on the LHV, can be used to facilitate the calculation of total enthalpy. In the case of combustors and gasiiers, the energy carried by exiting low of unconverted solid fuel should also be accounted for. The composition of that residue can be computed by the reasoning shown in Section B.7, as well. The above procedure allows solid fuels and residues to be treated like any other chemical species, and it allows the energy balance equations to be applied (Section 5.2) without any special modiications.

5.4.1

EXAMPLE 5.1

Here, an example for computation of equilibrium conditions is presented. The steam reform of natural gas is a well-known process, described basically by CH 4 + H 2O ↔ CO + 3H 2 .

(5.38)

This reaction is usually promoted in tubular pressurized reactors at 20 bar, and the products leave at 900°C as illustrated by Figure 5.2. 1 Steam 3 Reformer Products 2 Methane

FIGURE 5.2 Scheme of tubular reformer of natural gas.

114

Solid Fuels Combustion and Gasification

If methane and water enter the reactor at the stoichiometric proportion, compute the composition of the exiting stream, assuming that equilibrium was reached. Ideal gas behavior is assumed as an approximation. Solution Equation 5.18 should be used. Taking the basis of 1 kmol/m3 of methane (actually, one may depart from any basis because only the relative concentrations are required), we have the following: 3 • ρ CH4,0 = 1 kmo1/m . 3 • ρ H2O,0 = 1 kmol/m . 3 • ρ CO,0 = 0 kmol/m . 3 • ρ H2,0 = 0 kmol/m . • The inal concentration of methane ρ CH4 is unknown.

Let us now assume that methane concentration decreases by a. Therefore, the inal concentrations would be as follows: • • • •

ρ CH 4,0 = ρ CH 4,0 − a = 1 − a ρ H2 O = ρ H2 O,0 − a = 1 − a ρ CO = a ρ H2 = 3a

Next, let us compute the equilibrium coeficient by Equation 5.18, where xj = and

ρ j n j = ρ n

(5.39)

n CP

ρ = ∑ ρ j = 2(1 − a ) + 4a = 2(1 + a ).

(5.40)

j

Therefore, x CH4 =

1− a 2(1 + a )

(5.41)

x H2 O =

1− a 2(1 + a )

(5.42)

x CO =

a 2(1 + a )

(5.43)

x H2 =

3a 2(1 + a )

(5.44)

115

Zero-Dimensional Models

and Ki =

P 27a 4 4(1 − a )2 (1 + a )2  P0 

2

(5.45)

where P/P0 = 20. The variation of Gibbs free energy is detailed in Appendix B and computed by Equation B.11, or T T n CP n CO       0 = ∑ ν j,i g 0 , j (T) = ∑ ν j,i   h f + c j dT  − T  s 0 , j + c j dT   . (B.11) ∆G ∫ ∫ T j=1 j=1 298 298      

The values for formation enthalpy and absolute entropy can be found in Appendix B (Table B.1), leading to T

∫ c dT = a (T − 298) + a 1

298

2

1  T 2 − 2982 T3 − 2983 1 + a3 − a4  −  T 298  2 3

(5.46)

and T

c 1  T 2 − 2982 a 4  1  T  = dT a a T a ln + ( − ) + −  2 − 298 1 2 3   . ∫298 T 298 2 2 T 2982 

(5.47)

In the present case, T = 1173.15 K, and  0 = g 0 ,CO + 3g 0 ,H2 − g 0 ,CH4 − g 0 ,H2O ∆G = [(−110600 + 26292) − 1173.15 × (197.68 + 40.90)] + 3 × [(0 + 27267) − 1173.15 × (130.61 + 41.89)] − [(−74860 + 49299) − 1173.15 × (186.44 + 71.27)]

(5.48)

− [(−241980 + 33520) − 1173.15 × (188.85 + 50.95)] = −364198 − 525304 + 327893 + 489781 = −71828 kJ/kmol and

  −71828 K i = exp −  = 1578.3.  8.3142 × 1173.15 

(5.49)

This last result, applied to Equation 5.45, leads a single solution between 0 and 1, or equal to 0.75482. Therefore: x CH4 =

1− a = 0.06986 2(1 + a )

(5.50)

116

Solid Fuels Combustion and Gasification

x H 2O =

1− a   = 0.06986 2(1 + a )

(5.51)

x CO =

a   = 0.21507 2(1 + a )

(5.52)

x H2 =

3a   = 0.64521. 2(1 + a )

(5.53)

As we have seen, the natural gas reforming with steam might lead to a product with a relatively high concentration of hydrogen. The heating value of the stream is much higher than that of the reactant mixture. However, the process requires energy to be provided in form of heat transfer* to the CV, which can be computed by the energy balance as illustrated in the following example.

5.4.2

EXAMPLE 5.2

Assume a reforming reactor into which 1 kmol/min of methane at 20 bar and 298 K and the same rate of steam at 400 K and 20 bar are injected. The products leave at 900°C, or 1173.15 K. Compute the required rate of heat transfer to the reactor. As steady-state conditions prevail, no work is involved, and neglecting the variations in kinetic and potential energy of the streams, from Equation 5.4 it is possible to write nSR

∑F

i SR

i =0. h iSR I iSR,i CV + Q CV

i SR =1

(5.54)

The enthalpy of each stream is computed as h iSR =

1  nCP h iSR =  ∑ x j,iSR M iSR M iSR  j=1

TiSR    f , j + c jdT  . h   ∫    298

(5.55)

The numbers deining each stream are shown in Figure 5.2. Using Table B.1, the enthalpies can be calculated as follows h1 =

1 M H 2O

400    f,H O + c H O dT = 1 (−241980 + 3456) h 2  18.015  2 ∫   298

(5.56)

= −13240.3 kJ/kg

*

Energy for the process could also be provided by work transfers, such as those created by using mixing devices.

117

Zero-Dimensional Models

h2 =

1 M CH4

298    f,CH4 + c CH4 dT = 1 (−74850) h  16.043  ∫   298

(5.57)

= −4665.5kJ/kg 4

h 3 = ∑ x j,3 j=1

=

1 Mj

1173.15    f ,j + h c j dT  ∫   298

0.06986 0.06986 (−74860 + 49299) + (−241980 + 33520) 16.043 18.015 0.21507 0.64521 + (0 + 27267) (−110600 + 26292) + 28.010 2.0018

(5.58)

= −111.3 − 808.38 − 647.35 + 8718.0 = 7151.0 kJ/kg as: F1 = 18.015/60 kg/s, I1 = 1, F2 = 16.043/60 kg/s, I2 = 1, F3 = F1 + F2 = 34.058/60 kg/s, I3 = –1, nSR

 i = − ∑ Fi h i I i ,i Q SR SR CV CV SR i SR =1

−13240.3) + 16.043(−4665.5) − 34.058 × (7151.0)] /60. = −[18.015 × (−

(5.59)

= 9282 kW ≅ 9.3 MW As can be seen, the process is highly endothermic and requires a considerable rate of energy input in the form of heat. Some processes take advantage of this by recovering energy from a turbine exit to reform the natural gas, which would be injected into the combustion chamber of the same turbine. These processes are called thermochemical cycles, and more details can be found elsewhere [4].

5.5

SUMMARY OF THE 0D-S MODEL

Once the following are set, the mass low, composition, and temperature of exiting stream can be determined by solution of system formed by Equations 5.2, 5.10, 5.12, 5.17, and 5.18: • Composition, mass low, and temperature of all streams entering the combustor or gasiier • A group of selected independent reactions representing the process • Value for eficiency (Href ) (Section 5.2.3.1) • Pressure of leaving stream (often, isobaric conditions throughout the combustor or gasiier are assumed)

118

Solid Fuels Combustion and Gasification

Excluding the inert components, the reactions should include all components of the exiting stream. Therefore, the system would be formed by nCP (mass lows of all components) variables plus 1 (the temperature). It is a reiterative process, in which the equilibrium conditions should be computed after each guess at the temperature of exiting stream. This would provide a result for equilibrium 0D-S model of a combustor or gasiier. As an example, suppose carbon and air are injected into a combustor. Air is assumed to be composed only of oxygen and nitrogen, and the nitrogen is assumed to be inert. Equations 3.1 and 3.2 can be used to form the system of reactions. If the pressure of the exiting stream is known and a guessed value for its temperature is taken, the mass balance and equilibrium conditions would provide the concentrations of CO, CO2, O2, C, and N2 in that stream. These can now be used in Equation 5.10 to provide the enthalpy of exiting stream. Since the enthalpy is a function of temperature (Equation 5.34 of this chapter and Table B.1 of Appendix B), a new exiting stream temperature can be estimated. The process is repeated until an acceptable deviation between two successive temperatures is reached. Of course, this is an approximation and should be regarded with reservations for several reasons: irst, because gasiiers and combustors do not deliver streams at equilibrium condition; second, because the temperature computed by 0D model is an average and not a real representative of the leaving stream; and third, because the fuel conversion is not complete and the correct values depend too much on the geometry, residence time, and several other operational process parameters. It should be stressed that the equilibrium assumption usually leads to unrealistically high conversions of the fuel. Therefore, only the inclusion of dimensions in the model would allow a better representation of the process. Despite that, the 0D-S model might provide some initial information and is usually valuable for further development. Among the information it provides is the commonly referred to lame temperature, as presented below.

5.6

FLAME TEMPERATURE

The concepts of lame temperature and adiabatic lame temperature are widely used. However, those parameters have very restricted uses, and care should be exercised in their application. This is so because for every combustion process, the temperature varies throughout the chamber. On the other hand, the adiabatic lame temperature indicates a theoretical limit that, although not obtainable, may be used as a reference. Its deinition requires the following to be assumed: a) Fuel (gas, liquid, or solid) and oxidant streams are injected into a chamber surrounded by adiabatic walls. b) Those streams are perfectly mixed. c) The reactants are at exactly stoichiometric proportions. d) The combustion is complete or represented by irreversible reactions. In this way, the average temperature of the products would be the maximum possible value, or the adiabatic lame temperature (AFT). One should be careful because the usual term adiabatic lame temperature refers to the value using air as the stream

119

Zero-Dimensional Models

carrying oxygen. If a stream with a higher concentration of oxygen, or even pure O2, is used, the AFT would be much higher. From the above maximum value, others that might approach a more realistic average temperature of a postlame mixture of gases can be obtained by dropping one or more of the above assumptions. In all these cases, the general name of lame temperature is used. However, any computed lame temperature would always be lower than the maximum, as deined above. It is always advisable to perform a careful veriication of the precise deinition. Computation of these temperatures is straightforward.

5.6.1

EXAMPLE 5.3

Suppose the case of natural gas, assumed again to be pure methane. The initial conditions for the stream (methane and air) should be set. Let us assume air and methane streams injected at 1 bar and 100°C, an air excess of 30%, and an adiabatic combustor. Figure 5.3 illustrates the problem. The solution is as follows: • The air ratio would be 1.3. • Mass balance: According to our notation, F1 is the mass low of stream 1, or air. As this is not set directly, we may work on the basis of the stoichiometry and compensate with the air ratio. Let us use the basis, for instance, of a molar low unit of methane, i.e., 1 kmol/s. Therefore, if stoichiometric conditions prevailed, the mass low of air would be 2 kmol/s and molar low of air would be equal to 2 × 1.3, or 2.6 kmol/s. The molar low of stream 3 is shown below. • Composition of the products: Using Equation 5.28 and air ratio 1.3, the mass fractions component j in stream 3 would be given by wj = xj

Mj M3

(5.60)

and the molar fractions by xj =

nj n3

(5.61)

1 Air 3 Combustor Products 2 Methane

FIGURE 5.3 Scheme to illustrate example of energy balance for a combustor.

120

Solid Fuels Combustion and Gasification

where nj is the number of moles of each component j in the formula of the stream 3. Looking at Equation 5.20, nCO2 = 1 and nH2O = 2. As air is involved, nN2 = 1.3 × 7.52 and the amount of oxygen left unreacted would be nO2 = 0.3 × 2. Therefore, the total number of moles in the products would be n 3 = 13.376

(5.62)

and xCO2 = 0.07476, xH2O = 0.14952, xN2 = 0.73086, and xO2 = 0.04486. • Energy balance: Equation 5.10 or its molar basis equivalent may be applied. The solution would provide the enthalpies of the leaving mixture that can be used to ind its temperature. As adiabatic conditions are set, one should assume eficiency equal to 1.0. Of course, more realistic computations would be possible if losses could be estimated. As said before, for most combustors, it is usual to assume losses in the range of 2% of the combustion enthalpy of the incoming fuel. In these cases, Href should be computed by combustion enthalpy of the entering fuel. One should remember to apply the concept of total enthalpy (formation plus sensible) when computing the enthalpies of each stream. • Enthalpy of methane at 373.15 K and 1 bar computed by Equation 5.34 (see Appendix B, Table B.1): It should be remembered that in Equation 5.34, T0 is the temperature for the basis at which the formation enthalpy is given, or 298 K, and in the present case, T = 373.15 K. Therefore, h 2 = −74.86 × 10 3 + 5812.7 = −69.05 × 10 3 kJ/kmol.

(5.63)

• The same computation for air leads to h 1 ≅ 0.79h N2 + 0.21h O2

(5.64)

= 0.79(0 + 1656.3) + 0.21(0 + 4520.7) = 2.62010 3 kJ/kmol. • The enthalpy of the leaving stream is approximately given by h 3 = ∑ x j  h f , j + c j (T3 − 298)  = ∑ x j h f , j + (T3 − 298)∑ x jc j j

j

(5.65)

j

where the speciic heat of each component is assumed to be constant and the same value as a guessed value for T3. A irst guess for that temperature would allow the enthalpy to be computed. • From the energy balance, 1     (F1 h1 + F 2 h 2 ) h 3 =  F3 1 (2.6 × 2.620 × 10 3 − 69.050 × 10 3 ) = −4.523 × 10 3 kJ/kmol. = 13.76

(5.66)

As we have seen, a reiterative procedure would show that temperature around 2200 K would provide the enthalpy value given by Equation 5.66.

Zero-Dimensional Models

5.6.2

121

A MORE REALISTIC APPROACH

The lame temperature can be computed under a more realistic approach if the assumption of irreversible reaction could be abandoned. This computation would be the same as illustrated before for a 0D-S combustor.

5.7 5.7.1

EXERCISES PROBLEM 5.1* Determine the adiabatic lame temperature for benzene at 298 K and 1 bar in air. Repeat for pure hydrogen under the same conditions.

5.7.2

PROBLEM 5.2* Assume coal with HHVd (or at dry basis) of 30.84 MJ/kg and proximate and ultimate analysis as given by Tables 4.5 and 4.6. Assume the following approximations: • Representative formation enthalpy as given in Appendix B, Section B.7 • LHVd as estimated in Appendix B, Equation B.41 Apply the reasoning in Appendix B, Section B.7, for a representative formula for the fuel and calculate the following: a) Composition of leaving stream from complete combustion b) Adiabatic lame temperature in pure oxygen injected at 298 K

5.7.3

PROBLEM 5.3** Pulverized graphite (almost pure carbon) is fed into a combustor operating at 1 MPa. Air is supplied at 10% excess. If the leaving stream is leaving the reactor at 1000 K, determine its equilibrium composition.

5.7.4

PROBLEM 5.4** Develop a block diagram to compute composition and temperature from a combustor or gasiier using a zero-dimensional model. All conditions of injected streams are known. Follow the instructions given in Section 5.5.

5.7.5

PROBLEM 5.5** Develop a small computer program to calculate speciic heat, enthalpy, entropy, and speciic heats of real gases. Assume temperature and pressure as input. Compare the values obtained for hydrogen and steam at 800 K and 0.1 MPa with those found in Table B.1 of Appendix B. Repeat for pressure of 10 MPa.

5.7.6

PROBLEM 5.6** Develop a routine to compute the Gibbs free energy speciic value and the variation of Gibbs free energy (see Appendix B, Equations B.8b and B.11) for a

122

Solid Fuels Combustion and Gasification mixture where the composition (molar or mass basis), temperature, and pressure are given. Then, for the same inputs, write another routine to compute the equilibrium constant of a given reaction i (therefore, with values of νj,i).

5.7.7

PROBLEM 5.7*** Improve the routines developed for Problems 5.4 and 5.5 in order to provide corrections for nonideal behavior of gases. Apply, for instance, Redlich-KwongSoave corrections (Appendix B) and data from the literature [10].

5.7.8

PROBLEM 5.8** Repeat Problem 5.2 for a case in which the combustor eficiency could be assumed as 98% on the basis of the total combustion enthalpy of the entering fuel.

5.7.9

PROBLEM 5.9** Two streams are continuously fed into a reactor, which operates at a steady-state regime. The irst stream, consisting of pure hydrogen at 1000 K and 2 MPa, is injected at 10 kg/s. The second stream, a mixture of 40% oxygen and 60% carbon dioxide (molar percentages) at 500 K and 2 MPa, is injected at 20 kg/s. If the three overall reactions take place, 2H 2 + O 2 ⇔ 2H 2 O CO 2 + H 2 ⇔ CO + H 2 O 2CO + O 2 ⇔ 2CO 2 , and the exit stream is at equilibrium, determine its composition and temperature. Assume isobaric and adiabatic reactor, as well as ideal gas behavior. Note that any of the reactions listed above can be written as a combination of the other two. Therefore, for calculation of the equilibrium composition, one of the above reactions should be eliminated.

5.7.10

PROBLEM 5.10**

In the previous problem, compute the variation of entropy in the reactor and the total energy generated by the reactions.

5.7.11

PROBLEM 5.11**

Estimate the adiabatic lame temperature for natural gas in oxygen and air burning in a combustor at 3 MPa. Assume the following composition (molar percentages): • Methane: 92.47% • Ethane: 3.57% • n-Propane: 0.79%

123

Zero-Dimensional Models Compressor

Turbine

1

3

1 Air

4

2

5 3

2

Natural gas Combustor Exhaust

FIGURE 5.4 Scheme of a simple power system. • • • • •

5.7.12

n-Butane: 0.35% n-Pentane: 0.07% n-Hexane: 0.07% Nitrogen: 2.01% Carbon dioxide: 0.67%

PROBLEM 5.12**

A stream of 1 kmol/s of steam at 1000 K carries same amount of pulverized graphite (initially at 298 K) into a well-insulated reactor operating at a constant pressure of 1 MPa. Determine the composition and temperature of the leaving stream. Assume that the two following reactions are present in the process. C + H 2 O ⇔ CO + H 2 CO 2 + H 2 ⇔ CO + H 2 O

5.7.13

PROBLEM 5.13**

Using the general procedure shown for zero-dimensional simulation of a power unit of various linked equipment, shown in Appendix B (Section B.5), try to set the system of equations to obtain temperatures, mass lows, and compositions of all streams of a simple process shown Figure 5.4. How many input conditions would be necessary to ensure that a square system of equations could be obtained?

5.7.14

PROBLEM 5.14***

Write a program to simulate the process described in Problem 5.13. Use any computational language.

124

Solid Fuels Combustion and Gasification

REFERENCES 1. Moran, M.J., and Shapiro, H.N., Fundamentals of Thermodynamics, 3rd Ed., John Wiley, New York, 1996. 2. van Wylen, G.J., and Sonntag R.E., Fundamentals of Classical Thermodynamics, John Wiley, New York, 1973. 3. Kestin J., A Course in Thermodynamics, Vols. I and II, Hemisphere, New York, 1979. 4. de Souza-Santos, M.L., A study on thermo-chemically recuperated power generation systems using natural gas, Fuel, 76(7), 593–601, 1997. 5. Perry, J.H., Green, D.W., and Maloney, J.O., in Perry’s Chemical Engineers Handbook, 7th Ed., McGraw-Hill, New York, 1997. 6. Carnahan, B., Lutter, H.A., and Wilkes, J.O., Applied Numerical Methods, John Wiley, New York, 1969. 7. Bakhvalov, N.S., Numerical Methods, Mir, Moscow, 1977. 8. Kopachenova, N.V., and Maron, I.A., Computational Mathematics, Mir, Moscow, 1975. 9. Lapidus, L., Digital Computation for Chemical Engineers, McGraw-Hill, New York, 1962. 10. Reid, C.R., Prausnitz, J.M., and Poling, B.E., The Properties of Gases and Liquids, 4th Ed., McGraw-Hill, New York, 1987. 11. Poling, B.E., Prausnitz, J.M., and O’Connell, J.P., The Properties of Gases and Liquids, 5th Ed., McGraw-Hill, New York, 2000.

to 6 Introduction One-Dimensional Steady-State Models CONTENTS 6.1 6.2 6.3

Introduction .................................................................................................. 125 Deinitions..................................................................................................... 126 Fundamental Equations ................................................................................ 127 6.3.1 Boundary Conditions ........................................................................ 131 6.3.1.1 A Classiication for Boundary Conditions ......................... 131 6.3.1.2 The Present Case................................................................ 133 6.3.2 A Few Comments on the Continuity Equations ............................... 133 6.3.3 Simpler Level of Attack .................................................................... 134 6.3.4 Solutions for Equations with Second Derivatives............................. 136 6.4 Final Comments............................................................................................ 138 6.5 Exercises ....................................................................................................... 138 6.5.1 Problem 6.1 ....................................................................................... 138 6.5.2 Problem 6.2 ....................................................................................... 138 6.5.3 Problem 6.3 ....................................................................................... 138 6.5.4 Problem 6.4 ....................................................................................... 139 6.5.5 Problem 6.5 ....................................................................................... 139 References .............................................................................................................. 139

6.1

INTRODUCTION

As explained in Chapter 1, there are situations in which a one-dimensional model is not enough to describe the processes occurring inside equipment with reasonable accuracy. However, in many situations, that approach may lead to a reasonable representation of a real situation, even for cases involving pulverized combustion and gasiication of solid fuels [1, 2]. This chapter introduces one-dimensional steady-state models. To accomplish that, simple plug-low gas combustion is used. As discussed below, even that process involves complex aspects. For now, the purpose is just to illustrate a methodology for dealing with fundamental equations. There is no preoccupation with presenting numerical solutions. Considerations regarding the assumptions, as well as ways to improve that irst treatment, are made.

125

126

6.2

Solid Fuels Combustion and Gasification

DEFINITIONS

Let us assume the plug-low of a fuel gas in a cylindrical chamber, as shown in Figure 6.1. Given the appropriate nature of the gas and the presence of an oxidant, the combustion process may occur. However, there are marked differences among the possible characteristics of such a process. The region of space called the lame zone or lame front concentrates almost all reactions of the combustion process, and there are two basic types of lames: • Premixed lame, in which the reactants (fuel and oxidant) are perfectly mixed before the reaction zone • Diffusion lame, in which the reactants diffuse into each other before combustions as well other reactions may occur The speed of the combustion front through a reacting mixture determines the characteristics of the process. There are two basic situations: 1. Delagration, where the lame front travels through the combustion media at subsonic speed. In this process, there is a slightly decrease in the pressure, i.e., P2 < P1. This, combined with the great increase in temperature, leads to a considerable increase in the velocity, i.e., u2 >> u1. That is the main reason behind the eficiency of gas turbines used in aviation, where high kinetic energy of the exhausting gases leads to high gains of momentum by the aircraft. 2. Detonation, where the lame front travels through the combustion media at supersonic speed. In such a process, there is a considerable increase in the pressure, i.e., P2 >> P1. Despite the great augmentation of temperature, the increase in pressure leads to a decrease in the velocity, i.e., u2 < u1. It is dificult to decide the path a given situation will follow. Experimental veriication or coherent modeling and solution of the derived equations may show the answer. However, most industrial combustors operate under delagration. Combustion chamber

r z

Insulation

u1, T1, P1, x11, x21, x31 ...

u2, T2, P2, x12, x22, x32 ...

Unburned gas

Burned gas

Flame front

FIGURE 6.1 One-dimensional lame.

Introduction to One-Dimensional Steady-State Models

6.3

127

FUNDAMENTAL EQUATIONS

Before setting the system of equations that governs the process, let us make some considerations. Signiicant changes in temperature, velocity, pressure, and composition occur in the combustion chamber. Therefore, density, viscosity, thermal conductivity, diffusivity, or any other property cannot be assumed to be constant throughout the chamber. Mass, energy, and momentum balances, represented by a system of differential equations, should be solved using numerical methods. During that process, the equations are solved using a great number of small control volumes. The values of variables (such as temperature, pressure, velocity, and concentration) are used to set the boundary conditions for the next small volume. Thus, the assumption of constant values for the properties at each individual cell is usually a good approximation, as long as numerical methods are applied to small inite volumes. Note that this does not hamper the possibility of large variations of properties throughout the combustion chamber. Any method and its respective computational program used to solve the system of differential conservation equations needs assistance from a bank of physicalchemical properties. For any given chemical species, such a bank should contain equations describing the dependence of each property on temperature and pressure [3–6]. It should also include correlations to compute physical-chemical properties of mixtures of such species and would be assessed several times during the solution of a mathematical model to provide the values of properties at each point of the simulated equipment. Equations listed in Appendix A assume a Newtonian luid. This is usually the case for processes involving combustion of gases and most liquids. For solids, this is also not a problem because dissipations by shear stresses are negligible in almost all situations. The example shown below assumes laminar low. This is a feeble hypothesis in most cases of combustion processes because drastic variations in velocity and thermal ield are present. Localized increases in the temperature of the luid greatly affect the velocity and pressure ields. Therefore, turbulent low is almost inevitable. Of course, the conservation equations (mass, momentum, and energy, as shown in Appendix A) are also valid for turbulent low. However, this regime introduces luctuations in the velocity ield, and the Newtonian treatment for the shear stresses is not enough to represent the reality of momentum transfers. Special formulations for the viscosity, called closure strategies, may model such lows. A brief introduction to these and a few other aspects related to turbulence is presented in Appendix E. At this point, the objective is just to introduce the method applied throughout the book for building up a model from fundamental transport equations and not to develop a general approach for combustion in chambers. The subject of lames, especially turbulent ones, is beyond the scope of this book. As stressed in the Introduction, this text is dedicated mainly to introducing models for packed and luidized beds. Therefore, it does not include general models for combustion of gases or pulverized solid fuels. The literature on models for such cases is vast, and a few of them are listed here [1, 2, 7–9].

128

Solid Fuels Combustion and Gasification

For the present, suppose the following simplifying conditions and hypotheses: A. A steady-state regime is assumed. B. Plug-low is assumed for the gas throughout the chamber without rotational movements. Hence, no other velocity component is included besides the one in the z or axial direction. Since the velocity of lowing luids vanishes for positions at the reactor cylindrical wall, one may argue whether the hypothesis of lat velocity proile is reasonable. For instance, in nonreacting, isothermal laminar lows, the proile is parabolic; hence, the radial velocity component is present. However, the present irst approach assumes laminar low in which the velocity at any radial position will be equal to the average value at a given axial position z. This is also called inviscid low. The assumption is used only for a irst approximation and is applied by many researchers for the same purpose. An improvement would consider two velocity components: axial and radial. C. Tubular reactor is perfectly insulated at the side walls. No heat transfer is veriied in the radial direction. This leads to lat temperature proiles in the radial direction. D. Heat dissipations by viscous effects and by diffusion are negligible, as are the inluences of the change from one basic transport phenomenon to another (Dufour and Soret effects) [10]. E. Heat transfer by radiation is also neglected. This assumption deserves some discussion, which is presented at a later point in this chapter. F. Gravity effects are negligible. G. All reactions occur at slow rates. The fuel gas and/or the oxygen are assumed to be much diluted at the injection point. Therefore, turbulence might be avoided because of the slow oxidation rate. Adopting a cylindrical system of coordinates (Appendix A, Table A.2), the governing equations follow: • Continuity for the mixture. From simpliication A, the irst term of Equation A.7 (Appendix A) is equal to 0. Assumption B forces all velocity components to vanish except the one in the low or z direction, which will be called u. Thus, all derivatives in other directions are also equal to 0, and it is possible to write d(ρu) = 0. dz

(6.1)

• Continuity for each individual species j (among the n chemical components present). In Equation A.8: a) According to simpliication A above, the derivative against time is 0. b) According to simpliication B, there is no other velocity component except for that in the axial direction. Therefore, that equation becomes

Introduction to One-Dimensional Steady-State Models

u

dρ j d 2ρ j = D j 2 + R Mj , 1 ≤ j ≤ n. dz dz

129

(6.2)

Here the approximation of an average diffusivity (Dj) of component j in the mixture of gases is assumed. As seen, the concentration proiles will be lat as well, and therefore they do not vary at radial positions. • Momentum conservation or motion equations can be obtained from Equations A.10, A.11, and A.12 of Appendix A. However here, one should keep in mind that: a) Simpliication A leads to no derivative in time. b) Simpliication B eliminates all terms with velocity components other than in the axial (z) direction. It also leads to pressure variations only in that direction. c) Simpliication F eliminates all terms with gravity components. d) Simpliication G might avoid creation of turbulent conditions in the vicinity of the oxidation front and beyond. Therefore, Equations A.10 and A.11 (Appendix A) are rendered trivial, and A.12 can be rewritten as ρu

du d 2 u dP =µ 2 − . dz dz dz

(6.3)

• A simple differential equation for the energy balance can be obtained from Equation A.9 by making the following changes: a) Eliminating the time derivative due to condition A. b) Eliminating all terms with velocity components in the radial and angular directions due to condition B. c) Eliminating all terms multiplied by viscosity due to condition D. Therefore, it is possible to write uρc

dT d2T = λ 2 + RQ . dz dz

(6.4)

Special attention should be paid to the last term, which represents the rate of energy transformation of chemical potential into internal energy of the luid. In other words, exothermic chemical reactions would provoke an increase in the temperature of the control volume, and endothermic reactions would provoke a decrease. It is also important to stress that in combustion processes, both exothermic and endothermic reactions are usually present. The exothermic oxidation of fuel is a much faster reaction than the others, such as the one between carbonaceous fuel and water. Therefore, the overall effect is an increase in temperature and volume. In addition, as the temperature varies in the axial direction, heat transfer by radiation in that

130

Solid Fuels Combustion and Gasification

direction should be expected. Nevertheless, the fundamental energy transfer equations presented in Appendix A do not explicitly show those terms. The inclusion of radiative transfers is not a trivial task. Emissions, absorption, and scattering by interfering media should be considered. The lux method allows such a treatment [11–22], but it is beyond the scope of the present book. On the other hand, in some situations the effects of radiative heat transfer might be mimicked by an overestimation of thermal conduction. Even convection terms might be added to the fundamental equations, either as isolated terms or as part of the energy source (or sink) term RQ. Let us further illustrate how the system formed by Equations 6.1 through 6.4 can be solved to produce the proiles of velocity, temperature, and concentration (which are here called primary variables). That system provides 3 + n equations, which is also the number of primary variables. The auxiliary parameters or secondary variables (global density, viscosity, speciic heat, thermal conductivity, diffusivity, energy, and mass sources) can be calculated using the primary variables. Starting with Equation 6.1, the global density ρ can be expressed in terms of the concentrations of all chemical species by n

ρ = ∑ ρj.

(6.5)

j =1

The thermal conductivity, viscosity, and diffusivities can be computed as functions of temperature, pressure, and composition. For this, methods for evaluation of properties of pure components, as well as their mixtures, should be applied [4, 5]. As a irst approximation, averages based on molar or mass fractions might be used. The sources of each chemical species j or R Mj, appearing in the last term on the right side of Equation 6.2, can be evaluated using the information given by the kinetics and stoichiometry of the reactions involved in the process, or  Mj , 1 ≤ j ≤ n R Mj = M j R

(6.6)

and 60

 Mj = M j v i, jri , 1 ≤ j ≤ n. MjR ∑

(6.7)

i = 41

The last term in the summation is the rate of reaction i, and the term before it is the stoichiometry coeficient of component j in that reaction i. The numbers 41 and 60, shown in the last summation, follow the convention of numbering reactions, which is introduced in Chapter 8 (Table 8.5). For now, it is enough to know that any number of gas–gas reactions can be included. The energy source term, or the last term of Equation 6.4, can be computed by n

R Q = − ∑ h jR M , j j=1

(6.8)

Introduction to One-Dimensional Steady-State Models

131

where the total rate of production (if negative is consumption) of chemical component j is given by Equation 6.7, and the total enthalpy of a component or chemical species by Equation 5.34, or hj =

6.3.1

T  1   h f , j + ∫ c jdT  . Mj  298 

(6.9)

BOUNDARY CONDITIONS

6.3.1.1 A Classification for Boundary Conditions The system of differential equations formed by Equations 6.1 through 6.4 can provide a general solution for the problem but not one applicable to a particular real situation. For this, a coherent set of boundary conditions is necessary. In order to allow discussions of the boundary conditions for that or any other model, it is useful to organize and set a classiication for boundary conditions, as follows: 1. First-kind boundary conditions are those that set values for the transport variable (velocity, temperature, or concentration) at certain values of the independent variable or variables. This category is subdivided into two others: 1.1. The initial condition sets the value at a position or time set as 0, represented by T(z = 0) = T0 .

(6.10a)

1.2. The intermediate or inal condition is when the value is set at any other time or position different from the initial point or origin marked as 0, or T(z = a ) = Ta , a ≠ 0.

(6.10b)

2. Second-kind boundary conditions are those that set values for the transport lux or low (rate of momentum transfer, heat lux, or mass lux). In other words, a derivative of the transport property is set. An example is given by dT =a dz z= 0

(6.11)

where parameter a is constant. Again, this kind of boundary condition can be subdivided into initial and inal/intermediate. 3. Third-kind boundary conditions are those that impose a relationship between values of lux and transport variable at a given point in space

132

Solid Fuels Combustion and Gasification

or time, such as when the heat luxes by conduction and convection are assumed to be equal at the interface between a solid and a luid, or −k

dT = α[T(z = 0) − T∞]. dz z = 0

(6.12)

This kind of boundary condition (BC) can also be subdivided into initial and inal/intermediate. A boundary value problem (BVP) is the combination of differential equations and the set of necessary and suficient boundary conditions to solve that problem. In order to obtain a coherent single solution for a BVP, one boundary condition should be assigned to each derivative of the corresponding variable. Needless to say, the dificulty of solving a BVP usually increases for higher classes of the boundary condition. For example, if the BVP is one-dimensional with just one dependent variable, a single BC is required. If, in addition, the BC is a irst-class and initial type (or kind 1.1 above), the numerical solution is generally easily achieved. This is so because the numerical solution would start from a given initial value of the dependent variable and progress obtaining values for that dependent variable at all values of the independent variable. To visualize this, let us take Equation 6.4 and imagine a simpliied problem in which density and velocity are constants and thermal conduction could be neglected. Therefore, Equation 6.4 would become a one-dimensional ordinary differential equation. The temperature of the entering mixture in the combustion chamber can readily be measured. Consequently, the initial condition is known, or T equals T0 at z equals zero. If the reaction rate can be simpliied as solely a function of temperature, that differential equation can be easily solved. However, if one-dimensional boundary condition type 1.2 is the only one available, even this apparently simple problem could lead to complications. For instance, if only the temperature (T = Ti) at a given intermediate position (say, z = zi) of the reactor is known, the solution would require the following: 1. A irst guess at the value of the temperature at z = 0 2. The numerical solution until the position z = zi is reached 3. Comparison between the computed value of temperature at that position and the actual one (Ti) 4. If the difference between the computed and actual temperature is above an expected or acceptable level, returning to step 1 for a new guess of T at z=0 This is an example of a convergence problem. Commercially available computational libraries provide eficient numerical procedures that lead to fast convergence. Dificulties for convergence would increase if second-kind boundary or thirdkind conditions at remote positions of the reactor are imposed. Since derivatives are usually more sensitive than the main variable to changes of trials values (step 1 above), the convergence would be even more dificult to achieve. The complexity

Introduction to One-Dimensional Steady-State Models

133

might rise to considerable levels if one has to solve more complex systems of differential equations, such as those given by Equations 6.1 through 6.4. Useful advice on how to progress toward a solution in such situations is given below. 6.3.1.2 The Present Case In order to set a complete BVP for the system formed by Equations 6.1 through 6.4, the following BCs should be assigned: a) b) c) d)

Two boundary conditions for the velocity (Equations 6.1 and 6.3) 2n boundary conditions for the concentrations (Equation 6.2) One boundary condition for the pressure (Equation 6.3) Two boundary conditions for temperature (Equation 6.4)

Of course, depending on the actual conditions of the speciic case, the modeler should do the following: • Simplify the problem even further to establish a gradual approach to more complete and realistic models • Recognize the boundary conditions at each level of approach

6.3.2

A FEW COMMENTS ON THE CONTINUITY EQUATIONS

Before progressing to further simpliication of the above problem, it is important to recognize and understand the contributions of each term in differential equations, as follows: • Apart from the global mass (Equation 6.1), all continuity equations (Equations 6.2, 6.3, and 6.4) have a similar form, i.e., on the left side is the irst derivative of the main variable, and on the right side is the second derivative of that variable added by the last term, which represents a source (or sink) term. • Usually, the irst derivative (on the left) and the second derivative (on the right) represent opposing inluences. The former relects increases or decreases in the main dependent variable due to the source or sink term, whereas the latter moderates those changes. This last is also called the dissipative term. As an example, in cases of progressing combustions in which the commanding reactions are exothermic, the irst term of Equation 6.4 would be a positive derivative. It would be a very high value unless a dissipative or smoothing factor is present. This smoothing effect is represented by the second derivative, on the right side, which decreases the rate at which temperature increases. This happens because a temperature that is too high at a given position would lead to an increase in the conduction process in the vicinity. Similarly for Equation 6.2, high values of concentration would increase the diffusion to the vicinity. In Equation 6.3, a high value of velocity would increase viscous dissipation of the momentum. As seen in all cases, fast increases or decreases of the main

134

Solid Fuels Combustion and Gasification

variable have their rates reduced by the inluence of dissipative terms. In other words, the second derivatives represent the action of smoothing the proile of main variables throughout the reactor and decrease their localized peak values.

6.3.3

SIMPLER LEVEL OF ATTACK

The present treatment is too simple to describe a gas–gas combustor. However, this example is interesting to demonstrate a typical modeling procedure. As recommended before, one should start with the simplest model and evolve to ones that are more complex. Comments on the usual criteria to decide the acceptable level of deviations have already been made in Chapter 1. Nonetheless, it is important to stress that for most situations, the very process of obtaining the numerical solution of differential equations might provide a good indication of model coherence. Cases of dificult convergences or impossible computed values, among other factors, might be signs of unrealistic models. As one might expect, it is not common to achieve a good model in the irst trial. Modeling is an iterative process, and the following discussion might illustrate this characteristic. Several simpliications have already been made in order to write the equations above. However, let us go even further by assuming the following: 1. A luid with no viscosity. In this case, Equation 6.3 becomes ρu

du dP =− . dz dz

(6.13)

According to the comment above, the second derivative in Equation 6.3 provides a decrease in the value of the irst derivative in the left side. Hence, it represents a dissipative factor, where viscosity is the physical parameter. The suppression of the second derivative leads to higher values of the irst derivative and therefore sharper variations of the velocity. This may or may not provide an acceptable representation of reality. Only comparisons against experimental values can determine that. As the viscosity of gases is much lower than that of liquids, it is understandable that the assumption of inviscid lows in the case of gases is less critical than the same for the case of liquids. 2. No thermal conductivity. In this case, Equation 6.4 becomes ρcu

dT = RQ . dz

(6.14)

Again, the term of conductivity acts as a smoothening factor on the temperature variations due to possible sudden surges caused by exothermic (or endothermic) chemical reactions. The second derivative provides a decrease in the value of the irst-order derivative. On the other hand,

Introduction to One-Dimensional Steady-State Models

135

thermal conductivity of gases is much smaller than that of liquids. That is why not including this term is usually less critical for gases than for liquids. Another aspect concerns the heat exchange by radiation, which has been neglected here. This might be reasonable for processes involving relatively low temperatures or when the gases are transparent within the wavelength of thermal radiation. At low temperatures—usually below 500 K—radiative heat transfers are negligible compared with the conductive and convective transfers. Nonetheless, for high concentrations of gases such as carbon dioxide and water, the inclusion of radiative heat transfer might become mandatory. In addition, radiative transfers would help smooth the temperature proiles. 3. Neglecting the diffusion effects allows Equation 6.2 to be written as u

dρ j = R j, 1 ≤ j ≤ n . dz

(6.15)

The same considerations as before can be made. Eventual surges in the concentrations of products due to chemical reactions are avoided by diffusion or second derivative terms. In other words, when the reactions lead to fast increases in the production of a chemical species, diffusion transport moderates the eventual concentration peaks. However, the diffusion also provides a velocity ield that superimposes the velocity of the mass center. Large average velocity ields decrease the effect of diffusion in the concentration proiles. Consequently, the second derivative might be neglected in cases of relatively high global mass low. After the above simpliications, the system to be solved is formed by Equations 6.1, 6.13, 6.14, and 6.15. The number of variables (T, P, u, ρj, j = 1 . . . n) and equations continues to be the same, or 3 + n, but the required number of boundary conditions drops to just 3 + n. These would be, for instance, the values of average velocity, temperature, and pressure plus the concentrations at the reactor entrance. These are boundary conditions type 1.1, and they are simpler to set because they are usually known at the injection point of gases in the reactor. Nonetheless, one should be aware that this simplicity might be misleading. For instance, if the temperature set at the reactor entrance is too low, an equally low temperature derivative would be computed at z = 0. Below a certain value of derivative, the temperature may remain constant or low throughout the reactor. In other words, ignition conditions should be provided to achieve minimum combustion reaction rates. During the start-up of combustor chambers, ignition—represented by a relatively high temperature derivative at the fuel injection point—is set by an electrical spark, pilot lame, or other. Once a steady state is achieved, heat from the lame is transferred in all directions, including backward. Therefore, the spark or pilot can be turned off, and the hot regions ahead provide the necessary positive derivative of temperature at the reactor entrance. However, the present simpliied model assumes plug-low and no second temperature derivative. Hence, no backward lows or heat transfers to positions behind the combustion front are considered. In order to escape

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from this situation, one might imagine changing the irst-type boundary condition for temperature by applying a second-type boundary condition, similar to Equation 6.11. In other words, one would impose a positive derivative for the temperature at the reactor entrance (z = 0). However, without the complete solution of the model, the value of that derivative is not known, and an arbitrary value should be guessed.* That guess could be checked if another value of temperature is known. For instance, if the temperature of the exiting stream from the combustion chamber known, a derivative at z = 0 would be guessed, and after each solution, the temperature at the exiting condition would be compared with the previous guess. The procedure could even use sophisticated methods to set new guesses at z = 0 until convergence is achieved. It might be argued that the exiting temperature is not known when one is trying to predict the operation of a combustor. Of course, every simpliication has a price, and the results may not be rewarding. In some cases, such as lame modeling, only more realistic approaches—which would include second derivatives, turbulence, backward lows, etc.—would be able to predict combustor operations without artiicial guesses. In addition to the above, for most combustion cases, the present level of modeling leads to stiff differential systems of equations. Since the combustion reactions progress, the temperature becomes too high, leading to further increases in the rates of exothermic combustion reactions. Consequently, the temperature increases even faster, and a snowball effect takes place. If derivatives become too high, the convergence of the numerical solution would demand decreasing increments of step sizes on the space variable (z). Very often, commercial packages for numerical solutions of differential equations provide some warning or even safety devices that stop the computations with too stiff differential equations. The problem can be avoided only in cases of much diluted fuel gases. An improvement in situations involving higher concentrations of fuel and oxidant would be to include the second derivatives, as shown in Equations 6.2 through 6.4, not to mention the probable need to add a second dimension, radiative heat transfers, as well as turbulent low regime (Appendix E).

6.3.4

SOLUTIONS FOR EQUATIONS WITH SECOND DERIVATIVES

From the above, it is clear that including second derivatives might provide more realistic modeling or better predictions of velocity, temperature, and concentration proiles throughout the combustion chamber. However, this may have a price in terms of the computational effort necessary to solve the system of differential equations. On the other hand, adding second derivatives could be beneicial. As we have seen, the conductivity term on the energy balance tends to smooth the temperature proile. Consequently, its introduction may avoid stiffness during the solution of the differential system. The discussion below has the objective of showing a possible route to the solution of systems involving second-order derivatives in the space coordinate z. That system

*

A rough guess of the temperature of the exiting gas stream can also be obtained using a zero-order model.

Introduction to One-Dimensional Steady-State Models

137

can be transformed into another with only irst-order differential equations through the application of auxiliary functions. For instance, if du = ξu , dz

(6.16)

dξ u 1 dP ρu = + ξu . dz µ dz µ

(6.17)

Equation 6.3 would lead to

The same can be applied to the temperature, and Equation 6.4 would be written by the combination of the two following equations, dT = ξT dz

(6.18)

dξ T R ρc uξ T . =− Q + dz λ λ

(6.19)

and

Similarly, Equation 6.2 can be rewritten as dρ j = ξ M, j , 1 ≤ j ≤ n dz

(6.20)

dξ M , j R u ξ M, j , 1 ≤ j ≤ n . = − M, j + dz Dj Dj

(6.21)

and

The system (Equations 6.1 and 6.16 through 6.21, added by 6.5 and 6.9) would comprise 5 + 2n irst-order differential equations and the same number of unknowns, or T, P, u, ξu, ξT, ξM,j (j = 1 . . . n), and ρj (j = 1 . . . n). The obstacle now is to obtain all necessary boundary conditions. Note that initial (or at z = 0) conditions for primary variables T, P, and ρj (j = 1 . . . n) could be easily measured. However, their derivatives cannot. Again, if one sets a relatively cold initial temperature and derivatives as 0, ignition would not occur. The consequences of that have already been explained. As before, the alternative is to start from a relatively low positive value for the temperature derivative. After a irst run of the program, the conditions could be compared with the real measures, and a trialand-error procedure might indicate the minimum derivative that allows reasonable

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representation of the combustion process. As we have seen, the dependence on at least one experimental measurement might continue, but the solution of the differential system would progress more smoothly than before.

6.4

FINAL COMMENTS

A relatively simple model for the combustion of gases in a tubular plug-low reactor has been introduced. Despite its apparent simplicity, the above differential system is highly coupled, i.e., the properties, as well as the reaction rates, are strongly inluenced by temperature and pressure. Solution of such systems is possible through numerical procedures and methods easily found in the literature [23–25]. In addition, commercial packages of mathematical routines, such as IMSL®, MATLAB®, etc., are very convenient. Of course, even considering the second derivatives or diffusion terms, the present model still represents a strong simpliication of a real process. The low of gases in an industrial combustor is far from laminar and plug-low. Turbulence, vortices, and reverse velocities are the norm. As we have seen, the returning of hot streams to positions near the fuel entrance or injection point ensures fuel ignition and stability of the lame. Of course, more realistic approaches would require a considerable increase in the complexity of differential equations. This is beyond the scope of the present introductory text, and for such cases, the use of commercially available packages on computational luid dynamics is recommended. Nonetheless, one should be aware that even for these applications, understanding the phenomena and processes involved is always required.

6.5

EXERCISES

6.5.1

PROBLEM 6.1* Show that in the case of ideal gases, Equation 6.1 can be written as follows.  1 dT 1 dP  du = u − dz  T dz P dz 

6.5.2

(6.22)

PROBLEM 6.2**

Equation 6.22 is not necessary to solve the system of Equations 6.1 through 6.4. Nevertheless, if a mixture of ideal gases is lowing through the chamber, how could Equation 6.22 be reconciled into the system of differential equations shown in the text?

6.5.3

PROBLEM 6.3**

From the simplest model (represented by Equations 6.1, 6.13, 6.14, and 6.15) and the more realistic one (represented by Equations 6.1 through 6.4) intermediate models can be devised. In this way, what should be the irst simplifying assumptions to be dropped? Write the system of equations that represents this intermediate model.

Introduction to One-Dimensional Steady-State Models

6.5.4

139

PROBLEM 6.4**

Imagine that a rotational motion is imposed on the gas stream injected into the tubular steady-state reactor. Therefore, simpliication B (Section 6.3) should be modiied, and the radial velocity component could be assumed to be 0. Maintaining all other simplifying hypotheses, write the system of equations to allow computations of velocities, temperature, pressure, and concentration proiles in the reactor.

6.5.5

PROBLEM 6.5***

If the steady-state condition for the operation of the reactor could no longer be applied, how would this affect the system of equations shown in Section 6.4? Indicate the boundary conditions for such a model. Those processes are called dynamic and are very useful in studying the behavior of a reactor under progressive or sudden changes in one or more variables. These studies are central to the development of controlling devices.

REFERENCES 1. Smoot, L.D., and Pratt, D.T., Pulverized-Coal Combustion and Gasiication, Plenum Press, New York, 1979. 2. Smoot, L.D., and Smith, P.J., Coal Combustion and Gasiication, Plenum Press, New York, 1985. 3. Perry, J.H., Green, D.W., and Maloney, J.O., in Perry’s Chemical Engineers Handbook, 7th Ed., McGraw-Hill, New York, 1997. 4. Poling, B.E., Prausnitz, J.M., and O’Connell, J.P., The Properties of Gases and Liquids, 5th Ed., McGraw-Hill, New York, 2000. 5. Reid, C.R., Prausnitz, J.M., and Poling, B.E., The Properties of Gases and Liquids, 4th Ed., McGraw-Hill, New York, 1987. 6. Beaton, C.F., and Hewitt, G.F., Physical Property Data for the Design Engineer, Hemisphere, New York, 1989. 7. Glassman, I., Combustion, 3rd Ed., Academic Press, San Diego, CA, 1996. 8. Warnatz, J., Maas, U., and Dibble, R.W., Combustion, Springer, Berlin, Germany, 1999. 9. Kuo, K.K., Principles of Combustion, John Wiley, New York, 1986. 10. Bird, R.B., Stewart, W.E., and Lightfoot, E.N., Transport Phenomena, John Wiley, New York, 1960. 11. Viskanta, R., Radiation transfer and interaction of convection with radiation heat transfer, in Advances in Heat Transfer, Vol. 3, Academic Press, New York, 1966, 175–251. 12. Gibson, M.M., and Monahan, J.A., A simple model of radiation heat transfer from a cloud of burning particles in a conined gas stream, Int. J. Heat Mass Transfer, 14, 141–147, 1971. 13. Siddal, R.G., Flux methods for the analysis of radiant heat transfer, in Proc. 4th Symposium on Flame Research Committee and Inst. of Fuel, 1972, pp. 169–177. 14. Shah, N.G., An improved lux model for the calculation of radiation heat transfer in combustion chambers, in Proc. 16th National Heat Transf. Conf., St. Louis, MO, August 1976. 15. Lockwood, F.C., and Shah, N.G., Evaluation of an eficient radiation lux model for furnace prediction procedures, in Proc. 16th International Heat Transfer Conference, Toronto, Ontario, Canada, August 1978. 16. Siddal, R.G., and Seçuk, R., Evaluation of a new six-lux model for radiative transfer in retangular enclosures, Trans. IchemE, 57, 163–169, 1979.

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17. Brewster, M.Q., Effective absorption and emissivity of particulate media with application to a luidized bed, Transactions of the ASME, 108, 710–713, 1986. 18. Brewster, M.Q., Thermal Radiative Transfer and Properties, John Wiley, New York, 1992. 19. Tien, C.L., Thermal radiation in packed and luidized beds, Transaction of the ASME, 110, 1230–1242, 1988. 20. Rabi, J.A., and de Souza-Santos, M.L., Incorporation of a 2-lux model for radiative heat transfer in a comprehensive luidized bed simulator. Part I: Preliminary model equations, in Proc. ENCIT-2002, 9th Brazilian Congress on Engineering and Thermal Sciences, Caxambu, Minas Gerais, Brazil, October 15–18, 2002. 21. Rabi, J.A., and de Souza-Santos, M.L., Incorporation of a 2-lux model for radiative heat transfer in a comprehensive luidized bed simulator. Part II: Numerical results and assessment, ENCIT-2002, 9th Brazilian Congress on Engineering and Thermal Sciences, Caxambu, Minas Gerais, Brazil, October 15–18, 2002. 22. Rabi, J.A., Usage of Flux Method to Improve Radiative Heat Transfer Modelling inside Bubbling Fluidized Bed Boilers and Gasiiers, PhD thesis, Faculty of Mechanical Engineering, State University of Campinas, Campinas, São Paulo, Brazil, 2002. 23. Bakhvalov, N.S., Numerical Methods, Mir, Moscow, 1977. 24. Kopachenova, N.V., and Maron, I.A., Computational Mathematics, Mir, Moscow, 1975. 25. Lapidus, L., Digital Computation for Chemical Engineers, McGraw-Hill, New York, 1962.

Combustion 7 Moving-Bed and Gasification Model CONTENTS 7.1 7.2 7.3

Introduction .................................................................................................. 141 Downdraft Moving-Bed Reactors................................................................. 142 The Model..................................................................................................... 143 7.3.1 The Model Chart .............................................................................. 143 7.3.2 Summary of Model Assumptions ..................................................... 146 7.3.3 Basic Equations................................................................................. 148 7.3.3.1 Global Continuity............................................................... 148 7.3.3.2 Species Mass Continuity .................................................... 149 7.3.3.3 Continuity of Energy ......................................................... 151 7.3.3.4 Momentum Continuity ....................................................... 152 7.3.3.5 Summary of the Problem ................................................... 152 7.3.3.6 Further Simpliications ...................................................... 153 7.3.4 Boundary Conditions ........................................................................ 157 7.4 Updraft Moving-Bed Reactors .................................................................... 158 7.5 The Model..................................................................................................... 159 7.5.1 The Model Chart .............................................................................. 159 7.5.2 Summary of Model Assumptions ..................................................... 160 7.5.3 Basic Equations................................................................................. 161 7.5.4 Boundary Conditions ........................................................................ 161 7.6 Exercises ....................................................................................................... 164 7.6.1 Problem 7.1 ....................................................................................... 164 7.6.2 Problem 7.2 ....................................................................................... 164 7.6.3 Problem 7.3 ....................................................................................... 164 7.6.4 Problem 7.4 ....................................................................................... 164 7.6.5 Problem 7.5 ....................................................................................... 164 References .............................................................................................................. 165

7.1

INTRODUCTION

As mentioned before, the basic philosophy of the present course is to teach through example. Having this in mind, a one-dimensional model for moving-bed reactors is shown. The present case is focused on gasiiers and combustors; however, the present model is not limited to this equipment, and any other kind of process might use the same approach. Of course, in such cases, the set of kinetics shown in Chapters 8, 9, and 10 may receive additions or modiications. 141

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As also previously noted, a combustion process is a particular case of gasiication. The model presented here considers the important reactions that usually take place in these processes. Therefore, this model can be used to build simulation programs for either combustors or gasiiers. The same approach has been used by several other works [1–7]. There is a long list of publications on the modeling of ixed- or moving-bed combustors and gasiiers. Most of them employ a one-dimensional model. Good reviews and convenient tables showing the main aspects of each model have been published [1, 2]. However, differences among models cannot be fully appreciated without basic experience in the subject. As seen before, there are two major conigurations for a moving bed: downdraft and updraft. Both are described here.

7.2

DOWNDRAFT MOVING-BED REACTORS

Figure 7.1 illustrates a schematic view of this sort of equipment. As one may recall, in the cocurrent or downdraft version, the solid carbonaceous particles are fed at the top of the reactor and slowly low to the base, where the residual solid is withdrawn. The combustion and gasiication agents* are injected through a position near the top of the bed. The gas that is produced—composed Carbonaceous solid feeding

z=0

z

Main injection of combustion or gasifying agents (z = 0)

Bed

Possible intermediate injection of gases (0 < z < zD) Insulation

z = zD Main withdraw of gases (z = zD) Ash withdrawal (z = zD)

FIGURE 7.1 Scheme of downdraft reactor.

*

Usually, these agents are air and steam, but any mixture of gases that react with the solid fuel may work.

Moving-Bed Combustion and Gasification Model

143

mainly of carbon monoxide (if operating as a gasiier), hydrogen, carbon dioxide, and nitrogen—is withdrawn through the bottom. As noted previously, moving-bed gasiiers and furnaces have been widely used because of their simplicity and degree of controllability. Several large units, as well small units, can be found throughout the globe.

7.3

THE MODEL

A great number of phenomena are involved in the gasiication process, such as drying, devolatilization, gasiication, and combustion. They include several combined processes, such as homogeneous and heterogeneous chemical reactions; heat, mass, and momentum transfers; particle attrition; etc. Of course, if one starts locking at the process from the side of complexities, the task of modeling would seem terrifying. However, several mathematical models for gasiiers and combustors have been built, and they continue to provide excellent reproductions and predictions of industrialscale operational conditions. In this way, they have proved to be useful and important tools for equipment design and optimization. Hence, it is important to assume an optimistic approach and imagine the process in a simpliied way. The degree of simplicity or complexity is relative, but this is exactly the point that makes modeling a creative task. A general discussion of the level of modeling (zero-, one-, or three-dimensional) to be chosen has been presented in Chapter 1. In the case of moving-bed gasiiers or combustors, many have successfully adopted one-dimensional modeling. This level provides the highest beneit-cost ratio among all possible levels of modeling, at least for moving-bed reactors [3–5, 8]. In addition, it offers almost all the information needed for engineering design and process optimization.

7.3.1

THE MODEL CHART

The irst task is to develop a model chart, which is a scheme where the basic features of the model are shown. For that, let us imagine (Figure 7.2) that the combustor or gasiier can be divided into two basic streams: gas and solid lowing in the same direction. The two streams would exchange heat and mass through their common interface, which can be imagined as a single continuous surface with an area equivalent to the real area separating the two phases. This area would be given by total surface area of all solid particles in the bed. As the size of particles usually varies throughout the bed, the area per unit of bed height would also vary along with the bed height. In addition, models for the hydrodynamics of each phase should be adopted. The simplest model is to imagine each phase (gas and solid) lowing through the reactor in plug-low regimes. Some important points should be made about this simpliication: 1. From the strict point of view of low, the assumption of overall plug-low regime is reasonable for any luid that percolates through a bed of particles in a main direction. This is illustrated in Figure 7.3. As gas percolates through

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Solid Fuels Combustion and Gasification

the bed, it is forced to pass through small corridors or channels between particles. No matter what the low regime is (laminar or turbulent), the overall combination of these small streams mimics a plug-low for the gas phase. If the gas velocity can be assumed to be even throughout the bed crosssection, the temperature and concentration proiles can also be modeled as Solid particles

Gas injection

z=0

Virtual interface between gas and solid phases

Solid phase z Combustor or gasifier

Heat & mass

Gas phase

z = zD Solid residue

Exiting gas

FIGURE 7.2 Scheme of idealized downdraft moving-bed reactor. Fluid in

Particles

Fluid fronts

FIGURE 7.3 Illustration to justify the adoption of plug-low regime for the luid phase.

145

Moving-Bed Combustion and Gasification Model

lat proiles. Therefore, the rates of gas–solid reactions do not vary too much in the radial or horizontal direction. This last aspect leads to an almost uniform consumption of the solid particles at a given cross-section of the bed. In this way, it would be reasonable to assume the particulate solid phase lowing in an almost uniform downward movement. Thus, plug-low regime can be also taken as a valid approximation of the overall movement of solid phase. Another factor that should be ensured to allow acceptable one-dimensional treatment is the relatively large differences between the reactor cross-section diameter and the particle diameters. This situation is found in almost all industrial or pilot units. Even in cases where relatively large particles (on the order of 2 cm) are fed to the bed, the usual moving-bed diameters (or equivalent diameters) are on the order of 1 m to several meters. As rough rule, the one-dimensional approach would be reasonable for ratios above 30 between reactor and average particle diameter. This provides a picture even closer to the macroscopic point of view behind the reasoning justifying the assumption of a plug-low regime for the solids and gases. Consequently, negligible wall effects on lows and temperature proiles are ensured. 2. From the side of mass exchange between phases, the simpliication of the plug-low regime deserves some more elaborate discussion. As oxygen, water, and other gaseous chemical species react with the carbonaceous fuel, mass is necessarily transferred to the solid phase. At the same time, carbon dioxide, carbon monoxide, hydrogen, and several other gaseous components leave the solid phase and migrate to the gas phase. Of course, this establishes a crosslow or perpendicular low to the vertical or main direction of the gas and solid streams. Nonetheless, there is no preferential overall low for the mass Carbonaceous solid particles

Gas injection z=0 z Heat & mass

Descending solid particles (plug-flow)

Descending gas (plug-flow)

Heat

Environment

Heat & mass

z = zD Solid residue (ash & unburned)

Exiting gas

FIGURE 7.4 Model chart adopted for the downdraft moving-bed reactor.

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Solid Fuels Combustion and Gasification

transfer between phases throughout the horizontal direction. In other words, gases enter and leave the particles, which are more or less evenly distributed in the bed. Therefore, it is impossible to decide whether the radial gas low is in the positive or negative r direction, or there is no clear overall velocity ield in the radial direction. This would be possible only if a geometric model were proposed in which, for instance, the solid would be assumed to low downward in a periphery annulus and the gas in the same direction, but through the center. Then, the problem would involve two coordinates, and the conservation equations would lead to a system of partial differential equations. Luckily, it is possible to steer clear of such a complication by including radial components of variables in the source or sink terms. This technique would preserve the integrity of mass and energy balances while keeping the problem at the ordinary differential level. The details are shown below. After the above, it is possible to draw Figure 7.4, which illustrates the adopted model chart.

7.3.2

SUMMARY OF MODEL ASSUMPTIONS

Before setting equations, it is advisable to develop a clear list of model assumptions. In the present case, they are the following: A. Steady-state operation is assumed. This can be assumed during operations of almost all industrial reactors. Of course, starting-up and shutting-down periods cannot be included in this category. However, moving-bed gasiiers and combustors operate for long periods with almost no major problems that might require interruptions for maintenance or repairs. The rates of solid and gas feedings and withdrawals are relatively constant or present relatively low luctuations. In some pilot units and small industrial units, feeding and withdrawal of solids are performed at intervals. Of course, from a rigorous point of view, this would take them out of the present category. Nevertheless, if those intervals are short enough and the variations of bed height are below 10% of its average value, the operation may be approached as steady state. B. Gas lows downward in a plug-low regime. This has been justiied above. In addition, no rotational velocity is present, and at a given cross-section of the bed, all variables are uniformly distributed. Therefore, no angular components are considered. C. Solid particles also low downward in a plug-low regime. Hence, only the axial velocity component is present in the solid phase. This is much easier to justify than in the case of gases because radial low of a solid is 0 or insigniicant. D. Momentum transfers between the two phases are negligible. In other words, the velocity proile of one phase is not affected by the low of the other phase. This assumption is very reasonable since the velocities found in moving-bed gasiiers are relatively small, not only for the solid phase

Moving-Bed Combustion and Gasification Model

147

but also for the gas. However, for completeness, the equations related to momentum transfers are shown below. E. Inviscid low is assumed for the two phases. The idea of that sort of approximation has already been discussed in Chapter 6. This seems very reasonable, mainly because of the approximation of plug-low regimes. In such lows, there are no shear stresses between layers. This, combined with the fact that layers of gas between particles are thin (see Appendix E), allows the assumption that there is no presence of turbulent lows, even in cases of relatively high gas velocity. F. At each phase, temperature and concentration proiles are lat. Of course, the values are different between the gas and the solid phase. Consequently, heat and mass transfers will be imposed between the two phases. It is obvious that, for instance, the difference of temperatures between the gas and solid phase at each position (z) of the bed would lead to a nonlat temperature proile within each phase. Let us imagine, for instance, that the solid particles at a given axial position (z) in the bed are hotter than the gas at the same height. At points near the particle surfaces, the temperature in the solid should be smaller than the values for positions nearer particle centers. At the same time, the temperature of the gas layers nearer the particle surfaces should be greater than the values found for layers farther from those surfaces. However, if those cooler layers within the solid particles are relatively thin compared with the average particle diameter, the temperature proile will remain close to lat for most of the particle volume. In other words, it is assumed that the heat transfer within each particle is fast enough to equalize the temperature throughout its volume. Of course, that assumption may be more critical in the case of large particles. The Biot number provides a parameter to verify how reasonable that approximation is. It is given by the ratio between the internal and boundary resistances to heat transfer in the particle, or N Bi =

α G, p Vp . λ pA p

(7.1)

Usually, if the Biot number is below 0.1, a nearly lat temperature proile can be assumed [9]. This is the case for most operations of gas–solid combustion and gasiication processes.* For the gas phase, lat temperature proiles can also be assumed because of thin layers between particles. The assumption of lat concentration proiles within each phase follows similar considerations. For that, the mass transfer coeficient between solid and gas would replace the convective heat transfer coeficient, and diffusivity would replace the thermal conductivity. In the case of gas phase, the lat temperature and concentration proiles can be assumed when the total mass low in the main direction z is much higher than the low exchanged between phases. This is also the case in most gas–solid combustion and gasiication *

Further discussion of this aspect is presented in Appendix C.

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Solid Fuels Combustion and Gasification

equipment operations. Exceptions can be found, and careful examination of each case should be made. G. Heat transfer by radiation inside each phase and between phases will also be neglected in this irst approach. This seems acceptable because the gas layers between particles are small. Actually, assumption F forces the radiative transfers between particles in the radial direction to be neglected. The same can be said within the gas phase or between gas portions. On the other hand, since temperature varies in the axial (z) direction, radiative transfers among particles or gases at different layers exist. Nonetheless, because of the relatively large lows of gases and solids, transfers by radiation and even conduction would be masked by the overall convections in the same direction. Those points will become clearer during the description of fundamental equations. H. Secondary terms such as dissipation of energy due to viscous effects and couplings among various transport phenomena (Dufour and Soret effects), are neglected. Discussions of those effects can be found in the literature [10]. Of course, any model should be tested and the assumptions veriied after running the simulations for speciic cases. Here, only the basic equations of a model will be presented, while auxiliary ones are shown in Chapters 8 through 11. Comparisons between simulation and real operation parameters are illustrated in Chapter 12.

7.3.3

BASIC EQUATIONS

The coordinate system is very simple because only the axial direction (z) is considered here (see Figure 7.1). Again, let us follow the procedure used in the last chapter and apply the various conservation equations. 7.3.3.1 Global Continuity Global mass conservation or continuity is given by Equation A.7 (Appendix A, Table A.2), or ∂ρ 1 ∂(ρru r ) 1 ∂(ρu θ ) ∂(ρu z ) + + + = 0, ∂t r ∂r ∂z r ∂θ

(A.7)

which can be applied separately for the gas and solid phases. Because of condition A, the irst term does not exist. According to discussion in item B, the term involving the angular velocity component vanishes as well. In addition, the radial term (second on the left) will be written as a source term, and for the gas phase, the equation becomes d(ρG u G ) = R M,G , dz

(7.2)

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Moving-Bed Combustion and Gasification Model

whereas for the solid it will be written as d(ρSuS ) = − R M , G = R M ,S . dz

(7.3)

The value of the total rate of production* from the gas phase (R M,G) needs to be equal to that of consumption in the solid phase (R M,S). Therefore, the integrity of the mass conservation is preserved. If these two equations are added, the source terms cancel each other out, restoring the classical form of the overall continuity equation for a steady-state, plug-low regime, or d(ρu) = 0. dz

(7.4)

As a consequence, the product of total density and velocity is now deined as ρu = ρG u G + ρSuS.

(7.5)

7.3.3.2 Species Mass Continuity Species mass continuity is also called mass balance for each chemical species j and is described by Equation A.8 (Appendix A, Table A.2), or  1 ∂  ∂ρ j  1 ∂ 2ρ j ∂ 2ρ j  ∂ρ ∂ρ u ∂ρ ∂ρ j + 2  + R M , j . (A.8) + u r j + θ j + uz j = D j  r + 2 2 ∂z  r ∂θ ∂z ∂r ∂t  r ∂r  ∂r  r ∂θ Condition A eliminates the irst term on the left. Condition B provides a simpliication for the case of the gas phase and C for the solid, eliminating the radial and rotational velocity components. However, the radial transfers do exist and are given by the second term on the left and the irst inside the brackets. It is interesting to note that these terms represent the contribution of convective and diffusive mass transfers between the phases, respectively. In this case, as the one-dimensional approach was chosen, the radial transfers should be seen as sources or sinks of chemical species j and therefore similar to the last term, RM,j. For now, let us understand them as source and sink terms that are simply added to R M,j. After that, for the gaseous phase, Equation A.8 is written as uG

dρG , j d 2 ρG , j = DG, j + R M ,G , j , 1 ≤ j ≤ n G . dz dz 2

(7.6)

For the solid phase, a similar equation is provided by uS * †

dρS, j d 2ρS, j = DS, j + R M,S, j , 1 + n G ≤ j ≤ n. dz dz 2

(7.7)†

Or consumption, if a negative sign is used. Although the diffusivity coeficient of solid components in the solid phase is negligible, it is included in Equation 7.7 for the sake of completeness and coherence with Equation 7.6. That dissipating term will be omitted below.

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The notation adopted here assumes a total of n chemical components, from which the irst nG are gaseous ones and the remaining (n – nG) belong to the solid phase. Hence, n = nG + nS. Another aspect of the simpliication regarding the radial transfers between phases (or simpliication B) is that the migration of a chemical species from the gas to the solid phase does not require a radial velocity component or a diffusion process. For instance, the rate at which oxygen disappears from the gas phase is given by the rate at which it is consumed by homogeneous (gas–gas) and heterogeneous (gas–solid) reactions. The same is valid for gases produced in the solid phase and transferred to the gas phase. In other words, the rates of combustion and gasiication reactions govern the rates of chemical species transfers between phases. Consequently, RM,G,j represents the total rate of production (or consumption if negative) of gas component j by either gas– gas or gas–solid reactions. Moreover, the units for RM,G,j are kilograms (of gas component j) per unit of time per unit of gas phase volume (kg s–1 m–3). In the same way, RM,S,j represents the total rate of production (or consumption if negative) of solid component j, but this time only by gas–solid reactions. The units for RM,S,j are kilograms (of solid component j) per unit of time per unit of solid phase volume (kg s–1 m–3). As in Chapter 6, the diffusivity coeficient of a component j is represented by its value in relation to the mixture of components in the phase (G or S) where that component is present. As the composition, temperature, and pressure of each phase varies with the axial position, those diffusivities—as well as all other physical and chemical properties—would be recalculated at each point during the computation. Another important aspect of the diffusion terms is that nG

∑D

G, j

j =1

d 2ρ G , j =0 dz 2

and n

∑

DS, j

j =1+ n G

d 2ρS, j = 0. dz 2

(7.8)

(7.9)

Of course, components with greater diffusivity in relation to the average of the mixture tend to low ahead the average front, whereas the ones with lower diffusivity lag behind that front. Nonetheless, the net contribution of diffusion for the global low is 0. The total mass produced (or consumed) due to chemical reactions in each phase is given by adding the rates of production (or consumption) of all components in the respective phase, or nG

R M,G = ∑ R M,G , j

(7.10)

j =1

and R M ,S =

n

∑

j =1+ n G

R M,S, j .

(7.11)

Moving-Bed Combustion and Gasification Model

151

One should also remember that nG

ρG = ∑ ρG , j

(7.12)

j =1

and ρS =

n

∑

ρS, j .

(7.13)

j=1+ n G

7.3.3.3 Continuity of Energy The energy balance is given by Equation A.9 (Appendix A, Table A.2), or  1 ∂  ∂T  1 ∂ 2 T ∂ 2 T  ∂T u θ ∂T ∂T   ∂T ρc  + ur + + uz = λ  r  + 2 2 + ∂z 2  ∂r ∂z  r ∂θ  ∂t  r ∂r  ∂r  r ∂θ  2 2  ∂u 2  1  ∂u    ∂u   + 2µ  r  +   θ + v r   +  z      ∂z    ∂r   r  ∂θ

(A.9)

2  ∂u 1 ∂u 2  ∂u ∂u 2  1 ∂u ∂  u θ    z z r r + µ  θ + + + r + +  + RQ .   ∂z   r ∂θ ∂r  r     ∂z r ∂θ   ∂r 

As already discussed, energy transfers between phases are assumed to occur in the radial direction. They can be recognized in Equation A.9. Convective transfer is given by the second term in the brackets on the left side of the equation, and conductive transfer is given by the irst term in the irst brackets on the right side. Note that what is commonly known as convection heat transfer is not a fundamental phenomenon but simply a combination of heat conduction and velocity ields, or massconvective transfers. The dificulty of solving the partial differential equations with all these terms reafirms the convenience of empirical and semiempirical relations to compute combinations of transfer phenomena. These relations are available in the literature for various situations. Here, the combination of heat conduction and mass convection is going to be computed by the sum of two terms: 1. The convective heat transfer (or simply convection) (RC) 2. The enthalpy addition (or subtraction) (R h) to one phase due to the mass migration from (or to) the other phase Fortunately, these transfers can be seen as energy source or sink terms to be added or subtracted from RQ,G and RQ,S. The present approach also brings another advantage because radiative heat transfers between phases (not shown in Equation A.9), RR, could be also included as part of the source (or sink) terms. A suggestion for further improvement would be to include the radiative transfer in the axial direction as part of the R R term. The combination of heat transfers by conduction (the third term in the irst brackets of the right side of Equation A.9) and radiation in the axial direction acts as a dissipative effect, thereby

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Solid Fuels Combustion and Gasification

preventing extreme peaks of temperature caused by surges in combustion rates. The details of how to compute the source or sink terms are presented in Chapter 11. Using the above discussion on radial contributions and considering assumptions A, B, C, F, and G, the above equation can be written for the gas phase as ρG c G u G

dTG d 2 TG = λG + R Q , G + R C , G + R h , G + R R ,G . dz dz 2

(7.14)

For the solid phase, one gets ρ Sc S u S

dTS d2T = λ S 2S + R Q,S + R C,S + R h,S + R R ,S. dz dz

(7.15)

7.3.3.4 Momentum Continuity The conservation of momentum is described by a set of three equations. However, the only relevant momentum conservation equation is the one in the axial direction, and from Table A.2 (Appendix A), u ∂u ∂u ∂u  ∂u ρ  z + ur z + θ z + uz z r ∂θ ∂r ∂z  ∂t  1 ∂  ∂u z +µ r  r ∂r  ∂r

∂P   = − ∂z 

 1 ∂ 2uz ∂ 2uz   + r 2 ∂θ2 + ∂z 2  + ρg z .  

(A.12)

Again, the terms due to transfers between phases in the radial direction can be recognized as follows: • The momentum transfer due to the global mass convection is the second term in the left-side brackets. • The momentum transfer due to viscous transfer is the irst term in the rightside brackets. Because of assumptions A, B, C, D, and E, for the gas phase, the above equation becomes du dP ρG u G G = − G − ρG g , (7.16) dz dz and the equation for the solid becomes ρS u S

duS dP = − S − ρSg . dz dz

(7.17)

7.3.3.5 Summary of the Problem The system formed by Equations 7.2, 7.3, 7.6, 7.7, 7.14, 7.15, 7.16, and 7.17 involves two velocities (uG, uS), n concentrations (ρG,j, ρS,j, n = nG + nS), two temperatures

Moving-Bed Combustion and Gasification Model

153

(TG, TS), and two pressures (PG, PS). The two global densities (ρG, ρS) can be computed from Equations 7.12 and 7.13. Therefore, their values can be regarded as simply consequences of the above system solution. Thus, there are 6 + n equations and the same number of variables. Once the proper boundary conditions are set, the numerical computation will provide the proiles of velocities, concentrations, temperatures, and pressures for the two phases. 7.3.3.6 Further Simplifications Despite the possibility of solving the above system, the answers required for engineering design and industrial production can be better provided if forms based on mass lows instead of concentrations are available. In addition, further simpliications can be adopted to write a irst simpliied model. 7.3.3.6.1 Mass Balances The irst approximation is to neglect the diffusion term in Equations 7.6 and 7.7. As noted in Chapter 6, this is justiiable because in moving-bed combustors or gasiiers, the main convective terms, given by the left sides of those equations, are much greater than the diffusion additions. Moreover, the diffusivities of solid components are really negligible. Therefore, these equations become dρG, j = R M,G , j , 1 ≤ j ≤ n G dz

(7.18)

dρS, j = R M,S, j , 1 + n G ≤ j ≤ n . dz

(7.19)

uG and uS

On the other hand, the mass lux of a chemical species j in the z or axial direction is given by FG, j = ρG, ju GSG.

(7.20)

Similarly, the mass low of a chemical species j in the solid phase is FS, j = ρS, juSSS.

(7.21)

Here, SG and SS are the fractions of total transversal or cross-section reactor area, respectively, available to the low of gas and solid phases. The variations for these areas and on the overall velocity in each phase would be computed at each axial position. One should also be aware that Equations 7.18 and 7.19 were obtained from Equation A.8, which was derived assuming constant global density. Equation 7.4 shows that this would lead to constant global velocity. If all densities were constants combined with the fact of independent (or nearly independent) velocities in each phase, Equation 7.5 would lead to constant velocities uG and uS. Of course, this is not true, and likewise Equation 7.2, a more general and rigorous form of Equation 7.18, should present the product of velocity uG and the concentration of component j inside

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Solid Fuels Combustion and Gasification

the derivative against the axial coordinate. A similar argument can be applied to the solid phase, and therefore, Equations 7.18 and 7.19 can be written as dFG, j = SG R M,G, j , 1 ≤ j ≤ n G dz

(7.22)

dFS, j = SSR M,S, j , 1 + n G ≤ j ≤ n . dz

(7.23)

and

It is dificult not to notice the simplicity of Equations 7.22 and 7.23. For instance, the mass low (in kg/s) of a given chemical component j of the gas phase increases or decreases in the low direction if it is being produced (RM,G,j positive) or consumed (RM,G,j negative), respectively. The same is valid for the solid phase. A similar argument can be applied to the total mass low at each phase. Therefore, Equation 7.2 can be written as dFG = SG R M,G dz

(7.24)

dFS = SSR M,S . dz

(7.25)

and Equation 7.3 as

On the other hand, by deinition the total mass low in the gas phase is also given by nG

FG = ∑ FG, j .

(7.26)

j =1

Therefore, once the entire set of differential Equation 7.22 is available, Equation 7.26 would provide the total mass low of that phase at each height z in the bed. This allows us to discard Equation 7.24 from the system of differential equations to be solved. The same reasoning can be made for the solid phase: FS =

n

∑

FS, j .

(7.27)

j=1+n G

Likewise, the solution of Equation 7.23 would render unnecessary the effort to solve Equation 7.25. In Equations 7.22 and 7.23, the cross-sectional areas SG and SS through which the gas and solid phases should travel are dificult to measure and therefore dificult to verify. However, they can be related to the total cross-sectional area S of the reactor, which is known or easily measured. This can be achieved by applying the concept of void fraction ε. This is deined as the ratio between the volume occupied by the gas in the bed and the total volume of the bed, or ε=

VG . V

(7.28)

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Moving-Bed Combustion and Gasification Model

Now, for the gas phase it is possible to write S V 1 = = , SG VG ε

(7.29)

S V V 1 . = = = SS VS V − VG 1 − ε

(7.30)

and for the solid phase,

Therefore, Equation 7.22 can be written as dFG , j = εSR M ,G , j , 1 ≤ j ≤ n G, dz

(7.31)

and Equation 7.23 can be written as dFS, j = (1 − ε)SR M ,S, j , 1 + n G ≤ j ≤ n. dz

(7.32)

It is important to note that in Equation 7.31, the rate of production (or consumption) of gaseous component j is computed by the gas–gas (or homogeneous) reactions, as well as by the gas–solid (or heterogeneous) reactions in which j is involved. Again, it should be remembered that this is possible only because of simpliication B, i.e., that all gas components produced (or consumed) by gas–solid reactions leave (or enter) the solid phase and are immediately added to (or subtracted from) the gas phase. In Equation 7.32, the rate of production or consumption of solid phase component j is computed only by the gas–solid (or heterogeneous) reactions. Let us use this opportunity to introduce the concepts of supericial velocities UG and US of the gas and solid phases, respectively. A supericial velocity represents the velocity of a phase if it could occupy the entire cross-sectional area of the reactor. Supericial and real average velocities can be related by the following equations:

and

U G = εu G

(7.33)

U S = (1 − ε) u S .

(7.34)

The demonstration of those equations is left as an exercise. Using Equations 7.26 through 7.30, 7.33, and 7.34, it is also possible to write

and

FG = ρG u GSG = ρG U GS

(7.35)

FS = ρSuSSS = ρS U SS.

(7.36)

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Solid Fuels Combustion and Gasification

It should be noted that Equations 7.35 and 7.36 allow the mass lows to be written without reference to the particular cross-sectional area of their respective phases. Of course, the problem of varying areas is transferred to the supericial velocities. Nevertheless, as shown in Chapter 11, this treatment presents some conveniences because it facilitates the use of correlations from the literature. One must remember that the void fraction ε varies along the bed and should be reevaluated at each axial position or height z (method is shown in Chapter 11). Nonetheless, in most cases, that variation is mild, and some simpliications might even assume void fraction as a constant throughout the bed. 7.3.3.6.2 Energy Balances Departing from Equations 7.14 and 7.15, an approximation can be made where heat transfers by conduction in the axial direction are neglected. This is possible because the irst terms of the right sides are relatively small compared with the respective left sides. In other words, the energy carried by the main low in the axial direction is much higher than that transferred by conduction in the same direction. Combining the above with Equations 7.35 and 7.36, it is possible to write Equations 7.14 and 7.15 as dTG = εS(R Q,G + R C,G + R h,G + R R ,G ) dz

(7.37)

dTS = (1 − ε)S(R Q,S + R C,S + R h,S + R R ,S ). dz

(7.38)

FG c G and FSc S

7.3.3.6.3 Momentum Balances For relatively small variations of the cross-sectional area of the bed occupied by each phase, as well as for the void fraction, Equations 7.16 and 7.17 can be rewritten in terms of mass lows, or dPG F dU G = − 2G − ρG g dz ε S dz

(7.39)

dPS FS dU S =− − ρ Sg . 2 dz (1 − ε) S dz

(7.40)

and

In the above equations, the irst terms on the right sides are the dynamic pressure losses, whereas the second terms are the static pressure losses. The solution of such equations would not bring any additional dificulty to the problem. On the other hand, most moving-bed operations do not present considerable variation in pressure throughout the equipment. In addition, concentrations and temperature proiles are not severely affected by relatively small pressure variations.

Moving-Bed Combustion and Gasification Model

157

Unlike the case of gas, variation of pressure in the solid phase is considerable, mainly because of the static contribution. However, that phase should be considered incompressible, i.e., properties present almost no variation regarding changes in the pressure. In view of the above, and at least as a irst approximation, isobaric operation can be assumed, and Equations 7.39 and 7.40 can be discarded from the system to be solved. 7.3.3.6.3.1 Notes on the Simplified Problem The apparent simplicity of Equations 7.31, 7.32, 7.37, and 7.38 is deceptive. For instance, the rate of component generation for each chemical component j must be calculated at each axial position z of the reactor. There are several simultaneous endothermic and exothermic reactions involving the same component. As seen before, the rates of those reactions vary exponentially with the temperature and depend on the concentrations of several other components besides j. Therefore, one should expect a strong coupling between composition and temperature throughout the bed.

7.3.4

BOUNDARY CONDITIONS

The solutions of Equations 7.31, 7.32, 7.37, and 7.38 would provide the mass lows and temperatures throughout the bed. In addition, at each point in the bed the following relations FG, j , 1 ≤ j ≤ nG FG

(7.41)

FS, j , 1 + nG ≤ j ≤ n FS

(7.42)

wG, j =

wS, j =

would allow drawing proiles of concentrations at each gaseous and solid species throughout the bed. The solutions for the proposed system with 2 + nG + nS (or 2 + n) differential equations require the same number of boundary conditions. The boundary conditions are those at the top of the bed, or z = 0, and are given by the following: 1. 2. 3. 4.

The mass low of each gas component Fj(0), for 1 ≤ j ≤ nG The gas, or even the gas mixture, inlet temperature, i.e., TG(0) The mass low of each solid component Fj(0), for nG+1 ≤ j ≤ n The inlet temperature of solid phase, i.e., TS(0)

Therefore, at irst glance, all conditions are known because they represent the very inputs set for the reactor operation. Nonetheless, that is a bit deceiving, since if the solid fuel is fed at ambient temperature, no ignition would be possible. Similarly to the case described in Chapter 6, those conditions would lead to lat proiles of

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Solid Fuels Combustion and Gasification

temperature and composition throughout the bed. In real operations, if ignition is not started, the combustor or gasiier would not work as intended. This is provided at the beginning of operation by heating the bed with a gas torch or any other means; once the fuel starts burning near the top of the bed, more fuel is added along the gas stream. A well-designed combustor or gasiier would be self-suficient, and a real steady-state regime would be established. The ignition temperature for the solid phase can be found either by setting a value TS(0) or by inding its derivative. Of course, near the point where air or oxidant gas mixture is injected into the downdraft bed, many processes take place, including combustion. The rate of exothermic reactions is much higher than that of endothermic ones. The surplus of energy released by the reactions should be enough to provide the ignition. That would be achieved when the rate of overall energy production by reactions in the solid phase is capable of maintaining the process and increasing the temperature of both phases for positions just above the bed base (z = 0). Using the differential equations shown above, a convergence routine to compute the derivative of solid phase temperature would determine the ignition (Tig). It would impose increasing values of TS(z = 0) until the derivative the left side of Equation 7.38 becomes 0 or positive. The last set value for TS would be Tig. Note that RQ,S should be the driving force to allow a positive derivative of temperature. A similar procedure may be applied using the gas temperature derivative instead of that related to the solid phase. Despite a known value for injected gas temperature, the derivative for gas phase at z = 0 could be found to allow ignition of solid phase. Few tests have shown that ignition has been achieved for values above 106 K/m. This method has proved to be more robust and thus preferable to inding conditions for the solid-phase positive derivative as described above. On the other hand, it has been noted that for a wide range of experiments with various carbonaceous solids and conditions, ignition occurs around 700 K. That value can be tried as TS(0) and the results used to calibrate the simulator. Nevertheless, this requires at least one experimental determination in using the equipment, and that may not be available if one is trying to design a gasiication or combustion unit. Hence, the convergence described above is the preferred method. The solution of the complete boundary condition problem would provide all information regarding the concentrations and temperatures of all gaseous and solid species throughout the bed. Of course, all engineering variables, such as quality of produced gas and discharged residue, their temperatures, overall reactor performance, and many others would be known. This is the complete simulation of any downdraft moving-bed equipment. Of course, another task is to set correlations and methods for computing the auxiliary parameters involved in the process and asked by differential Equations 7.31, 7.32, 7.37, and 7.38. Among them are chemical and physical properties of individual species and their mixtures, kinetics of all involved reactions, etc. Such details are described in Chapter 11.

7.4

UPDRAFT MOVING-BED REACTORS

Figure 7.5 shows a schematic view of an updraft moving-bed reactor.

Moving-Bed Combustion and Gasification Model

159

Carbonaceous solid feeding

z = zD

Main withdraw of gases (z = zD)

Bed

Possible intermediate injection of gases (0 < z < zD) Insulation

z

Main injection of combustion or gasifying agents (z = 0)

z=0 Ash withdrawal (z = zD)

FIGURE 7.5 Scheme of updraft reactor.

As one may recall, in the countercurrent or updraft type, the solid carbonaceous particles are fed at the top of the reactor and slowly low to the base where the solid residual is withdrawn. Therefore, the main difference between the updraft and downdraft reactor rests on the direction of the gas stream. All other aspects have been already presented, and the remainder of this chapter concentrates only on the modiications to be made to the previous approach in order to model the updraft version.

7.5

THE MODEL

The fundamental hypothesis or set of assumptions, as well as the equations, is the same as for the case of downdraft equipment. The differences between the two types of moving bed are in the boundary conditions.

7.5.1

THE MODEL CHART

Again, the reactor is imagined to be divided into two basic streams, gas and solid, which low in a parallel vertical countermovement, as illustrated by Figure 7.6. These two streams would exchange heat and mass through their common interface. The interface can be imagined as a single continuous surface with area equivalent to the real area separating the two phases. Again, that area would be given by the total surface area of solid particles in the bed. As the size of particles usually varies

160

Solid Fuels Combustion and Gasification Exiting gas

Solid particles

z = zD

Virtual interface between gas and solid phases

Solid phase

Combustor or gasifier

Heat & mass

z

Gas phase

z=0 Solid residue

Gas injection

FIGURE 7.6 Scheme of idealized updraft moving-bed reactor.

throughout the bed, the area per unit of bed height would also vary from point to point in the bed. For clarity, the assumptions are reviewed below: 1. Again, the assumption of an overall plug-low regime for luids percolating through a bed of particles in one main direction is reasonable, as illustrated Figure 7.3. This point is discussed in the respective section for downdraft reactors. 2. The consequences of the adopted plug-low regime on mass transfers between phases should also be consulted in the previous section on downdraft reactor modeling. However, Figure 7.6 should be consulted regarding the low orientations. After those basic considerations are taken into account, it is possible to draw Figure 7.7, which illustrates the adopted model chart for moving-bed updraft reactors.

7.5.2

SUMMARY OF MODEL ASSUMPTIONS

Before setting equations, it is advisable to develop a clear list of model assumptions. Most of them are exactly the same as in the case of downdraft reactors, and the reader should ind additional details in the previous section. However, for the sake of clarity, the list of assumptions is briely presented, as follows: a) Steady-state operation is assumed. b) Gas lows upward in a plug-low regime. Here, the same assumption as in the previous downdraft case is taken, with the exception of the direction of the stream.

161

Moving-Bed Combustion and Gasification Model Carbonaceous solid particles

Exiting gas

z = zD

Heat & mass Descending solid particles (plug-flow)

Ascending gas (plug-flow)

Heat

Environment

Heat & mass

z=0 Solid residue (ash & unburned)

Gas injection

FIGURE 7.7 Model chart adopted for the updraft moving-bed reactor.

c) d) e) f) g)

Solid particles low downward in a plug-low regime. Momentum transfers between the two phases are negligible. Inviscid low is assumed for the two phases. At each phase, temperature and concentration proiles are lat. Heat transfer by radiation inside each phase and between phases will also be neglected in this irst approach. h) Secondary inluences, such as dissipation of energy due to viscous effects and the inluences of the change of one basic transport phenomenon into another (Dufour and Soret effects), are neglected.

Comparisons between simulation and real operation parameters are shown in Chapter 12.

7.5.3

BASIC EQUATIONS

The same basic equations apply for the updraft and downdraft versions. However, attention should be paid to the change in the orientation of axial or vertical coordinate (z). For instance, for the downdraft version, the velocity of the solid phase (uS) was positive or in the direction of coordinate z, whereas now it acquires negative values, or in the opposite direction of coordinate z.

7.5.4

BOUNDARY CONDITIONS

Again, the solutions of Equations 7.31, 7.32, 7.37, and 7.38 would provide the mass lows and temperatures throughout the bed. Using the relationships in Equations 7.41 and 7.42, the mass fractions, as well as any other form of concentration proiles, can be drawn.

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Solid Fuels Combustion and Gasification

The solutions for the proposed system with 2 + nG + nS (or 2 + n) differential equations would require an equal number of boundary conditions. Nevertheless, in the case of the updraft version, setting the boundary conditions for the solid phase becomes a problem. Neither its temperature nor its composition is known at z = 0. On the other hand, both are established at the feeding point, or z = zD. This is a typical inal, irst-kind boundary condition (Equation 6.10b). In all, the following boundary conditions are necessary to solve the complete problem: • The mass low of each gas component Fj(0), for 1 ≤ j ≤ nG. Those values are known. • The inlet temperature of gas mixture TG(0). This is known as well. • The mass low of each solid component withdrawn from the reactor Fj(0), for nG + 1 ≤ j ≤ n. Note that, differently from the downdraft version, in updraft beds, the solids move against the z direction; therefore, their mass lows acquire negative values. Those values are not known. • The discharge temperature of solid phase TS(0). That is not known either. As we have seen, the boundary conditions for the solid phase are not known at z = 0. On the other hand, those values are set at the feeding point of the solid particles in the bed, i.e., at z = zD. The strategy used in this case of inal boundary condition can be summarized by the following steps: 1. Guess the values of the temperature and the mass low of each component (which would allow the composition computation) of the solid phase at z = 0. 2. Solve the system of differential equations. 3. Obtain the computed values of the composition and temperature for the solid phase at the top of the bed (z = zD). 4. Compare those values with those known for solid feeding. 5. If the absolute differences between the guessed and computed values are above a certain maximum value, return to step 1. A method for setting a new guess is provided by the adopted convergence procedure. This procedure is not as simple as it may sound. The basic dificulty is related to the number of convergence variables. There are several components and one temperature to be converged. However, the usual methods accept just one variable as a convergence parameter. Consequently, if the above procedure is to be implemented, it would need several nested convergence loops. This certainly would lead to a cumbersome computational procedure with a very slim chance of success. One method that works fairly well in several situations of combustion and gasiication modeling [2–6, 11–23] is to select one reference component and set the whole convergence strategy around it. This procedure can be described as follows: a) Choose a reference chemical component typical of the solid fuel, such as carbon. b) Assume a fractional carbon conversion in the process. Therefore, the mass low of carbon at z = 0 would be set.

Moving-Bed Combustion and Gasification Model

163

c) Assume the fractional conversions of all other solids of the fuel equal to the carbon conversion (of course, this is not applicable to ash). This can be assumed only for the irst iteration. For subsequent iterations, a simple substitution of the last computed value works well. d) The solution of the differential mass and energy equations throughout the bed would lead to the value of the mass low of carbon at the top. That value should be compared with the mass low of carbon at the feeding. If the difference is above a certain tolerance, return to step A. Carbon usually is the best choice as the reference component for the convergence procedure because it is the one always present in solid fuels and at a higher concentration in the feeding fuel. Thus, carbon conversion is less sensitive than conversion of other components of the solid matrix. The experience has shown that when convergence of carbon conversion is achieved, convergences for all other component conversions are reached as well. The other variable to be converged is the temperature of the solid phase at z = 0, which allows determination of the ignition temperature for the solid fuel. The methods are similar to those described for the case of a downdraft reactor. However, in the case of the updraft version, the solid leaving the bed at z = 0 should have some amount of unconverted fuel to allow ignition. Nonetheless, too much unconverted fuel is an indication of low gasiication eficiency. Well-designed and well-operated moving beds present values of 95%–99% for the fraction of unconverted carbon. In spite the seemly narrow range of such conversions, differences of just 1% lead to signiicant differences in operational behavior. For instance, the position and value of peak temperatures (Figure 3.6) are very sensitive to carbon conversion, and only dimensional models can determine that position. Some considerations related to the real situation of updraft version provide an alternative method to avoid the use of convergence procedures. Since the injected gas mixture has to travel through a layer of ash deposited at the base of the bed, one may speculate that the gas and solid phase temperatures would be equal before gases start reacting with the solid phase at z = 0. In addition, Figure 3.6 shows a sharp peak of temperature not very far from the base of the bed. This provides a criterion to set the ignition temperature as the values for the solid and gas phases at z = 0. The greater the assumed value for that temperature, the lower the height in the gasiier where the temperature peak would occur. Having this in mind, an appropriate minimum value can be adjusted by matching the position of the peak predicted by simulation to one that has been experimentally veriied. It has been seen that for a wide range of experiments with various carbonaceous solids and conditions, ignition occurs around 700 K. On the other hand, the temperature at the top of the bed must be equal to the solid feeding temperature. This last piece of information is the inal test for the procedure described here, and it might help to calibrate the program by adjusting the ignition temperature at z = 0. Nevertheless, again, this last method requires at least one experimental determination in using the equipment, and that may not be available if one is trying to design a gasiication or combustion unit. Hence, the convergence described above is the preferred method unless real operational conditions require the gas to pass through the layer

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of deposited ash. In that case, the alternative last method, based on setting the ignition temperature, may be preferable.

7.6 7.6.1

EXERCISES PROBLEM 7.1* Deduce Equations 7.33 through 7.36.

7.6.2

PROBLEM 7.2* Show that Equation 7.16 can be written as  F dρ dPG F dF  = 2 G2  G G − G  − ρG g. dz dz  ε S ρG  ρG dz

7.6.3

(7.43)

PROBLEM 7.3** Discuss the effects on the above model (which led to system formed by Equations 7.31, 7.32, 7.37, and 7.38) if the diffusion effect in the axial or vertical (z) direction is included. A. What would be the inal form of the mass balance for individual chemical species? B. What would be the effect on the composition proiles compared with those obtained from Equations 7.31 and 7.32? C. Would it be possible to assume constant diffusion coeficients throughout the bed? Explain. D. Would it be possible to assume constant densities for gases or for solids throughout the bed? Explain. E. Show the inal form for the gas phase equation assuming an ideal behavior. F. What would the necessary boundary conditions be?

7.6.4

PROBLEM 7.4** Add Equations 7.39 (or the alternative form Equation 7.43) and 7.40 to the set formed by Equations 7.31, 7.32, 7.37, and 7.38. Assuming ideal behavior for the gas phase and incompressibility for the solid phase, describe the system of differential equations to be solved in order to provide proiles for the mass lows, temperatures, and pressures throughout the bed. Set the necessary boundary conditions.

7.6.5

PROBLEM 7.5*** If the model either for updraft or downdraft versions proved to require the inclusion of the radial (r) direction, the reader is asked the following: A. What would be the set of differential equations for the mass and energy balances throughout the bed? B. Assuming isobaric conditions, how many boundary conditions would now be necessary to arrive at concentration and temperature proiles throughout the bed? C. Try to write those boundary conditions.

Moving-Bed Combustion and Gasification Model

165

REFERENCES 1. Smoot, L.D., and Smith, P.J., Coal Combustion and Gasiication, Plenum Press, New York, 1985. 2. de Souza-Santos, M.L., Modelling and Simulation of Fluidized-Bed Boilers and Gasiiers for Carbonaceous Solids, PhD thesis, University of Shefield, Shefield, United Kingdom, 1987. 3. de Souza-Santos, M.L., Development of a Simulation Model and Optimization of Gasiiers for Various Fuels (Desenvolvimento de Modelo de Simulação e Otimização de Gaseiicadores com Diversos Tipos de Combustíveis), IPT-Inst. Pesq. Tec. Est. São Paulo, SCTDE-SP, Report No. 20.689, DEM/AET, São Paulo, Brazil, 1985. 4. de Souza-Santos, M.L., and Jen, L.C., Study of Energy Alternative Sources; Use of Biomass and Crop Residues as Energy Source: Part B: Development of a Mathematical Models and Simulation Programs for Up-Stream and Down-Stream Moving Bed Gasiiers. (Estudo de Fontes Alternativa de Energia, Parte B), IPT—Institute for Technological Research of São Paulo, Report No. 16.223-B/DEM/AET, São Paulo, Brazil, 1982. 5. Jen, L.C., and de Souza-Santos, M.L., Modeling and simulation of ixed-bed gasiiers for charcoal, Brazilian Journal of Chemical Engineering (Modelagem e simulação de gaseiicador de leito ixo para carvão, Revista Brasileira de Engenharia Química), 7(3–4), 18–23, 1984. 6. DeSai, P.R., and Wen, C.Y., Computer Modeling of the MERC Fixed Bed Gasiier, U.S. Department of Energy Report, MERC/CR-78/3, Morgantown, WV, 1978. 7. Yoon, H., Wei, J., and Denn, M.M., A model for moving-bed coal gasiication reactors, AIChE J., 24(5), 885–903, 1978. 8. Hobbs, M.L., Radulovic, P.T., and Smoot, L.D., Combustion and gasiication of coals in ixed-beds, Prog. Energy Combust. Sci., 19, 505–556, 1993. 9. Schmidt, F.W., Henderson, R.E., and Wolgemuth, C.H., Introduction to Thermal Sciences, 2nd Ed., John Wiley, New York, 1984. 10. Bird, R.B., Stewart, W.E., and Lightfoot, E.N., Transport Phenomena, John Wiley, New York, 1960. 11. de Souza-Santos, M.L., Comprehensive modelling and simulation of luidized-bed boilers and gasiiers, Fuel, 68, 1507–1521, 1989. 12. de Souza-Santos, M.L., Application of comprehensive simulation to pressurized bed hydroretorting of shale, Fuel, 73, 1459–1465, 1994. 13. de Souza-Santos, M.L., A study on pressurized luidized-bed gasiication of biomass through the use of comprehensive simulation, in Proc. Fourth International Conference on Technologies and Combustion for a Clean Environment, Lisbon, Portugal, July 7–10, 1997, paper 25.2, vol. II, pp. 7–13. 14. de Souza-Santos, M.L., A study on pressurized luidized-bed gasiication of biomass through the use of comprehensive simulation, in Combustion Technologies for a Clean Environment, Gordon and Breach, Amsterdam, Netherlands, 1998. 15. de Souza-Santos, M.L., Application of comprehensive simulation of luidized-bed reactors to the pressurized gasiication of biomass, Journal of the Brazilian Society of Mechanical Sciences, 16(4), 376–383, 1994. 16. de Souza-Santos, M.L., Search for favorable conditions of atmospheric luidized-bed gasiication of sugar-cane bagasse through comprehensive simulation, in Proc. 14th Brazilian Congress of Mechanical Engineering, Bauru, São Paulo, Brazil, December 8–12, 1997. 17. de Souza-Santos, M.L., Search for favorable conditions of pressurized luidized-bed gasiication of sugar-cane bagasse through comprehensive simulation, in Proc. 14th Brazilian Congress of Mechanical Engineering, Bauru, São Paulo, Brazil, December 8–12, 1997.

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18. de Souza-Santos, M.L., A feasibility study of an alternative power generation system based on biomass gasiication/gas turbine concept, Fuel, 78, 529–538, 1999. 19. Rabi, J.A., and de Souza-Santos, M.L., Incorporation of a 2-lux model for radiative heat transfer in a comprehensive luidized bed simulator. Part I: Preliminary model equations, in Proc. ENCIT-2002, 9th Brazilian Congress on Engineering and Thermal Sciences, Caxambu, Minas Gerais, Brazil, October 15–18, 2002. 20. Rabi, J.A., and de Souza-Santos, M.L., Incorporation of a 2-lux model for radiative heat transfer in a comprehensive luidized bed simulator. Part II: Numerical results and assessment, in ENCIT-2002, 9th Brazilian Congress on Engineering and Thermal Sciences, Caxambu, Minas Gerais, Brazil, October 15–18, 2002. 21. Rabi, J.A., Usage of Flux Method to Improve Radiative Heat Transfer Modelling inside Bubbling Fluidized Bed Boilers and Gasiiers, PhD thesis, Faculty of Mechanical Engineering, State University of Campinas, Campinas, São Paulo, Brazil, 2002. 22. Costa, M.A.S., and de Souza-Santos, M.L., Studies on the mathematical modeling of circulation rates of particles in bubbling luidized beds, Power Technology, 103, 110– 116, 1999. 23. Rabi, J.A., and de Souza-Santos, M.L., Incorporation of a two-lux model for radiative heat transfer in a comprehensive luidized bed simulator. Part I: Preliminary theoretical investigations, Thermal Engineering, 3, 64–70, 2003.

8 Chemical Reactions CONTENTS 8.1 8.2 8.3 8.4

Introduction .................................................................................................. 167 Combustion and Gasiication Reactions ....................................................... 167 Numbering Chemical Components .............................................................. 168 A System of Chemical Reactions ................................................................. 168 8.4.1 Nitrogen Oxides ................................................................................ 170 8.5 Stoichiometry ................................................................................................ 172 8.6 Kinetics ......................................................................................................... 173 8.6.1 Homogeneous Reactions ................................................................... 175 8.6.1.1 Equilibrium ........................................................................ 175 8.7 Final Notes on Kinetics ................................................................................ 175 8.8 Independent Set of Reactions ....................................................................... 179 8.8.1 Mathematical Criterion..................................................................... 179 8.9 Exercises ....................................................................................................... 180 8.9.1 Problem 8.1 ....................................................................................... 180 8.9.2 Problem 8.2 ....................................................................................... 180 References .............................................................................................................. 180

8.1

INTRODUCTION

This chapter presents the following: • A system for numbering chemical species and reactions • Various comments on the proposed system of reactions • Methods by which to calculate rates of homogeneous reactions Techniques to compute rates of most of the heterogeneous reactions are left for Chapter 9, and those related to pyrolysis are presented in Chapter 10.

8.2

COMBUSTION AND GASIFICATION REACTIONS

As seen in the last chapter, the continuity equations for chemical species require the description of the rates of production or consumption of each component j. For the purposes of the present text, there are basically two kinds of reactions: 1. Homogeneous, or gas–gas, reactions 2. Heterogeneous, or gas–solid, reactions For each kind of reaction, the rate of production or consumption of a given component j can be described by 167

168

Solid Fuels Combustion and Gasification

R kind , j = M j ∑ νij ri.

(8.1)

i

As always, the stoichiometric coeficient νij is positive if the chemical species j is produced and negative if consumed by each reaction i. The reaction rate ri is described by its kinetics, which depends on the temperature, pressure, and concentrations of the chemical components involved, as well as heat and mass transfer resistances. In addition, if catalysts are present, their concentrations play an important role in the rates of reactions affected by them. Mass transfers play a very important role in rates of production and consumption of chemical species during heterogeneous reactions. This is detailed in Chapter 9. Although important for fast reactions, gas–gas reaction rates may be also be inluenced by the mass transfer process. This transfer is necessary to bring the molecules of reacting gases close enough to react. If the reaction is relatively slow, the mass transfer inluence is usually negligible. In this introductory text, this has been assumed with reasonable to good results when the simulations are compared against data from real plants. However, the user may ind it interesting to search for texts on the subject. It is also important to notice that in a more complete model for gas– gas reactions, mass transfer resistances may not only improve the whole model but also avoid stiff differential equations. This is easily understood if one thinks about the high rate of energy released during the combustion of, for example, hydrogen. As seen in Chapters 2 and 3, during devolatilization, hydrogen is released from the particle and added to the gas phase, where oxygen may be present. If the kinetics is allowed to play unchecked, the computation might by interrupted because of too stiff energy differential balance equations. Therefore, more realistic approach not only leads to better representations of reality but also contributes to faster and more robust simulation programs.

8.3

NUMBERING CHEMICAL COMPONENTS

A numbering system for the chemical components is useful in order to allow shorter notations for the mathematical and computational treatment. Tables 8.1 through 8.3 list the components that are the most common in the combustion and gasiication process. This numbering will be useful throughout the book. It should be noted that tar is assumed to be a single component, as are ash and volatile. As seen before, tar is a complex mixture of organic and inorganic components in vapor and liquid phases. The liquid fraction is released as a mist suspended in the gas stream leaving the solid fuel during the devolatilization process. However, it is included in the gas phase. This brings some convenience and simplicity because the inclusion of another physical phase (liquid) can be avoided.

8.4

A SYSTEM OF CHEMICAL REACTIONS

A reasonable set of chemical reactions for a general problem of carbonaceous gasiication is described in Tables 8.4 and 8.5.

169

Chemical Reactions

TABLE 8.1 Numbers for the Gas Components Used Throughout the Text Name Hydrogen Water Hydrogen sulide Ammonia Nitric oxide Nitrogen dioxide Nitrogen Nitrous oxide Oxygen Sulfur dioxide Carbon monoxide Carbon dioxide Hydrogen cyanide Methane Ethylene Ethane Propylene Propane Benzene Naphtalene n-Dodecane

Molecular Formula

Number

H2 H2O (gas) H2S NH3 NO NO2 N2 N2O O2 SO2 CO CO2 HCN CH4 C2H4 C2H6 C3H6 C3H8 C6H6 C10H8 C12H26

19 20 21 22 27 28 29 30 32 33 44 46 51 61 88 100 120 132 242 398 433

TABLE 8.2 Numbers for the Components of Carbonaceous Solid Phase Name Carbon Hydrogen Nitrogen Oxygen Sulfur Asha Moisture Volatile a

Molecular Formula

Number

C H N O S SiO2 H2O (liquid) A mixture

514 531 546 551 563 824 700 1000

As seen before, ash is a complex mixture of several oxides. Usually, the most abundant is SiO2, which is chosen here as the representative formula.

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TABLE 8.3 Numbers for the Components of Sulfur Absorbents and Inert Phase Name Calcium Calcium carbonate Calcium oxide Calcium sulite Calcium sulfate Moisture Silicon oxide Magnesium Magnesium carbonate Magnesium oxide Magnesium sulite Magnesium sulfate

Molecular Formula

Number

Ca CaCO3 CaO CaS CaSO4 H2O SiO2 Mg MgCO3 MgO MgS MgSO4

515 629 633 622 645 700 824 543 733 736 737 738

This set has of reactions been successfully used in previous works [1–11].* Although solid fuels contain several other components, they are represented here by carbon, hydrogen, oxygen, nitrogen, sulfur, and ash because those are the most important species. All other components of char (such as metal and alkaline oxides) are included in the ash fraction. Reactions R.1 to R.6 involve char, and Chapter 10 shows how its composition (aj, j = 514, 531, 551, 546, and 563) can be obtained after drying and devolatilization of the original fuel. The products from Reaction R.1 and the others differ because of the following basic rule: Oxidant reactions should produce oxidized components, whereas reducing reactions preferably lead to reduced molecules. On the other hand, other alternatives may be proposed. For instance, in Reaction R.3, nitrogen from solid fuel could lead to HCN instead NH3. In this system, the production of HCN has been left to the pyrolysis, or Reaction R.8. It is important to stress that the forms presented here are just a proposal, but one that is the most commonly used in the literature. The present system has been favorably compared with real operations [1–3, 6, 9–14], as shown in Chapters 12, 16, and 18.

8.4.1

NITROGEN OXIDES

Despite the small inluence of nitrogen-related reactions in the overall mass and energy balances of a combustor or gasiier, their products are important because of aspects of pollutant generation. The number of publications on kinetics related to nitrogen oxides relects that importance. *

In those previous works, the form for reaction R.1 is a bit different and includes the distribution between CO and CO2. However, the discussion shown in Chapter 2, Section 2.5, and the results of simulations to be shown in Chapters 12 and 16 show that the results from both forms are similar.

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Chemical Reactions

TABLE 8.4 List of Chemical Reactions Considered by the General Model (Part 1) Reaction Number R.1

Representation a a 1 a  CH a 531 Oa 551 N a 546 Sa 563 +  + 531 − 551 + 546 + a 563  O 2 → CO 2  4 2 2 +

R.2

 a  CH a531 Oa551 N a546 Sa563 + (1 − a 551 )H 2 O ↔  1 + 531 − a 551 − a 563  H 2 + CO 2   +

R.3

a 531 H 2 O + a 546 NO + a 563 SO 2 2

a 546 N 2 + a 563H 2S 2

CH a 531 Oa 551 N a 546 Sa 563 + CO 2 ↔ 2CO + a 551H 2 O  a 3a +  531 − a 551 − 546 − a 563  H 2 + a 546 NH 3 + a 563 H 2S 2   2

R.4

  a 3 CH a 531 Oa 551 N a 546 Sa 563 +  2 − 531 + a 551 + a 546 + a 563  H 2 ↔ CH 4 2 2   + a 551H 2 O + a 546 NH 3 + a 563 H 2S

R.5

a  CH a531 Oa551 N a546 Sa563 + (2 − a 551 )NO ↔  531 − a 563  H 2 + CO2  2  a   a +  1 + 546 − 551  N 2 + a 563H 2S 2 2  

R.6

a  CH a531 Oa551 N a546 Sa563 + (1 − a 551 )N 2 O ↔  531 − a 563  H 2 + CO  2   a  +  1 + 546 − a 551  N 2 + a 563H 2S 2  

R.7

Fuel daf → Volatile + Char1

R.8

Volatile → Gases + Tar

R.9

Tar → Char2

The nitrogen oxides are NO, NO2, and N2O; however, during combustion and gasiication, the concentration of NO is usually much higher than the other two oxides. The generation and consumption of that oxide is represented in Reactions R.1, R.5, R.46, R.48, and R.49. In addition, Reactions R.51 through R.53 can be included in a set forming a model of a combustor or gasiier [15]. This should be seen as added sophistication, because the set without those has given reasonable results regarding NO generation [1, 2]. In addition, it has been shown [15, 16] that substantially higher concentrations of NO than other nitrogen oxides are found in gases from

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Solid Fuels Combustion and Gasification

TABLE 8.5 List of Chemical Reactions Considered by the General Model (Part 2) Reaction Number

Representation

R.10

Wet − Carbonaceous − Solid ↔ Dry − Carbonaceous − Solid + H 2 O

R.21

CaCO3 ⇔ CaO + CO2

R.22

2CaO + 2SO2 + O2 ⇔ 2CaSO 4

R.23

Wet − Limestone ↔ Dry − Limestone + H 2 O

R.24

CaO + H 2S ⇔ CaS + H 2 O

R.25

MgCO3 ⇔ MgO + CO2

R.26

2MgO + 2SO2 + O2 ⇔ 2MgSO4

R.27

Wet − Dolomite ↔ Dry − Dolomite + H 2 O

R.28

MgO + H 2S ⇔ MgS + H 2 O

R.31

Wet − Inert − Solid ↔ Dry – Inert – Solid + H 2 O

R.41

CO + H 2 O ⇔ CO2 + H 2

R.42

2CO + O2 ⇔ 2CO2

R.43

2H 2 + O 2 ⇔ 2H 2 O

R.44

CH 4 + 2O2 ⇔ CO2 + 2H 2 O

R.45

C2 H 6 + 7O2 ⇔ 4CO2 + 6H 2 O

R.46

4NH 3 + 5O2 ⇔ 4NO + 6H 2 O

R.47

2H 2S + 3O2 ⇔ 2SO2 + 2H 2 O

R.48

N 2 + O2 ⇔ 2NO

R.49

Tar + O2 → Combustion Gases

R.50

Tar → Gases

R.51

NO + CO ⇔ 1/2 N 2 + CO 2

R.52

CO + N 2 O ⇔ N 2 + CO2

R.53

N 2 O ⇔ N 2 + 1/2O 2

R.54

Tar + H 2 → Gases

combustion. Besides, most industrial and even pilot units measure only NO concentration [17–19]. Those interested in deeper aspects of nitrogen oxide generation will ind plenty of material in the literature [15, 20–30].

8.5

STOICHIOMETRY

In most cases, mainly for homogeneous reactions, the determination of stoichiometry is a straightforward task. For example, the coeficients for Reaction R.41 would be written as ν41,44 = –1, ν41,20 = –1, ν41,46 = 1, and ν41,19 = 1. The other coeficients are equal to 0.

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Chemical Reactions

Heterogeneous reactions require the composition of the solid for writing the stoichiometry (Table 8.5). A simpliication would be to assume the values of aj (for the carbonaceous solid) as constants, i.e., equal to those found for the feeding carbonaceous solid. This is a strong assumption because the apparent composition of the solid changes during the various processes occurring in the bed. Among these processes is pyrolysis. The composition of products released during that process is not equal to that of the entering or remaining solid matrix. Hence, the composition of char is usually very different from the feeding carbonaceous. In the particular case of moving beds, the present model allows a simpler estimation of carbonaceous solid composition at each position in the equipment. As seen in Chapter 7, the mass balance provides the mass low (Fj) of each component of the solid phase. Consequently, it is possible to write  Fj   M  j , 501 ≤ j ≤ 1000 a j = 1000 Fk ∑ k = 501 M k where aj =

8.6

aj . a514

(8.2)

(8.3)

KINETICS

Equation 5.15 (Chapter 5) generally describes the rate of a reaction, where kinetic k coeficients are given by the Arrhenius relationship (Equation 5.16). Various reactions follow that classical form, but many others are represented by complicated forms. Actually, reactions are combinations of several more fundamental ones, which occur in a connected series of steps known as chain-reaction mechanisms. For instance, among other possibilities, the combustion reaction of hydrogen (R.43, Table 8.5) can be represented [31] by: k

a1 H 2 + O 2  → 2OH (initiation)

k

a2 OH + H 2   → H 2 O + H (propagation)

(8.4) (8.5)

k

(8.6)

k

(8.7)

a3 H + O 2  → OH + O (branching)

a4 O + H 2   → OH + H (branching)

k

a5 H + O 2 + M  → HO 2 + M (breaking)

(8.8)

1 ka6 → H 2 + M (breaking). H + M  2

(8.9)

Here, M is a third particle or even the reactor wall.

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Solid Fuels Combustion and Gasification

The overall rate of a chain process may or may not be represented by typical Arrhenius forms. To show that, let us assume that each step of the above chain follows a classical form, as given by Equation 5.15. This is very reasonable, since the steps of a chain reaction are elementary ones. Therefore, it is possible to write the production rates of various ions involved as follows: ∂ρ OH = 2 k a1ρ H2 ρ O2 − k a 2 ρ H2 ρ OH + k a 3ρ H ρ O2 + k a 4 ρ O ρ H2 ∂t

(8.10)

∂ρ H = k a2 ρ H2 ρ OH − k a3ρ H ρ O2 + k a4 ρ O ρ H2 − k a5ρ H ρ O2 ρ M − k a 6 ρ H ρ M ∂t

(8.11)

∂ρ O = k a3ρ H ρ O2 − k a4 ρ O ρ H2 ∂t

(8.12)

∂ρ HO2 = k a 5ρ H ρ O2 ρ M . ∂t

(8.13)

The principle of a stationary state can be applied, which assumes a stationary concentration of ions within a certain interval of time, or ∂ρ OH ∂ρ H ∂ρ O ∂ρ HO2 = = = = 0. ∂t ∂t ∂t ∂t

(8.14)

Equation 8.14, combined with Equations 8.10 through 8.13, leads to a system with four algebraic relations that, when solved, can provide the concentrations of the four ions as functions of the concentrations of stable molecules (H2, O2, H2O, M). These can now be inserted into the following equation: ∂ρ H2 = − k a1ρ H2 ρ O2 − k a2 ρ H2 ρ OH − k a4 ρ O ρ H 2 + k a6 ρ H ρ M . ∂t

(8.15)

Once the kinetics coeficients of each individual reaction (Equations 8.10 through 8.13) is known, Equation 8.15 would provide the kinetics of H2 consumption. Of course, most of the time, this would not lead to a classical formula like that shown by Equation 5.15. Selected collections of chain mechanisms related to various combustion and gasiication reactions can be found in the literature [20–21]. Another and more frequently used approach is the empirical determination of kinetics. As an example, the overall representation for kinetics of hydrogen combustion is given by [32] r43 = k 43T −1.5ρ 1H.52 ρ O2,

(8.16)

which also does not follow the form given by Equation 5.15. All methods introduce deviations, and errors around 30% are not uncommon when computed rates are compared against experimental values. The selection of the representation of reaction kinetics to be included in a model depends on the type of process. For instance, combustion of gases and pulverized

175

Chemical Reactions

solids in lames (or suspensions) involves very fast processes, and the overall representation of kinetics does not work well because these phenomena involve high temperatures combined with very small time scales. In such conditions, the concentrations cannot be equated to 0; therefore, the system of chain reactions is the preferred one. On the other hand, moving and luidized beds do not involve too high temperatures and phenomena are much slower, leading to good representations by overall kinetics. Of course, the additional dificulty imposed by solution of a differential system of equations intrinsic to the chain representations should, if possible, be avoided. That is why almost all modelers of moving and luidized beds prefer overall representations of reactions.

8.6.1

HOMOGENEOUS REACTIONS

The methods used to compute the homogeneous reactions, usually involved in combustion and gasiication, are described below, whereas methods for heterogeneous ones are left to the next chapter. As an example, the rate of shift reaction rate is given by  ρ ρ r41 = k 41  ρ COρ H2 O − CO2 H2 K 41 

 . 

(8.17)

The necessary information for the computation of gas–gas reaction rates, along with some for the gas–solid ones, is given in Tables 8.6 through 8.9. Table 8.9, in particular, describes the formula used for each gas–gas reaction involved in the present approach. 8.6.1.1 Equilibrium Chapter 5 illustrates how the equilibrium constant can be computed for a given reaction. Correlations that allow the calculation of equilibrium parameters for a few reactions can be found in the literature [33–35]. However, most of them are too limited in the range of temperatures to which they are applicable. My advice is for the user to develop routines for those calculations. It does not take too much time and effort, and it is essential for the development of simulation programs in many areas of engineering.

8.7

FINAL NOTES ON KINETICS

This section presents a brief discussion of details regarding choices of reactions and kinetics. Most works in the area of combustion uses the following form for Reaction R.1: C* + βO2 → (2β − 1)CO2 + (2 − 2β)CO

(8.18)

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Solid Fuels Combustion and Gasification

TABLE 8.6 Kinetic Coefficients for the Chemical Reactions (Part 1) k0,i (s–1)a

Reaction

Te,i (K)

Reference

 17.67ρ 514 ,I RT G −3   6.05 × 10 ρ514 ,I RT G −3   1.9 × 10 ρ514 ,I RTG

13,600

42

21,150

43

–7,220

R.3 R.3

1.33 × 10 −8 3.13 × 10 −8  1.24 × 10 3 ρ 514 ,I RT G

R.3(2)b,c R.3(3)b,c R.4(1)d

1.24 × 10 −13 31.2  ) 2.345 × 10 −11 (p19ρ 514 ,I RT G

–22,900 15,100 13,670

R.4(2)d

2.85 × 10 −10

–11,100

R.5

5.24 × 10 5

R.1 R.2 R.3(1)b R.3(2)b b (3) b,c (1)

8.6 × 1014 i   E In general, k i k 0 i exp  − .   RT 

R.7 a

6 d p,m =1

16,840 44

–5,050 29,600 45

46

17,130

47

27,700

48

b

For Reaction R.3, k 3 = k 3(1) ⁄ (1 + k 3( 2) p46 + k 3(3) p19 ) , note that unit for k3(2) and k3(3) is Pa.

c

For charcoal or wood.

d

For Reaction R.4, k 4 = k 4 (1) p19 ⁄ (1 + k 4 ( 2) p19 ) , note that unit for k4(2) is Pa–1.

where C* represents char (with all its components), and the distribution coeficient β is given [36, 37] by 2 + β′ β= (8.19) 2 + 2β′ where  6240  . β′ = 2500 exp  −  Tm =1 

(8.20)

From those equations, one may verify that coeficient β approaches 0.5 for combustions at high temperatures (usually found in furnaces). Therefore, the main product is carbon monoxide and not carbon dioxide. Nonetheless, that does not account for the inluence of pressure. A more precise alternative for the stoichiometry of Reactions R.1 and R.49 is given by considering that only CO, and not a mixture of CO and CO2, is produced. This is possible when the following two conditions are satisied: 1. The oxidation of CO (given by Reaction R.42) is also included in the same system or model. 2. The equilibrium of those reactions is computed and included for the correct calculation of their rates.

177

Chemical Reactions

TABLE 8.7 Kinetic Coefficients for the Chemical Reactions (Part 2) k0,i (s–1)a

Reaction R.8 R.21 R.22b R.22c R.24

Te,i (K)

Assume the same as R.7 6.507 × 109 4.90 × 103 (–3.843 Tm=2 +5640)ι 4.90 × 103 (3.590 Tm=2 –3670)ι 6 1.023 × 10 −3 d p ,m = 2

Reference

Same as R.7 20,410 8,810 8,810

Same as R.7 49, 50 51 51

3,420

52

R.50d

3.57 × 1011

24,540

53

R.50e

4.28 × 10 6

12,930

54

R.50f

9.55 × 10 4

11,220

55

R.50

4.22 × 10

13,240

56

R.50h

1.9 × 10 6

11,980

57

R.50i

4.138 × 10 3

10,187

58

R.50j

4.0 × 10 4

9,210

59

g

a b c d e f g h i j

6

 T  In general: k i = k 0,i exp  − e,i   T  For Tm=2 < 1,253 K. For Tm=2 > 1,253 K. Cellulose tar between 773 and 1,023K. Wood tar between 723 and 873K. Coal tar between 873 and 1,073 K. Almond shell tar between 978 and 1,123 K. Municipal solid wastes between 973 and 1,123 K. Kraft lignin tar between 673 and 1,033 K. Wood tar from 720 and 1,260 K.

TABLE 8.8 Kinetic Coefficients for the Chemical Reactions (Part 3) Reaction R.41 R.42 R.43 R.44 and R.45 R.46 and R.47 R.47 R.48 R.49 R.51 R.52 R.53

k0,i

Units

Te,i (K)

Reference

2.78 × 103 1.3 × 1017 5.159 × 1013 3.552 × 1014 9.79 × 1011 8.26 × 1012 1.815 × 1013 59.8 5.67 × 103 2.51 × 1011 1.75 × 108

kmol–1 m3 s–1 kmol–0.75 m2.25 s–1 kmol–1.5 m4.5 K1.5 s–1 kmol–1 m3 K s–1 kmol–0.9 m2.7 s–1 kmol–1.158 m3.474 s–1 kmol–0.5 m1.5 s–1 kmol–0.5 m1.5 K–1 Pa–0.3 s–1 K–1 s–1 kmol–1 m3 s–1 s–1

1,510 34,740 3,430 15,700 19,655 18,956 67,338 12,200 13,952 23,180 23,800

60 41 32 32 61 62 63 64 15 15 15

178

Solid Fuels Combustion and Gasification

TABLE 8.9 Values to be Used for Computations of Gas–Gas Reaction Rates Reaction (i)a R.42 R.43 R.44 R.45b R.46 R.47 R.48 R.49 R.51 R.52 R.53 a b

a1

a2

a3

a4

a5

a6

a7

a8

Reference

0 –1.5 –1 –1 0 0 0 1 1 0 0

CO H2 CH4 C2H6 NH3 H2S N2 Tar NO N2O N2O

1 1.5 1 1 0.86 1.074 1 0.5 1 1 1

O2 O2 O2 O2 O2 O2 O2 O2

0.25 1 1 1 1.04 1.084 0.5 1 0 1 0

H2O

0.5

0 0 0 0 0 0 0 0.3 0 0 0

41 32 32 32 61 62 63 64 15 15 15

CO

—

0 0 0

Form to be used: ri = k i Ta1 ρ aa 32 ρ aa 54 ρ aa 67 P a8 . Assumed as following the form of the preceding reaction.

The oxidation of carbon monoxide has been extensively studied [32, 38–41].* The same is true for many others reactions, and this amount of recurring information presents a problem to the modeler who must make a choice among various possibilities. The following suggestions may help to orient the modeler in this decision: • Find the range of temperature, pressure, and concentrations of reactants for which the kinetics is to be determined. Of course, the range should be compatible with the model objective. • Study the methodology of experimental determinations. • Verify the coherence of the proposed kinetics parameters. A good procedure is to compare the various rates in one or more given situations. The rates should vary within a range. It is uncommon to ind deviations around 50%. It is also reasonable to choose one that does not provide values too far from the average. Table 8.10 illustrates an example. For the sake of the example, the rates have been computed at 1300 K and molar fractions as indicated in the table. Those are possible situations during combustion. As shown, the deviation between the highest and lowest values reach, in this particular case, 60%. The values provided by Yetter et al. [41] are the closest to the average. • Try to choose the kinetics found in the most recent work. Any good researcher carefully studies previous works in his or her area and tries to improve, not only regarding the technique but also in order to expand the range of applicability of the kinetics. However, such improvements are not guaranteed. *

To mention just a few studies.

179

Chemical Reactions

TABLE 8.10 Various Formulations for the Kinetics of CO Combustion k0,42 1.0 × 1015 2.2 × 1012 1.3 × 1011 4.8 × 108 1.3 × 1017 a

Te (K)

a1

a2

a3

a4

16,000 20,140 15,110 8,060 34,740

–1.5 0 0 0 0

1 1 1 1 1

0.25 0.25 0.5 0.3 0.25

0.5 0.5 0.5 0.5 0.5

r42 (kmol m–3 s–1)a Reference 2.395 × 10–2 1.022 × 10–1 5.605 × 10–2 1.744 × 10–1 8.011 × 10–2

32 38 39 40 41

a a2  a3  a4  ρO2 ρH2 O . Value computed at 1,300 K; 101.325 Form to be used: r42 = k 42 T 1 ρ CO kPa; and for xCO = 0.01, xO2 = 0.15, and xH2O = 0.02.

For all these reasons, the formulation by Yetter et al. [41] has been selected here. In any case, doubts may arise, and another alternative is to try several possibilities and compare the model simulation results against experimental ones.

8.8

INDEPENDENT SET OF REACTIONS

It is important to note that models based on equilibrium assumption may lead to false conclusions if the reactions representing the process are not independent. In other words, if one or more reactions could be written as combinations of at least two others, one may be computing recurrent information.

8.8.1

MATHEMATICAL CRITERION

There is a simple method to determine whether a system is composed of independent reactions. This can be illustrated with an example. Take the following set of reactions: R.41, R.42, and R.43. It is easy to see that each one of these reactions can be written as a combination of the other two. Mathematically, it can be shown by writing the matrix of the coeficients, as in the following table. If we take any 3 × 3 square matrix, the determinant is 0; that is, the system is linearly dependent. On the other hand, any two of the above reactions are independent.

TABLE 8.11 Matrix Representing the Coefficients of a Reaction Set H2

H 2O

CO

O2

CO2

1 0 –2

–1 0 2

–1 –2 0

0 –1 –1

1 2 0

180

Solid Fuels Combustion and Gasification

This can be veriied by calculating the determinants of all 2 × 2 matrices. The system is independent if at least one is not equal to 0. Despite the simplicity of the example, the above criterion can be very convenient when it is necessary to analyze complex systems where equilibrium is assumed. Nevertheless, this criterion is not mandatory for models where the kinetics of competing reactions is considered, simply because equilibrium is not imposed on those more realistic models.

8.9 8.9.1

EXERCISES PROBLEM 8.1* The following molar concentrations have been found at a point inside a gasiier: • Oxygen = 10% • Hydrogen = 1% • Water = 50% If the temperature is 1000 K, compute the direct rate of Reaction R.43. Use Tables 8.8 and 8.9.

8.9.2

PROBLEM 8.2** Write a general routine to compute gas–gas reaction rates, once the following inputs are set: composition of the gas mixture (on either a molar or a mass basis), temperature, and pressure of the mixture. Use the routines developed for Problems 5.4 and 5.5.

REFERENCES 1. de Souza-Santos, M.L., Modelling and Simulation of Fluidized-Bed Boilers and Gasiiers for Carbonaceous Solids, PhD thesis, University of Shefield, Shefield, United Kingdom, 1987. 2. de Souza-Santos, M.L., Comprehensive modelling and simulation of luidized-bed boilers and gasiiers, Fuel, 68, 1507–1521, 1989. 3. de Souza-Santos, M.L., Application of comprehensive simulation to pressurized bed hydroretorting of shale, Fuel, 73, 1459–1465, 1994. 4. de Souza-Santos, M.L., A study on pressurized luidized-bed gasiication of biomass through the use of comprehensive simulation, in Proc. Fourth International Conference on Technologies and Combustion for a Clean Environment, Lisbon, Portugal, July 7–10, 1997, paper 25.2, Vol. II, pp. 7–13. 5. de Souza-Santos, M.L., A study on pressurized luidized-bed gasiication of biomass through the use of comprehensive simulation, in Combustion Technologies for a Clean Environment, Gordon and Breach, Amsterdam, Netherlands, 1998. 6. de Souza-Santos, M.L., Application of comprehensive simulation of luidized-bed reactors to the pressurized gasiication of biomass, Journal of the Brazilian Society of Mechanical Sciences, 16(4), 376–383, 1994. 7. de Souza-Santos, M.L., Search for favorable conditions of atmospheric luidized-bed gasiication of sugar-cane bagasse through comprehensive simulation, in Proc. 14th Brazilian Congress of Mechanical Engineering, Bauru, São Paulo, Brazil, December 8–12, 1997.

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8. de Souza-Santos, M.L., Search for favorable conditions of pressurized luidized-bed gasiication of sugar-cane bagasse through comprehensive simulation, in Proc. 14th Brazilian Congress of Mechanical Engineering, Bauru, São Paulo, Brazil, December 8–12, 1997. 9. de Souza-Santos, M.L., Development of a Simulation Model and Optimization of Gasiiers for Various Fuels (Desenvolvimento de Modelo de Simulação e Otimização de Gaseiicadores com Diversos Tipos de Combustíveis), IPT-Inst. Pesq. Tec. Est. São Paulo, SCTDE-SP, Report No. 20.689, DEM/AET, São Paulo, SP, Brazil, 1985. 10. de Souza-Santos, M.L., and Jen, L.C., Study of Energy Alternative Sources; Use of Biomass and Crop Residues as Energy Source: Part B: Development of a Mathematical Models and Simulation Programs for Up-Stream and Down-Stream Moving Bed Gasiiers. (Estudo de Fontes Alternativa de Energia, Parte B). IPT-Institute for Technological Research of São Paulo, Report No. 16.223-B/DEM/AET, São Paulo, Brazil, 1982. 11. Jen, L.C., and de Souza-Santos, M.L., Modeling and simulation of ixed-bed gasiiers for charcoal (Modelagem e simulação de gaseiicador de leito ixo para carvão), Brazilian Journal of Chemical Engineering (Revista Brasileira de Engenharia Química), 7(3–4), 18–23, 1984. 12. Rabi, J.A., and de Souza-Santos, M.L., Incorporation of a 2-lux model for radiative heat transfer in a comprehensive luidized bed simulator. Part I: Preliminary model equations, in Proc. ENCIT-2002, 9th Brazilian Congress on Engineering and Thermal Sciences, Caxambu, Minas Gerais, Brazil, October 15–18, 2002. 13. Rabi, J.A., and de Souza-Santos, M.L., Incorporation of a 2-lux model for radiative heat transfer in a comprehensive luidized bed simulator. Part II: Numerical results and assessment, ENCIT-2002, 9th Brazilian Congress on Engineering and Thermal Sciences, Caxambu, Minas Gerais, Brazil, October 15–18, 2002. 14. Rabi, J.A., Usage of Flux Method to Improve Radiative Heat Transfer Modelling inside Bubbling Fluidized Bed Boilers and Gasiiers, PhD thesis, Faculty of Mechanical Engineering, State University of Campinas, Campinas, Sao Paulo, Brazil, 2002. 15. Chen, Z., Lin, M., Ignowski, J., Kelly, B., Linjewile, T.M., and Agarwal, P.K., Mathematical modeling of luidized bed combustion; 4: N2O and NOx emissions from the combustion of char, Fuel, 80, 1259–1272, 2001. 16. Desroches-Ducarne, E., Dolignier, J.C., Marty, E., Martin, G., and Delfosse, L., Modelling of gaseous pollutants emissions in circulating luidized bed combustion of municipal refuse, Fuel, 77(13), 1399–1410, 1998. 17. Babcock and Wilcox, SO2 Absorption in Fluidized Bed Combustor of Coal—Effect of Limestone Particle Size. Report EPRI FP-667, Project 719-1, 1978. 18. NCB (IEA Grimethorpe) Ltd., Fluidized Bed Combustion Project, Grimethorpe Experimental Facility, Report GEF/84/12, Test Series 2.2, Vol. 1, Main Report, 1985. 19. NCB (IEA Grimethorpe) Ltd., Fluidized Bed Combustion Project, Grimethorpe Experimental Facility, Report GEF/84/12, Test Series 2.2, Vol. 2, Appendices, 1985. 20. Glassman, I., Combustion, 3rd Ed., Academic Press, San Diego, CA, 1996. 21. Warnatz, J., Maas, U., and Dibble, R.W., Combustion, Springer, Berlin, Germany, 1999. 22. Kuo, K.K., Principles of Combustion, John Wiley, New York, 1986. 23. Fenimore, C.P., Formation of nitric oxide in premixed hydrocarbon lames, in Proc. of International Symposium on Combustion, 13, 1970, Utah, Combustion Institute, Pittsburgh, PA, 1971, pp. 373–380. 24. Glarborg, P., Miller, J.A., and Kee, R.J., Kinetic modeling and sensitivity analysis of nitrogen oxide formation in well-stirred reactors, Combustion and Flame, 65, 177–202, 1986. 25. Miller, J.A., and Bowman, C.T., Mechanism and modeling of nitrogen chemistry in combustion, Progress in Energy and Combustion Science, 15(4), 287–338, 1989.

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26. Meunier, P., Costa, M., and Carvalho, M.G., The formation and destruction of NO in turbulent propane diffusion lames, Fuel, 77(15), 1705–1714, 1998. 27. Westenberg, A.A., Kinetics of NO and CO in lean, premixed hydrocarbon-air lames, Combustion Science and Technology, 4(2), 59–64, 1971. 28. Wood, S.C., Select the right NOx control technology, Chemical Engineering Progress, 90(1), 32–38, 1994. 29. Desroches-Ducarne, E., Dolignier, J.C., Marty, E., Martin, G., and Delfosse, L., Modelling of gaseous pollutants emissions in circulating luidized bed combustion of municipal refuse, Fuel, 77(13), 1399–1410, 1998. 30. Liu, H., and Gibbs, B.M., Modelling of NO and N2O emissions from biomass-ired circulating luidized bed combustors, Fuel, 81, 271–280, 2002. 31. Pachenkov, G.M., and Lebedev, V.P., Chemical Kinetics and Catalysis, Mir, Moscow, 1976. 32. Vilienskii, T.V., and Hezmalian, D.M., Dynamics of the combustion of pulverized fuel, Energia (Moscow), 11, 246–251, 1978. 33. Hottel, H.C., and Howard, J.B., New Energy Technology, MIT Press, Cambridge, MA, 1971. 34. Parent, J.D., and Katz, S., Equilibrium Compositions and Enthalpy Changes for the Reaction of Carbon, Oxygen, and Steam, IGT-Inst. Gas Tech. Research Bulletin 2, 1948. 35. Kanury, A.M., Introduction to Combustion Phenomena, Gordon and Breach Science, London, 1975. 36. Arthur, J.A., Reaction between carbon and oxygen, Trans. Faraday Soc., 47, 164–178, 1951. 37. Rossberg, M., Experimentelle Ergebnisse über die Primärreaktionen bei der Kohlenstoffverbrennung Z. Elektrochem., 60, 952–956, 1956. 38. Dryer, F.L., and Glassman, I., High-temperature oxidation of CO and CH4, in Proc. 14th Symposium (International) on Combustion, Combustion Institute, 1973, pp. 978–1003. 39. Howard, J.B., Williams, G.C., and Fine, D.H., Kinetics of Carbon Monoxide Oxidation in Post Flame Gases, Proc. 14th Symposium (International) on Combustion, Combustion Institute, 1973, pp. 975–986. 40. Hottel, H.C., Williams, G.C., Nerheim, N.M., and Schneider, G.R., Burning Rate of Carbon Monoxide, Proc. 10th Symposium (International) on Combustion, Combustion Institute, 1965, pp. 111–121. 41. Yetter, R.A., Dryer, F.L., and Rabitz, H., Complications of one-step kinetics for moist CO oxidation, Proc. 21st Symposium (International) on Combustion, Combustion Institute, 1986, pp. 749–760. 42. Sergeant, G.D., and Smith, I.W., Combustion rate of bituminous coal char in the temperature range 800 to 1700 K, Fuel, 52, 52-77, 1973. 43. Gibson, M.A., and Euker, C.A., Mathematical Modelling of Fluidized Bed Coal Gasiication, presented at the AIChE meeting, Los Angeles, CA, 1975. 44. Adánez, J., Miranda, J.L., and Gavilán, J.M., Kinetics of a lignite-char gasiication by CO2, Fuel, 64, 801–804, 1985. 45. Fredersdorff, C.G., and Elliott, M.A., Coal gasiication, in Chemistry of Coal Utilization, Supplementary Volume, Lowry, H.H., Ed., John Wiley, New York, 1963. 46. Johnson, J.L., Kinetics of Coal Gasiication, John Wiley, New York, 1979. 47. Oguma, A., Yamada, N., Furusawa, T., and Kunii, D., Preprint for the 11th Fall Meeting of the Soc. of Chem. Eng. Japan, 1977, p. 121. 48. Solomon, P.R., Hamblen, D.G., Carangelo, R.M., Serio, M.A., and Deshpande, G.V., General model of coal devolatilization, Energy and Fuels, 2, 405–422, 1988. 49. Asaki, Z. Fukunaka, Y., Nagase, T., and Kondo, Y., Thermal decomposition of limestone in a luidized bed, Metallurgical Transactions, 5, 381–390, 1974.

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50. Borgwardt, R.H., Calcination kinetics and surface area of dispersed limestone particles, AIChE J., 31 (1), 103–111, 1985. 51. Rajan, R.R., and Wen, C.Y., A comprehensive model for luidized bed coal combustors, AIChE J., 26(4), 642-655, 1980. 52. Efthimiadis, E.A., and Sotirchos, S.V., Sulidation of limestone-derived calcines, Ind. Chem. Eng. Res., 31(10), 10, 2311–2321, 1992. 53. Antal, M.J., Jr., Effects of reactor severity on the gas-phase pyrolysis of celulose and kraft lignin-derived volatile matter, Ind. Eng. Chem. Prod. Res. Dev., 22(2), 366–375, 1983. 54. Liden, A.G., Berruti, F., and Scott, D.S., A kinetic model for the production of liquids from the lash pyrolysis of biomass, Chem. Eng. Commun., 65, 207–221, 1988. 55. Boroson, M.L., Howard, J.B., Longwell, J.P., and Peters, W.A., Product Yields and Kinetics from Vapor Phase Cracking of Wood Pyrolysis Tars, AIChE J., 35(1), 120–128, 1989. 56. Font, R., Marcilla, A., Devesa, J., and Verdu, E., Kinetic study of the lash pyrolysis of almond shells in a luidized bed reactor at high temperatures, Journal of Analytical and Applied Pyrolysis, 27(2), 245–273, 1993. 57. Garcia, A.N., Estudio Termoquimico y Cinetico de la Pirolisis de Residuos Solidos Urbanos, PhD thesis, University of Alicante, Alicante, Spain, 1993. 58. Caballero, J.A., Font, R., and Marcilla, A., Kinetic study of the secondary thermal decomposition of kraft lignin, Journal of Analytical and Applied Pyrolysis, 38, 131–152, 1996. 59. Morf, P., Hasler, P., and Nussbaumer, T., Mechanisms and kinetics of homogeneous secondary reactions of tar from continuous pyrolysis of wood chips, Fuel, 81, 843–853, 2002. 60. Biba, V., Mark, J., Klose, E., and Malecha, J., Mathematical Model for the Gasiication of Coal Under Pressure, Ind. Eng. Chem. Process Des. Dev., 17(1), 92–98, 1978. 61. Branch, M.C., and Sawyer, R.F., Ammonia oxidation kinetics in an arc heated low reactor, Proc. 14th Symposium (International) on Combustion, Pittsburgh, PA, 1973, pp. 967–974. 62. Leveson, P.D., Kinetic Studies to Determine the Rates of Ammonia and Hydrogen Sulphide Destruction Under Claus Plant Operating Conditions, PhD thesis, University of Shefield, Department of Chemical Engineering And Fuel Technology, Shefield, United Kingdom, 1997. 63. Quan, V., Marble, F.E., and Klingel, J.R., Proc. 14th Symposium (International) on Combustion, Pittsburgh, PA, August 20–25, 197, pp. 851–60. 64. Siminski, V.J., Wright, F.J., Edelman, R.B., Economos, C., and Fortune, O.F., Research on Methods of Improving the Combustion Characteristics of Liquid Hydrocarbon Fuels, AFAPLTR 72–74, Vol. I and II, Air Force Aeropropulsion Laboratory, Wright-Patterson Air Force Base, OH, 1972.

9 Heterogeneous Reactions CONTENTS 9.1 9.2

Introduction .................................................................................................. 185 General Form of the Problem ....................................................................... 188 9.2.1 Unexposed-Core Model .................................................................... 188 9.2.1.1 Effectiveness Coeficient ................................................... 193 9.2.2 Exposed-Core Model ........................................................................ 194 9.3 Generalized Treatment ................................................................................. 195 9.4 Other Heterogeneous Reactions ................................................................... 197 9.5 Exercises ....................................................................................................... 198 9.5.1 Problem 9.1 ....................................................................................... 198 9.5.2 Problem 9.2 ....................................................................................... 198 9.5.3 Problem 9.3 ....................................................................................... 198 9.5.4 Problem 9.4 ....................................................................................... 198 9.5.5 Problem 9.5 ....................................................................................... 199 9.5.6 Problem 9.6 ....................................................................................... 199 9.5.7 Problem 9.7 ....................................................................................... 199 9.5.8 Problem 9.8 ....................................................................................... 199 9.5.9 Problem 9.9 ....................................................................................... 199 9.5.10 Problem 9.10 ..................................................................................... 199 9.5.11 Problem 9.11...................................................................................... 199 9.5.12 Problem 9.12 .....................................................................................200 9.5.13 Problem 9.13 .....................................................................................200 9.5.14 Problem 9.14......................................................................................200 References ..............................................................................................................200

9.1

INTRODUCTION

The present chapter shows mathematical treatments to allow computations of heterogeneous chemical reaction rates. It is important to stress that these are just proposals because, as always, any model assumes approximations that, depending on the situation or application, might be acceptable or not. Therefore, it is advisable to make a critical review of the fundamental hypothesis before any application. However, the models presented here have been successfully applied to a wide range of situations [1–14], particularly to combustion and gasiication processes. As mentioned in Chapter 8, gas–gas reaction rates may be signiicantly inluenced by the mass transfer process. This transfer is necessary to bring the molecules of 185

186

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gases close enough for them to react. The mass transfer resistances are much more pronounced in cases of heterogeneous reactions because at least two distinct phases are involved. For the sake of an example, let us imagine that the gaseous species is oxygen and the active solid one is carbon. In order to reach the carbon within the solid, oxygen molecules should travel though the gas involving the particle and possibly an ash layer of spent material around the unreacted solid. Depending on the reaction, its rate might be controlled or limited by the pace of that mass transfer. As it is shown below, the resulting rate of reactant consumption will depend on the barrier imposed by several resistances. Let us start with two classical cases, which are limiting situations for heterogeneous gas–solid reactions and are described below: • The unexposed-core model or shrinking-core model. According to this model (Figure 9.1a), the core—where the reactions take place—is surrounded by a shell of spent inert material. Therefore, the core surface is not directly exposed to gases around the particle. Any reacting gas has to diffuse through the gas layer surrounding the particle and the inert shell in order to reach the core, through which it would diffuse while reacting. • The exposed-core model or ash-segregated model. According to this model (Figure 9.1b), as soon as the spent material forms at the particle surface, it detaches and disintegrates into very small particles. Hence, the core is always exposed to the surrounding gas. Reactions would be limited only by the resistances imposed by the surrounding gas layer and diffusion through the core. Take, for instance, the process of updraft moving-bed gasiication (Chapter 3, Figure 3.3), where each particle is in contact with its neighbors. As they move downward, attrition between particles imposes a stress on the outer shell. Ash or reacted material

Gas boundary layer

rA r rN

rN Core Core

(a)

FIGURE 9.1 reactions.

(b)

Unexposed-core (a) and exposed-core (b) models for the gas–solid particle

Heterogeneous Reactions

187

Moreover, as they approach the combustion region, a fast increase in temperature occurs, which is illustrated in Figure 3.6. Differential dilatation between the ash layer and nucleus material adds to the stress. Depending on the mechanical resistance of the ash formed, it might detach from the nucleus. It often occurs in luidized beds as well, due to intense attrition among particles with the thermal shocks. The deductions here can be applied to Reactions R.1 through R.6 (Table 8.4). Comments on possible adaptations for pyrolysis, drying, and sulfur absorption are discussed below in this chapter. As seen, besides the pure chemical kinetics, mass transfer processes inluence heterogeneous reaction rates. Of course, depending on the hypotheses, these models can face various degrees of complexity. In view of the present introductory text, the following are assumed: a) A steady-state regime exists. During its travel through a combustor or gasiier, the particles react with the gases, and therefore the temperature, density, and concentration of chemical components inside the particles vary according to their position in the equipment. Nonetheless, average characteristics of particles passing a given position in a reactor remain constant. b) Solid particles are homogeneous in all directions, except of course the one in which the main mass transfer occurs. For instance, in case of cylindrical particles, variations in concentration occur only in the radial (r) direction or perpendicular to the length. c) The velocity ield inside the solid matrix is negligible. As gases are entering and leaving the particles, their low in opposite directions tends to cancel out (at least approximately). Consequently, overall mass transfer does not create a substantial velocity component in the radial direction or any other. Even if an overall mass low of gas from the inside toward the particle surface exists, the overall convective term will be assumed to be negligible compared with source/sink (or production/consumption due to chemical reactions) terms. This assumption can be criticized in various situations, such as, for instance, application to devolatilization or drying, where gases escape from the particle interior with no counter incoming low. d) Particles are isothermal. This assumption can be also criticized, mainly when large particles with low thermal conductivity are involved. Roughly, this can be assumed when the Biot number (Chapter 7, Equation 7.1) is less than 0.1. In cases where this cannot be assumed, the temperature proile inside the particle should be obtained. This is proposed as exercises at the end of this chapter. e) The concentrations of any chemical species are assumed to be equal at each side of an interface (core-shell or shell-gas). Actually, the concentrations are near the equilibrium at each interface. It should be noted that even the equilibrium hypothesis is another approximation valid only when the thermodynamic process is quasistatic.

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f) Heat and mass transfers are independent. Therefore, the Soret effect is neglected.* This can be seen as a consequence of isothermal particle approximation. g) Particle pores do not suffer severe blockage by related processes. For instance, this is critical in the case of pyrolysis of various highly volatile coals, in which plastiication and swelling occur during the devolatilization. Therefore, for these cases, the models presented here might not produce reasonable results. h) Various reactions and processes taking part inside the particle can be treated independently. During the combustion or gasiication, processes might superimpose. For instance, while the particle outer layers are devolatilizing, internal ones might still be going through drying. The same might occur during other periods in relation to gasiication/combustion and pyrolysis. Actually, all of them might be occurring simultaneously. Nevertheless, almost all workers in the area use the independence assumption [15]. Usually, this superimposition is more critical in cases of slow heating rates, such as in moving- or ixed-bed combustion or gasiication. In cases of suspended or even luidized-bed combustion or gasiication, where the particles go through faster heating, the superimposition times are relatively shorter. A brief discussion of this aspect is presented in Appendix C. It is important to stress again that some if not all of these assumptions can be criticized. This is the very essence of modeling. A modeler should not be discouraged from making assumptions. Tests against experimental results are the inal yardstick of the validity of the assumptions. The application of the present model for heterogeneous reactions in comprehensive simulations has led to reasonable reproductions of real operations, as will be shown in Chapters 12, 16, and 18.

9.2 9.2.1

GENERAL FORM OF THE PROBLEM UNEXPOSED-CORE MODEL

Of course, particles are found in almost any shape. However, any particle can be classiied as having one of three basic shapes: plane (or plate-like), cylindrical (or needle-like), and spherical. The treatment below considers a spherical or near-spherical shape (see Figure 9.1a), and according to the unexposed-core model, the equivalent diameter (2rA) of a particle remains equal to the original value. Formulas to allow application to other forms are also presented below in this chapter. Equation A.14 written for a component j and on a molar basis becomes u φ ∂ρ j ∂ρ j ∂ρ j u θ ∂ρ j + ur + + = r ∂θ r sin θ ∂φ ∂t ∂r  1 ∂  2 ∂ρ j  ∂ 2 ρ j   ∂ρ j  1 ∂  1 + 2 2 Dj  2 r + 2 + R M, j . sin θ     ∂θ  r sin θ ∂φ2   r ∂r  ∂r  r sin θ ∂θ  *

(9.1)

The Soret effect is caused by interference of temperature gradients into mass transfer process [1].

189

Heterogeneous Reactions

Assumption A (see above) allows elimination of the irst term on the left side of Equation 9.1. Assumption C allows removal of all convective terms, represented by all other terms on the left side. Assumption B excludes the two last terms inside the brackets on the right side, and it is thus possible to write D jr −2

d  2 dρ j   M, j r = −R dr  dr 

(9.2)

where the total rate of production (or consumption) of component j due to competing reactions is given by Equation 8.1 (Chapter 8). It should be noted that coeficient Dj represents the diffusivity of component j into the phase in which the process takes place. If that phase is the particle core, or nucleus, the parameter is called the effective diffusivity of j in that porous structure, and it is represented by Dj,N. A similar notation is used for the diffusivity of j in the shell of inert porous solid coating the core, or Dj,A. Dj,G is the average diffusivity of a chemical species j in the gas layer surrounding the particles. As is shown in Chapter 11, the values for effective diffusivities can be correlated to the gas–gas diffusivity Dj,G. Without any interference of mass resistances, it is assumed that kinetics of any heterogeneous reaction can be written in the following form: k i (ρ j − ρ j,eq )n

(9.3)

where k i is the reaction rate coeficient. This is called the intrinsic reaction rate, as opposed to the effective reaction rate, which is the one calculated after the inluence of mass transfer resistances. In order to transform Equation 9.2 into a dimensionless form, the following new variables are used: r rA

(9.4)

ρ j − ρ j,eq . ρ j,∞ − ρ j,eq

(9.5)

x= y=

Using the above and generalizing to any fundamental shape, Equation 9.2 can be rewritten as ∇2y = Φ 2y n

(9.6)

where the Laplacian operator is described by ∇2 = x −p

d  p d  . x dx  dx 

(9.7)

The coeficient p takes the following possible values: 0 for plane geometry, 1 for cylindrical, and 2 for spherical particles. In the present case, p = 2. The Thiele coeficient is given by 1/ 2

 k (ρ − ρ j,eq )n −1  Φ = rA  i j,∞  . D j,N  

(9.8)

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Solid Fuels Combustion and Gasification

The above treatment is valid for most combustion and gasiication reactions, where n varies between 0 and 2 and most follow a irst-order behavior. Deductions for other situations are left as exercises. The objective now is to obtain the rate of mass transfer between the particle and the external gas layer. Since there are two main regions inside the particle, the solution of Equation 9.6 should be accomplished in two steps: it will irst be applied to the inert solid shell coating the core, and then it will be applied to the core. This is also necessary because different boundary conditions should be imposed for each region. For the coating inert shell, there is no reaction; therefore, Equation 9.6 becomes dy = A1x −2 , a ≤ x ≤ 1 dx

(9.9)

y = −A1x −1 + B1, a ≤ x ≤ 1

(9.10)

leading to

where rN . rA

a=

(9.11)

The surface concentration would be y(1) =

ρ j,S − ρ j,eq = − A1 + B1 . ρ j,∞ − ρ j,eq

(9.12)

The boundary conditions are as follows: 1. At the particle surface, the mass transfer is given by D j,G

dρ j dr

= D j,A r = rA ( + )

dρ j dr

= β G (ρ j,∞ − ρ j,surface ).

(9.13)

r = rA ( − )

As will be shown later, the mass transfer coeficient βG is given as a function of the transport parameters. The minus and plus symbols indicate whether the derivative refers to positions just below or above the speciied value, respectively. 2. The continuity of mass lux at the interface between the inert shell and the reacting gas is given by Equation 9.24. Equation 9.13 can be written as y′(1) = N Sh 1 − y(1) 

D j,G = N1 1 − y(1)  D j, A

(9.14)

where the Sherwood number is given by N Sh =

βG rA D j,G

N1 = N Sh

D j,G . D j, A

(9.15) (9.16)

191

Heterogeneous Reactions

Using Equations 9.9, 9.10, and 9.14, it is possible to write B1 = 1 −

1 − N1 A1 . N1

(9.17)

As we have seen, an extra relationship is necessary in order to compute the values of A1 and B1. This would be possible if the problem is solved for the core as well. In this case, Equation 9.6 can be written as xy ′′ + 2 y ′ − Φ 2 xy = 0.

(9.18)

This is a Bessel differential equation, with solutions given by x −1/ 2 I1/ 2 (xΦ ) and x −1/ 2 I −1/ 2 (xΦ). These can be transformed into an equivalent form to give the general solution y = A2

sinh(xΦ) cosh(xΦ) , 0≤x≤a + B2 x x

(9.19)

Since the concentration must acquire inite values at the particle center (as is the case anywhere), and because lim

sinh(xΦ ) =Φ x

(9.20)

lim

cosh(xΦ) = ∞, x

(9.21)

x →0

and x→ 0

coeficient B2 must be identical to 0, and y = A2

sinh(xΦ) , 0 ≤ x ≤ a. x

(9.22)

Because of assumption E, concentrations of reacting species at each side of the interface between the core and the shell are equal. Therefore, Equation 9.10 gives y(a ) = A 2

A sinh(aΦ) = − 1 + B1 . a a

(9.23)

As mentioned before, the mass lux must be conserved at the shell-core interface, or D j,A y ′(a +) = D j,N y ′(a −).

(9.24)

From Equations 9.10 and 9.22, one can write D j,A A1 = D j,N A 2 [aΦ cosh(aΦ) − sinh(aΦ)].

(9.25)

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Solid Fuels Combustion and Gasification

Combining Equations 9.17 and 9.23, it is possible to arrive at A2 = a

 1 1 − N1  1 − A1  + a N1  . sinh(aΦ)

(9.26)

The last two equations lead to A1 1 = a 1 + ψ1 + ψ 2

(9.27)

D j, A D j, N [aΦ coth(aΦ ) − 1]

(9.28)

where ψ1 = and ψ2 = a

1 − N1 . N1

(9.29)

The consumption rate of reacting gas j by reaction i is given by the rate of mass transfer through the external surface of the particle, or ri = D j,A

dρ j dr

= r = rA −

D j,A . (ρ j,∞ − ρ j,eq ) dy rA dx x=1−

(9.30)

After application of Equation 9.9, it becomes ri =

D j,A A1 (ρ j,∞ − ρ j,eq ). rA

(9.31)

The inal form is obtained using Equation 9.27 to give 2 ρ j,∞ − ρ j,eq . ri = 3 d p ,I U ∑ U ,k

(9.32)

k =1

Here, the three resistances are provided by 1 N Sh D j,G

(9.33)

1− a aD j, A

(9.34)

1 . aD j, N [aΦ coth(aΦ) − 1]

(9.35)

U U,1 =

U U ,2 = U U ,3 =

According to Equation 9.32, the rate of consumption (or production) of a gaseous component j by a solid-gas reaction i is given by a relatively simple formula. Consequently, the three resistances to mass transfers are as follows:

193

Heterogeneous Reactions

• UU,1 for the gas boundary layer • UU,2 for the shell, which surrounds the core • UU,3 for the core The last one combines mass transfer resistances and reaction kinetics. It is also interesting to note that all three parameters are inversely proportional to the irst power of the diffusivities of gases through the respective layers. The relative importance of these resistances determines the ruling or controlling mechanism related to each reaction. There is no predetermined rule for this, and the controlling factor would depend on the combined conditions to which the particle is subject at each point in the reactor. For example, at points near the base of a moving-bed gasiier, where relatively thick layers of ash usually surround the cores of particles, the ruling resistance would be offered by the ash shell. The picture can be different for particles near and above the combustion front, where the layer of ash is relatively thin, and the gas layer or the diffusion through the core would probably constitute the main resistance. This last picture is even more likely in regions of relatively low temperatures because the kinetics coeficient would decrease exponentially, leading to low values of the Thiele parameter (Equation 9.8). Of course, several correlations and data points would be required for proper computations of the parameters and coeficients involved. For instance, the kinetics coeficients can be found in Chapter 8, Table 8.6. The computation of equilibrium coeficients can be made by the equations shown in Chapter 5. Chapter 11 presents methods by which to compute other parameters. 9.2.1.1 Effectiveness Coefficient As we have seen, the external surface of the particle is exposed to the reacting gas at concentrations below the value found in midst of the gas phase (ρ j,∞). In addition, the surfaces of internal particle pores are exposed to the reacting gas, but at concentrations much lower than ρ j,∞ . On the other hand, the surface area of the internal pores of a particle is much greater than the external surface area. The treatment shown above takes into account all these effects and provides a formula to compute the real reaction rates. However, the same result provided by Equation 9.32 can be achieved through an alternative approach. This approach uses an effectiveness coeficient (ηI) to correct the rate of heterogeneous reaction i involving a gaseous species j, which is computed under the following simplifying assumptions: 1. The gas species j, at a concentration equal to the one found in the midst of the gas phase ( ρ j,∞), is in contact with the external surface of the particle. 2. There are no internal pores, or the external particle surface represents all available area for the reaction with gas j; therefore: ri =

d p ,I k i (ρ j,∞ − ρ j,eq ) ηi . 6D j,N

(9.36)

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Solid Fuels Combustion and Gasification

In the case of spherical geometry, the effectiveness coeficient is given by volume averaging, or a

ηi =

∫ y(x)dx 0

1

∫ dx

3

.

(9.37)

3

0

Using Equations 9.22, 9.26, and 9.27, the above equation can be written as ηi =

3 Φ2

1

.

3

∑U

(9.38)

U ,k

k =1

A few values for the case where N1 = 1 and Dj,N = Dj,A are shown in Table 9.1 to provide some idea of the role played by this factor.

9.2.2

EXPOSED-CORE MODEL

As noted above, this model assumes that no layer of spent material withstands fast variations of temperature combined with attrition with other particles. Therefore, as soon as it is formed, that layer breaks into small particles that detach from the original particle (Figure 9.1b). Adopting the same notation as before, the governing differential equation for the core is given by Equation 9.18 and the solution by Equation 9.22, or sinh(xΦ) , 0 ≤ x ≤ a. (9.39) x The inite value condition at the particle center has already been used. Similarly, as with Equations 9.13 and 9.14, at the gas-solid interface it is possible to write y = A3

y′(a ) = N 2 [1 − y(a )]

(9.40)

where N 2 = N Sh

D j,G . D j, N

(9.41)

TABLE 9.1 Effectiveness Factors for N1 = 1 and Dj,N = Dj,A a 0.001 0.1 0.2 0.5 1.0

Φ = 0.1 1.5 × 10 1.0 × 10–3 0.01 0.12 1.00

–9

Φ = 1.0

Φ = 10.0

1.2 × 10 9.9 × 10–4 0.01 0.11 0.72

1.0 × 10–9 7.1 × 10–4 3.1 × 10–3 0.01 0.03

–9

195

Heterogeneous Reactions

Using Equation 9.39, this last condition leads to ψ3 1 + ψ4

(9.42)

a sinh(aΦ )

(9.43)

aΦ coth(aΦ) − 1 . aN 2

(9.44)

A3 = where ψ3 = and ψ4 =

The consumption or production rate of the reacting gas by the particle is given in a similar way as before: D j,N

dρ j dr

= r = rN −

D j,N dy (ρ j,∞ − ρ j,eq ) . rA dx x=a −

(9.45)

Using Equations 9.39 and 9.42, it is possible to write 2 ρ j,∞ − ρ j,eq ri = . 3 d p ,I U X k , ∑

(9.46)

k =1

Here, the three resistance are given by U X,1 =

U X ,3 =

1 = U U,1 N Sh D j,G

(9.47)

U X,2 = 0

(9.48)

a = a 2 U U ,3 . D j,N [aΦ coth(aΦ) − 1]

(9.49)

As we have seen, the above formulas are similar to the case of the unexposed-core model. However, of course, no reference to the shell diffusivity can be made. One may ask, What about the resistance imposed by the spent material released from the original particles? Actually, this residue imposes a resistance to reactions because it mixes with the reacting solids, decreasing the number of active solid particles per volume of reactor. In other words, the concentration of active available solid phase declines inside the reactor. Additional details on how this decrease can be evaluated are described in Chapter 11.

9.3

GENERALIZED TREATMENT

The basic approach for the spherical particle can be generalized to plates (disks, chips) and cylinders (pellets). The formulas are summarized in Table 9.2, where

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Solid Fuels Combustion and Gasification

TABLE 9.2 Formulas to Compute the Mass Transfer Resistances in the Case of the Unexposed-Core Model Resistance Shape

UU,1

UU,2

UU,3

Plate

1 N Sh D j,G

1− a D j,A

coth(aΦ ) ΦD j,N

Cylinder

1 N Sh D j,G

Sphere

1 N Sh D j,G

−

ln(a ) D j,A

1− a aD j,A

I 0 (aΦ ) aΦD j,N I1 (aΦ ) 1 aD j,N [aΦ coth(aΦ) − 1]

indexes G, A, and N indicate the effective diffusivity through the gas boundary layer, the spent shell, and the nucleus, respectively. The resistances found for the exposed and unexposed core models can be related by the following equations: U X,1 = U U,1

(9.50)

U X,2 = 0

(9.51)

U X ,3 = a p U U ,3 .

(9.52)

The application of these relationships should be preceded by the following considerations: a) Equation 9.6 was written for one dominant direction. For instance, in the case of a plate or chip, the thickness is assumed to be much smaller than the other dimensions, and in the case of a cylinder, the diameter is much smaller than the length. The bulk of the mass transfer occurs through the smaller path or preferable transport dimension: the thickness in the case of chips and the diameter in the case of cylinders. As a rule, approximations are acceptable for ratios greater than 5 between the larger dimension of the particle and its preferable transport one. If that condition is not fulilled, the correlation developed for spheres, combined with concept of sphericity (see Chapter 2, Equation 2.1), usually provides a good approximation. b) It should be noted that from the concept of sphericity it is possible to obtain relationships between the equivalent diameter of a sphere and the preferable transport dimension. This is useful because the available values for the equivalent diameter are normally obtained from the screen analysis of particle size distribution. These correlation can be easily deduced for the following basic forms:

197

Heterogeneous Reactions

• Slab-like or plate-like shape, given by d p, I = ϕ

d screen , I 3

(9.53)

• Cylinder-like shape or needle-like shape, given by 2d screen , I 3

(9.54)

d p, I = ϕ d screen , I

(9.55)

d p, I = ϕ • Sphere-like shape, given by

c) In each case, the parameter a can be approximated by the following relationship 1

a = (1 − f514 ) p +1

(9.56)

where f514 is the fractional conversion of carbon or the most important solidphase reagent. d) It is important to stress that these solutions are valid for irst-order reactions, which is the case for most combustion and gasiication reactions. Approximations for orders not very far from that should be carefully examined.

9.4

OTHER HETEROGENEOUS REACTIONS

The above models of the exposed and unexposed core model can be also applied to reactions or processes other than the described above (Chapter 8, Reactions R.1 through R.6). For instance, unexposed or shrinking core model may be used for pyrolysis, drying, and sulfur absorption processes. It is easy to imagine that the deinition of spent inert material depends on the reaction. For instance, ash is the inert material for reactions involving the char and a gas, such as those in Reactions R.1 through R.6 (see Chapter 8, Table 8.4). However, as shown in Chapter 10, this model can be extended for devolatilization (Chapter 8, Reaction R.7). Then, char would be the inert material and undervolatilized fuel would constitute the core. Although not a reaction, the second drying period can also follow this model (see Chapter 10). Here, dry material would be the spent one, while original wet fuel would form the core. Finally, the unreacted-core model can also be applied to sulfur-absorption Reaction R.22 or R.26 (Chapter 8). This time, CaO or MgO would form the core, and the inert material would be CaCO3 or MgCO3, respectively. In this case, SO2 could be set as the representative reacting gas. On the other hand, in the case of devolatilization, drying, and sulfur absorption, there would little probability for the spent material to detach from its respective core. Consequently, the exposed core is not usually applied to such processes.

198

Solid Fuels Combustion and Gasification

9.5

EXERCISES

9.5.1

PROBLEM 9.1* Reproduce the same deductions made in this chapter for the case of a zero-order reaction, or n = 0 in Equation 9.3.

9.5.2

PROBLEM 9.2***

Repeat Problem 9.1 for the case of a second-order reaction.

9.5.3

PROBLEM 9.3*

Assume a carbonaceous solid with the following composition: • Ultimate analysis (wet basis): • Moisture: 5.00% • Volatile: 38.00% • Fixed carbon: 47.60% • Ash: 9.40 • Proximate analysis (dry basis): • C: 73.20% • H: 5.10% • O: 7.90% • N: 0.90% • S: 3.00% • Ash: 9.90% Assume the following: • • • •

Average particle diameter: 0.475 mm Mass low of injected air: 0.7 kg/s Unexposed core model Particles half converted, i.e., 50% of the original mass of carbon already consumed • A gas phase composed of a mixture (mass percentages) of 10% oxygen, 50% steam, and 40% of carbon dioxide at 1000 K and 10 bar

Determine whether the reaction rate (kmol m–2 [of particle surface area] s–1) between the carbon in the particles and oxygen in the gas is: A. B. C. D.

9.5.4

Below 10 –8 Between 10 –8 and 10 –6 Between 10 –6 and 10 –4 None of the above

PROBLEM 9.4* Under the same conditions as before, determine whether the reaction rate between the carbonaceous solid and water vapor is: A. B. C. D.

Below 10 –8 Between 10 –8 and 10 –6 Between 10 –6 and 10 –4 None of the above

199

Heterogeneous Reactions

9.5.5

PROBLEM 9.5**

Demonstrate the formulas shown in Table 9.2 for the case of cylindrical particles.

9.5.6

PROBLEM 9.6** Deduce Equations 9.53, 9.54, 9.55, and 9.56.

9.5.7

PROBLEM 9.7* Take the case of Reactions R.1 through R.6, as listed in Chapter 8, Table 8.4. Assuming char is composed only of carbon, to which of those reactions would hypothesis C (Section 9.1) be most critical? In other words, in which case would the overall radial velocity ur be farthest from 0, and therefore in which case would the second term in the left side of Equation 9.1 be most dificult to neglect?

9.5.8

PROBLEM 9.8**

Having in mind Problem 9.7, development of a solution was attempted for the unexposed core model in which that convective term in radial direction is included. For reaction i involving the component j, the following relation was proposed ur =

Ci ra

(9.57)

where Ci is a constant. Assuming for a moment that this is reasonable, and also assuming a steadystate process, isothermal and isobaric conditions, and low only in the radial direction, what would be the best integer value for exponent a? Use Equation A.13 (Appendix A, Table A.3).

9.5.9

PROBLEM 9.9*** In the previous problem, propose a relationship for parameter Ci as a function of the overall rate of consumption and production of gas components involved in the respective reaction i. Is such a proposal reasonable? Explain.

9.5.10

PROBLEM 9.10****

Assuming a constant value for Ci, deduce the solution for an unexposed core model in which the relationship shown in Equation 9.57 could be applied. Numerical methods may be used.

9.5.11

PROBLEM 9.11*

Assuming steady state and other conditions (list them), use Equation A.15 (Appendix A, Table A.3) to arrive at the following form: λ

1 ∂  2 ∂T  r = −R Q . r 2 ∂r  ∂r 

(9.58)

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Solid Fuels Combustion and Gasification

9.5.12

PROBLEM 9.12***

Use Equation 9.58 to obtain the temperature proile inside the particle, assuming a constant rate of energy generation RQ. In addition, take the temperature of the gas surrounding the particle and the convective heat transfer coeficient (αG) between gas and particle surface as known constants. Solve that problem in the following situations: a) Exposed core model with constant conductivity b) Unexposed core model where shell and core present different conductivities

9.5.13

PROBLEM 9.13***

Use the proiles determined in the last problem to arrive at the average temperature inside the particle. Apply the formula rp

Tav =

9.5.14

4π T(r )r 2 dr. Vp ∫0

(9.59)

PROBLEM 9.14****

An even more realistic situation than that assumed in Problems 9.11 and 9.12 would be to consider that RQ as function of temperature. A relatively simple situation would be  T  R Q = ∑ Ci exp  − e ,i   T  i

(9.60)

i E Te ,i =  , R

(9.61)

where

and the summation should include all i reactions simultaneously taking place between solid particle and gas reactants. Each parameter Ci would include the pre-exponential factor and reactant concentrations. In this case, try to set a numerical procedure to obtain the temperature proile inside the particle.

REFERENCES 1. de Souza-Santos, M.L., Modelling and Simulation of Fluidized-Bed Boilers and Gasiiers for Carbonaceous Solids, PhD thesis, University of Shefield, Shefield, United Kingdom, 1987. 2. de Souza-Santos, M.L., Comprehensive modelling and simulation of luidized-bed boilers and gasiiers, Fuel, 68, 1507–1521, 1989. 3. de Souza-Santos, M.L., Application of comprehensive simulation to pressurized bed hydroretorting of shale, Fuel, 73, 1459–1465, 1994. 4. de Souza-Santos, M.L., A study on pressurized luidized-bed gasiication of biomass through the use of comprehensive simulation, Proc. Fourth International Conference on Technologies and Combustion for a Clean Environment, Lisbon, Portugal, July 7–10, 1997, paper 25.2, Vol. II, pp. 7–13.

Heterogeneous Reactions

201

5. de Souza-Santos, M.L., A study on pressurized luidized-bed gasiication of biomass through the use of comprehensive simulation, in Combustion Technologies for a Clean Environment, Gordon and Breach, Amsterdam, Netherlands, 1998. 6. de Souza-Santos, M.L., Application of comprehensive simulation of luidized-bed reactors to the pressurized gasiication of biomass, Journal of the Brazilian Society of Mechanical Sciences, 16(4), 376–383, 1994. 7. de Souza-Santos, M.L., Search for favorable conditions of atmospheric luidized-bed gasiication of sugar-cane bagasse through comprehensive simulation, in Proc. 14th Brazilian Congress of Mechanical Engineering, Bauru, São Paulo, Brazil, December 8–12, 1997. 8. de Souza-Santos, M.L., search for favorable conditions of pressurized luidized-bed gasiication of sugar-cane bagasse through comprehensive simulation, in Proc. 14th Brazilian Congress of Mechanical Engineering, Bauru, São Paulo, Brazil, December 8–12, 1997. 9. Rabi, J.A., and de Souza-Santos, M.L., Incorporation of a 2-lux model for radiative heat transfer in a comprehensive luidized bed simulator. Part I: Preliminary model equations, in Proc. ENCIT-2002, 9th Brazilian Congress on Engineering and Thermal Sciences, Caxambu, Minas Gerais, Brazil, October 15–18, 2002. 10. Rabi, J.A., and de Souza-Santos, M.L., Incorporation of a 2-lux model for radiative heat transfer in a comprehensive luidized bed simulator. Part II: Numerical results and assessment, in Proc. ENCIT-2002, 9th Brazilian Congress on Engineering and Thermal Sciences, Caxambu, Minas Gerais, Brazil, October 15–18, 2002. 11. Rabi, J.A., Usage of Flux Method to Improve Radiative Heat Transfer Modelling inside Bubbling Fluidized Bed Boilers and Gasiiers, PhD thesis, Faculty of Mechanical Engineering, State University of Campinas, Campinas, São Paulo, Brazil, 2002. 12. de Souza-Santos, M.L., Development of a Simulation Model and Optimization of Gasiiers for Various Fuels (Desenvolvimento de Modelo de Simulação e Otimização de Gaseiicadores com Diversos Tipos de Combustíveis), IPT-Inst. Pesq. Tec. Est. São Paulo, SCTDE-SP, Report No. 20.689, DEM/AET, São Paulo, Brazil, 1985. 13. de Souza-Santos, M.L., and Jen, L.C., Study of Energy Alternative Sources; Use of Biomass and Crop Residues as Energy Source: Part B: Development of a Mathematical Models and Simulation Programs for Up-Stream and Down-Stream Moving Bed Gasiiers. (Estudo de Fontes Alternativa de Energia. Parte B), IPT-Institute for Technological Research of São Paulo, Report No. 16.223-B/DEM/AET, São Paulo, Brazil, 1982. 14. Jen, L.C., and de Souza-Santos, M.L., Modeling and simulation of ixed-bed gasiiers for charcoal (Modelagem e simulação de gaseiicador de leito ixo para carvão), Brazilian Journal of Chemical Engineering (Revista Brasileira de Engenharia Química), 7(3–4), 18–23, 1984. 15. Smoot, L.D., and Smith, P.J., Coal Combustion and Gasiication, Plenum Press, New York, 1985.

and 10 Drying Devolatilization CONTENTS 10.1 Drying ......................................................................................................... 203 10.2 Devolatilization ...........................................................................................206 10.2.1 Basic Kinetics ................................................................................207 10.2.1.1 Global............................................................................207 10.2.1.2 Combinations of Series and Parallel Reactions ............208 10.2.2 Distributed Activation Models ...................................................... 213 10.2.2.1 Comments ..................................................................... 216 10.2.3 Structural Models .......................................................................... 216 10.2.3.1 Gavalas-Cheong Model ................................................ 216 10.2.3.2 DISKIN and DISCHAIN.............................................. 217 10.2.3.3 Species Evolution/Functional Group ............................ 220 10.2.3.4 DVC and FG-DVC ........................................................ 223 10.2.3.5 FLASHCHAIN and CPD .............................................224 10.2.4 Adaptation to Present Model Approach ........................................224 10.2.5 Stoichiometry ................................................................................ 225 10.2.6 Kinetics.......................................................................................... 228 10.3 Exercises ..................................................................................................... 230 10.3.1 Problem 10.1 .................................................................................. 230 10.3.2 Problem 10.2 .................................................................................. 230 10.3.3 Problem 10.3 .................................................................................. 230 10.3.4 Problem 10.4 .................................................................................. 230 10.3.5 Problem 10.5 .................................................................................. 230 10.3.6 Problem 10.6 .................................................................................. 230 10.3.7 Problem 10.7 .................................................................................. 231 References .............................................................................................................. 231

10.1

DRYING

As briely described in Chapter 2, drying is a complex process. Although drying does not involve reactions, the treatment shown in Chapter 9 can be applied to provide a reasonable model for the drying of solid particles. For this model, the following assumptions should be applied: a) All drying is assumed to take place during the second period. Therefore, the irst drying period would be neglected, and all water would be transferred 203

204

Solid Fuels Combustion and Gasification

as steam from the wet core through the dry shell layer. The notation for the treatment below will indicate liquid water by the number 700 and as vapor by the number 20 (see Chapter 8, Tables 8.1 and 8.2). b) The particle is assumed to be isothermal. This is a more critical hypothesis because the dried shell usually attains higher temperatures than the wet core. However, for small particles, an average value may be assumed throughout the particle. Criteria for that involve the Biot number, as discussed in Chapter 7. c) The unexposed core is assumed to describe the particle drying. Of course, drying does not involve chemical reactions, but mass transfer plays an important role in the process, and the description shown in Chapter 9 can be adapted for the present case. In doing so, it will be assumed that all phase changes occur at the core-shell interface. As commented in item C of Section 9.1, the assumption of a negligible overall convective term (or radial velocity ur) for the case of drying can be criticized. Therefore, the adaptation shown below should be seen as a irst approximation. Despite those approximations, the present treatment is a reasonable approach for the following reasons: • It does not introduce any violation of mass or energy conservation laws. • Unlike devolatilization, drying involves just one chemical species (H2O); thus, no distribution of components resulting from the processes should be estimated. • In combustors or gasiiers, the drying of feeding solids is very fast, and small deviations in the computation of its rate do not affect the precision of the whole model. The adaptation of the unexposed-core model begins by setting y=

ρ 20 . ρ 20,∞

(10.1)

Hence, the governing differential mass balance equations can be taken from Chapter 9. In this way, for the shell of already dried material (a ≤ x ≤ 1), it is possible to write x −2 [ x 2 y′(x)]′ = 0

(10.2)

y ′(x) = A 4 x −2

(10.3)

y = − A 4 x −1 + B4 .

(10.4)

with the following solutions:

and

As explained in Chapter 9, the mass transfer from the particle surface leads to

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Drying and Devolatilization

y′(1) = N1[1 − y(1)]

(10.5)

where D 20 ,G D 20 , A

(10.6)

β G rA . D 20,G

(10.7)

N1 = N Sh and N Sh =

Applying Equations 10.3 through 10.5, it is possible to write B4 = A 4

N1 − 1 + 1. N1

(10.8)

In order to determine the coeficients A4 and B4, the change from liquid to vapor is assumed to occur at the interface between the core and the shell layers. Thus, the vapor is saturated at the interface temperature, or P ρ 20,a = sat ,a RTa

(10.9)

and y(a ) =

ρ 20,a P =  sat ,a  20,∞ ρ 20,∞ RT aρ

(10.10)

where a is given by Equation 9.11 (Chapter 9). As seen, y(a) is a function of the interface temperature (Ta). Since an isothermal condition in the particle has been assumed, the interface temperature is the average particle temperature. Thus, y(a) can be known at each point of the equipment because the energy and mass balances are solved simultaneously throughout the bed. Using Equations 10.3, 10.4, and 10.8, the following can be obtained: A4 = −

y(a ) − 1 . 1− a 1 + a N1

(10.11)

Then, the rate of steam transfer to surrounding gas at the external surface of the particle is given by rd = −D20, A

dρ 20 dr

=− r = rA

D20, Aρ 20,∞ y′(1) rA

(10.12)

Finally, from Equations 10.5, 10.11, and 10.12, the drying rate can be obtained as rd =

2 ρ 20,sat − ρ 20,∞ 3 d p, I ∑ U d, k k =1

(10.13)

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Solid Fuels Combustion and Gasification

where U d ,1 =

1 D20,G N Sh

U d ,2 =

= U U,1

1− a = U U ,2 aD20, A

U d,3 = 0.

(10.14)

(10.15)

(10.16)

The original particle diameter is indicated by dp,I.

10.2

DEVOLATILIZATION

As mentioned in Chapter 2, volatile is an important fraction of solid fuels and responsible for a series of processes during combustion and gasiication. Therefore, a good representation or model for devolatilization or pyrolysis is fundamental for reasonable modeling and simulation of any combustion and gasiication equipment. Pyrolysis of carbonaceous solid fuels, such as coal, charcoal, wood, and peat, among others, is a series of combined processes and reactions. The result is the release of gases and liquids into the atmosphere surrounding the particle. The composition of such fractions includes a large amount of chemical species, particularly tar [1]. Being such a complex process, pyrolysis has been investigated by many, and it has therefore been subjected to different levels of modeling. Some try extreme simpliications, while others use elaborate mechanisms to explain several details. The rate at which the various fractions are released is called the kinetics of devolatilization or pyrolysis. Nevertheless, devolatilization is not just a set of chemical processes; it involves phase changes, mass, and heat transfers. Thus, rigorously speaking, using the term kinetics to describe pyrolysis is not appropriate. In view of this dificulty, any model for pyrolysis is just a crude approximation of reality. Consequently, it is not surprising to ind models representing devolatilization within a wide range of precision when compared against the real process. The most important classes of models are the following: a) Isolated kinetics b) Distributed activation c) Structural The classiication shown above is generally accepted. Nonetheless, it is important to remember that those models evolved because of the need of better predictions for the yielding of products from pyrolysis. In many instances, the passage from a modeling category level to a more elaborate one was accomplished through slight improvements. In such cases, it may be dificult to distinguish one class of model from another.

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Drying and Devolatilization

10.2.1

BASIC KINETICS

This class of models tries to represent pyrolysis by a single chemical reaction or a series of them, and it is subdivided into the following: • Global • Combination of series and parallel reactions 10.2.1.1 Global The global kinetics model is the simplest level of representation. It assumes that volatile release can be represented by a single overall reaction, such as the following: Carbonaceous Solid(dry) → w V (volatiles) + (1 − w V )(char ).

(10.17)

Here, wV is the fraction of volatile matter at a given instant in the carbonaceous solid (coal, biomass, etc.). The total amount of volatile in the original carbonaceous (determined by proximate analysis, dry basis) is wV,I. Therefore, the kinetics is described by the following single equation: dw V = k (w V , I − w V ) n . dt

(10.18)

For instance, in the case of the coal present in Table 2.1 (Chapter 2), wV,I = 0.38/ (1 – 0.05) = 0.40. The kinetics coeficient is computed by classical Arrhenius law, or    E k = k 0 exp  −   .  RT 

(10.19)

~ The pre-exponential coeficient (k0) and the activation energy (E) are determined by experimental procedures, and the reaction order (n) varies between 1 and 3. The work of Badzioch and Hawksley [2] is an example of this approach. Arenillas et al. [3] present a careful review of the various kinetics parameters in a case of bituminous coal. After comparisons between experimental determinations of mass loss of that particular coal against time, they concluded that a third-order (n = 3) reaction with k0 ~ equal to 4 s–1 and E equal to 94.1 MJ kmol–1 provided the best representation. 10.2.1.1.1 Comments It should be noted that the value of WV,I in Equation 10.18 is determined under particular conditions of the experimental procedure: usually, slow heating rate, moderate temperatures, and ambient pressure. However, the actual conditions under which the pyrolysis occurs might be substantially far from those. As seen in Chapter 2, the total amount of released volatiles generally differs from WV,I. That approach would lead to mistakes, since the global kinetics method cannot a priori determine the actual total amount of volatile released from fuel. In addition, it does not provide information regarding the amount and composition of gas and

208

Solid Fuels Combustion and Gasification

tar evolved during the pyrolysis. These data are paramount for the development of consistent combustion or gasiication simulation models. 10.2.1.2 Combinations of Series and Parallel Reactions To overcome the previous lack of information, the next step in the model evolution includes combined parallel and series reactions. The simplest form is represented by a series of parallel irst-order reactions, or dw j = k j (w j,lim − w j ). dt

(10.20)

When the sample is kept at a given temperature (Tlim), the value wj,lim is the mass fraction of gas species j for each gas measured when no further variation is seen, or dw j dt

= 0.

(10.21)

Tlim

Notice that here the index j represents not only a chemical species but also a reaction. Hence, these models assume that each component of pyrolysis is produced (or consumed) according to an individual reaction. The kinetic coeficient kj is given by a typical Arrhenius form, or j   E k j = k 0, j exp  −   .  RT 

(10.22)

In view of that, the above class of models depends on parameters not just for the kinetics but also for values of wj,lim. This approach was used, for instance by Nunn et al. [4, 5]. They developed experiments for sweet gum hardwood and wood lignin. The tests used the captive technique (see Appendix G) in which samples of 100 g of carbonaceous solids were heated at 1000 K/s in an electrical batch reactor. Concentrations of various gases were determined while the temperature varied from 600 to 1400 K. Actually, for most measurements, that limit condition was found around 1200 K. A typical form of the curves obtained during tests is shown in Figure 10.1. The parameters adjusted for those conditions can be found in the published literature [4, 5]. In order to provide an idea of products from pyrolysis, Table 10.1 shows the composition of the gas from devolatilization of wood and wood lignin at approximately 1400 K obtained by those authors. The above model led to deviations of less than 1% between the predicted and measured concentrations of almost all gases, but somewhat higher deviations for CO and CH4 [4, 5]. However, deviations of around 6% were obtained for the accumulated weight loss. The main problems with application of the above approach are as follows: • Tar and other yields do not follow the typical curve shown in Figure 10.1. As shown in Figure 10.2, tar presents a maximum before stabilization. Therefore,

209

Drying and Devolatilization wj

T (K) 600

800

1000

1200

1400

FIGURE 10.1 Typical graph obtained for gas yield during tests by Nunn et al. [4–5].

a irst-order kinetics model cannot represent the release of tar. As seen before, this is mainly because, while releasing, tar cracks into lighter gases and cokes into char. Those processes are called secondary pyrolysis reactions. • Other gases do not reach a zero derivative (Equation 10.21) within the range of temperature set during experiments. This is the case of methane and several other gases, and the value of limit yield is not recognizable.

TABLE 10.1 Approximate Composition of Gas from Biomass Pyrolysis as Reported by Nunn et al. [4–5] Mass Percentagesa Gas Species H2O CO CO2 CH4 C2H4 C2H6 C3H6 Otherb Tar a b

Wood

Wood Lignin

5.9 19.5 7.0 2.6 1.5 0.2 0.5 9.9 52.9

4.5 22.7 4.9 3.8 1.1 0.4 0.4 6.2 56.0

Values deduced from Nunn et al. [4–5]. Mixture including HCHO, CH3OH, butene, ethanol, acetone, furan, acetic acid, and oxygenate components.

210

Solid Fuels Combustion and Gasification wTar

T (K) 600

800

1000

1200

1400

FIGURE 10.2 Typical graph for tar yield during tests by Nunn et al. [4–5].

• During the experiments of Nunn et al. [4, 5], hydrogen was not collected. However, several works [6–9] indicate that hydrogen is an important product of pyrolysis. As additional sophistication, others [10, 11] have proposed hydrogen as a product of secondary pyrolysis reactions. In any case, since tar starts decomposing when it is migrating through the pores inside the particle, it is almost impossible to separate the primary from secondary reactions of pyrolysis processes [12]. This is true especially in case of industrial combustion and gasiication. Therefore, a robust model should be able to provide the amounts of all gases (including hydrogen), as well as tar, at each condition of temperature, pressure, heating rate, and any other variable with a veriiable effect on pyrolysis. Despite those circumstances, the work of Nunn et al. [4, 5] gives important information about the pyrolysis stoichiometry, which might be applied in models as a irst simple approach. Shaizadeh and Chin [13] developed one of the earliest works having in mind combinations of series and parallel reactions to model coal pyrolysis. Variations on their model can be found throughout the literature [14, 15]. A similar approach for cellulose was used by Bradbury et al. [16], who proposed a mechanism in which: a) Cellulose decomposes into active cellulose. b) Then, active cellulose decomposes into char, gas, and volatiles (tar). As seen, their model does not allow the computation of the production of tar and individual gas species, nor does it consider the secondary reactions.

211

Drying and Devolatilization

Gas

1 4

Biomass

2

Tar

3

5

Char

FIGURE 10.3 Scheme for biomass devolatilization. (After Shaizadeh, F., and Chin, P.P.S., American Chemical Society Symp. Ser., 43, 57, 1977 and Thurner, F., and Mann, U., Ind. Eng. Chem. Progress Des. Dev., 20, 482–448, 1981.)

Thurner and Mann [17] researched cases of wood pyrolysis and followed the proposal by Shaizadeh and Chin [13]. That model was composed of two stages of reactions, as illustrated by Figure 10.3. In the primary stage, the carbonaceous decomposes into main fractions by three parallel reactions (Reactions 1–3). The secondary stage is composed of two parallel reactions—Reactions 4 and 5—through which tar can be cracked into light gases or cooked to char.* Thurner and Mann developed careful experiments and determined the kinetics of primary Reactions 1–3. They did not determine the kinetics of secondary Reactions 4 and 5 by assuming that tar was rapidly removed from reaction zone. Of course, they implicitly assumed that secondary reactions were much slower than Reaction 2. In addition, Reactions 1–3 were treated as independent and displaying irst-order behavior. Despite that, their model it the experiments reasonably well. Nevertheless, it does not provide for the distribution or production of individual gas species from pyrolysis. As mentioned before, the large majority of investigations on solid fuels concentrate on coal. Among them, the models by Solomon et al. [10, 18] have been widely used. That model involves the following main steps: *

Actually, there are two main routes for tar coking. If the tar is being formed and is still inside the particle pores, the coking leads to char and decomposition into light gases. If the tar is free from the particle (mainly suspended as a mist in the gas stream), it might crack into gases or form coke and soot. Soot and coke differ a lot in both physical and chemical characteristics. Soot can be formed only under reducing conditions, whereas coke can be formed only under oxidizing conditions. Coke particles are cenospheres, or hollow spheres with dimensions of around fractions of millimeters. Soot consists of lat particles with dimensions in the vicinity of angstroms, and therefore within the range of light wavelengths. That is why soot has special optical properties, such as very high absorption of light, nearing ideal black bodies. Coke does not present such properties.

212

Solid Fuels Combustion and Gasification

1. Raw coal decomposing into metaplast. Part of the raw coal may also form primary char and tar. 2. Metaplast decomposing into tar, primary gas, and char. 3. A inal stage in which: a) Tar cokes to form soot or cracks to form secondary gases. b) Primary gas decomposes into soot or secondary gases. c) Primary char leads to secondary char or secondary gases. As we have seen, coal is assumed to reach an intermediate condition called metaplast. This allows an explanation for swelling, which occurs during the processing for several coal ranks. Once coal is heated, it undergoes a reduction on hydrogen bonding and molecular structure breaking, leading to the formation of a liquid state. A malleable consistency is veriied in layers near the particle surface. Volatile liquids may deposit in these layers and coke, provoking the blockage of pores. Hence, the particle diameter increases when gases or tar, released from devolatilization, cannot leave the particle interior. For some types of coal, this swelling could be severe. 10.2.1.2.1 Comments Although this class of models includes the amounts of tar, gases, and char escaping during pyrolysis, they need special correlations to account for the stoichiometry or ratios of individual gas species obtained in the process. Notwithstanding the importance of the subject, there are relatively few published works on it and none encompassing the whole range of possible conditions. The irst notable work concerning stoichiometry was developed by Loison and Chauvin [6]. They determined the mass fractions of component species in the mixture released from fast devolatilization of several coals and arrived at the following correlations: (10.23) wH2 = 0.157 − 0869w V,daf + 1.338w2V,daf wH2 O = 0.409 − 2.389w V,daf + 4.554w2V,daf

(10.24)

wCO = 0.423 − 2.653w V,daf + 4.845w2V,daf

(10.25)

wCO2 = 0.135 − 0.900w V,daf + 1.906w2V,daf

(10.26)

wCH4 = 0.201 − 0.469w V,daf + 0.241w2V,daf

(10.27)

w Tar = −0.325 + 7.279w V ,daf − 12.884 w 2V ,daf .

(10.28)

Latter, Fine et al. [19] provided correlations to determine the amounts of H2S and NH3 obtained from pyrolysis. Those relations were developed just to evaluate the total amounts of species released after complete pyrolysis performed under a particular set of conditions. Therefore, they cannot relect the inluence of other factors, such as temperature, heating rate, pressure, etc. However, they are useful for simpliied modeling and are used in several works [20–26].

213

Drying and Devolatilization

Other approaches regarding devolatilization stoichiometry have also been taken. For instance, Fuller [7] reported the average overall composition of gas from coal pyrolysis as follows: 55% for H2, 29% for CH4, 5.5% for CO, 2.6% for CO2, 3% for aromatics, 2.6% for SOx, 0.9% for O2, and 4% for N2. Again, for the same reasons as noted for the work of Loison and Chauvin [6], these values are not useful for models encompassing a wide range of conditions. Howard et al. [27] observed that pyrolysis in the presence of calcium minerals leads to decreases in hydrocarbon yield and increases in CO production. Brage et al. [28] studied tar evolution from wood pyrolysis. They characterized and compared the tars released at different temperatures. From 970 to 1170 K, a decrease in the tar yield was observed. Asphaltene, phenol, ethane, and ethyne concentrations in tar diminished as well, whereas the concentrations of aromatics, such as naphthalene, increased. In spite of not providing correlations to estimate the concentration of species from pyrolysis, they veriied that yields were proportional to the dissociation energies of components. This had also been veriied previously by Bruinsma et al. [29]. More recently, Das [9] determined the rates of a few individual gas species from devolatilization of a few coals. All tests showed the following trend: • • • •

CH4 production rate peaks around 770 K. CO production rate peaks around 970 K. H2 production rate peaks around 1020 K. Overall volumetric concentrations of gas mixture from pyrolysis vary within the following ranges: 55%–70% for H2, 20%–35% for CH4, and 7%–22% for CO.

Das [9] also obtained correlations for H2, CH4, and CO yields. However, those correlations are dificult to apply because of dependence on parameters such as concentration of vitrinite in the original coal and mean maximum relectance of vitrinite in oil collected during pyrolysis, information that is not easily available.

10.2.2

DISTRIBUTED ACTIVATION MODELS

Distributed activation (DA) models also assume pyrolysis occurring through several parallel simultaneous irst-order reactions. This may sound as if the model is based on a series of parallel reaction, as shown above. However, the irst intent of DA models is to simplify or avoid experimental determination of species release kinetics. In addition, they include considerations of the internal physical-chemical structure of the carbonaceous fuel. This method departs from Equation 10.20. However, here index j might designate reactions producing not just gases but also liquids evolved and solids formed during pyrolysis. Moreover, since a species j can be produced by one or several reactions i, Equation 10.20 should now be written as dw j = ∑ k i (w j,lim − w j ). dt i

(10.29)

214

Solid Fuels Combustion and Gasification

It is interesting to note that the assumption of irst-order reactions allows a simpliication described by k j = ∑ ki .

(10.30)

i

Here, the summation should include only the irst-order reactions i involving each species j. Integration of Equation 10.29, provides   t  w j = w j,lim 1 − exp  − ∫ k jdt   .   0  

(10.31)

If an Arrhenius form for kinetics is assumed, or i   E k j = ∑ k 0, i exp  −   , i  RT 

(10.32)

and under isothermal conditions, Equation 10.31 could be written as     w j = w j,lim 1 − exp  −∑ k 0, i t exp  − E i     . RT     i 

(10.33)

As we have seen, that requires the estimation of all pre-exponential coeficients (k0,i) and activation energies (E˜ i). At least at the time of those developments, this could not be achieved. The DA method, irst proposed by Pitt [30], assumes the following simpliications: • All pre-exponential coeficients are equal, or k0,i = k0. • The number of reactions is large enough to permit the activation energy to be expressed as a continuous distribution function f(E ) with f (E )dE representing the fraction of the potential volatile loss (wv,lim) with activation energy between E and E + dE . Therefore, wj,lim becomes a differential part of the total volatile loss wv,lim or dw v ,lim = w v ,lim f (E ) dE with

∞

∫ f (E )dE = 1.

(10.34)

(10.35)

0

Using the above equations, it is possible to write the total amount of volatile released during pyrolysis under isothermal conditions from the start (t = 0) until a given time (t) as

215

Drying and Devolatilization

(

)

∞    )dE   . wv = wv,lim 1 − ∫ exp  − k 0 t exp −E   f (E RT    0 

(10.36)

Once the distributed function is chosen, the value of pre-exponential factor (k0) can be found by itting the theoretical and experimental curves of volatile released against time. Pitt [30] chose a simple step function where the term

(

)

 exp  − k 0 t exp −E   RT    3

1.33 × 10 7.67 × 10–7

–4.454 2.152

–3

Reference Coal

ffr,ref

Illinoisa

1.32

Ligniteb

1.93

a

The composition of the Illinois coal is reproduced from Yoon et al. [15]

b

The composition of Spanish lignite is taken from Adánez et al. [38]

245

Auxiliary Equations and Basic Calculations

k +i . (11.33) k +i ref The above value for k i may now be used to estimate the rate of heterogeneous reactions, using the correlations given in Chapter 8. Of course, Equations 9.32 or 9.46 (Chapter 9) should also be applied to compute the consumption or production rates of gases by those reactions. k i = k i ref

11.6

CORE DIMENSIONS

According to the exposed in Chapter 9, the dimension of the reacting core is essential for computations of production and consumption rates of species by heterogeneous reactions. Once the fractional conversion (f) of the reacting solid is known, the computation is straightforward and is given by Equation 9.56. A generalization for other gas–solid processes can be written as 1

a = (1 − f ) p +1.

(11.34)

It should be remembered that the parameter p takes the values 0, 1, or 2 depending on the basic form of the particle: plate, cylinder, or sphere, respectively. The deinition of fractional conversion depends on the process, and the following possibilities might arise: • In the case of Reactions R.1 through R.6, ixed carbon should be used as the reference component for fractional conversion computations. • For Reaction R.8, volatile is the reference component for conversion. As noted before, other heterogeneous pyrolysis-related reactions (R.7 and R.9) do not require the use of models shown in Chapter 9. • For drying (R.10, R.23, R.27, and R.31), moisture in the respective porous solid is the reference component. Therefore, the computation of fractional conversion is intrinsically coupled with mass and energy balances at each point of the reactor.

11.7

HEAT AND MASS TRANSFER COEFFICIENTS

The coeficients for convective heat and mass transfer coeficient allow estimation of the Nusselt and Sherwood numbers. Those are necessary for the computation of the transfers between the solid surface and the surrounding gas. The relationships for convective heat and mass transfers are largely equivalent, since the following rules are followed: 1. The Sherwood and Nusselt numbers, as the corresponding parameters for mass and heat transfer, are deined by N Sh =

βG d p DG

(11.35)

246

Solid Fuels Combustion and Gasification

and α G dp . (11.36) λG 2. The Schmidt and Prandtl numbers are the corresponding parameters for mass and heat transfers, respectively, and are given by N Nu =

N Sc = and

µG ρG D G

(11.37)

cG . (11.38) µGλ G 3. The corresponding Grashof numbers for mass and heat transfer are given by, respectively: N Pr =

2

N GrM =

2gd 3p ρ j,G,surf − ρ j,G,∞  ρG    ρ j,G,surf + ρ j,G,∞  µ G 

N Gr =

2gd 3p TG,surf − TG,∞  ρG   .  TG,surf + TG,∞  µ G 

and

(11.39)

2

(11.40)

The above are valid for ideal gases. The index surf indicates the value at or near the particle surface, and ∝ indicates the value far from it. 4. The Reynolds number, applied to both mass and heat transfers, should be computed as U dρ (11.41) N Re = G p G . µG The supericial velocity should be used for packed (ixed or moving) beds and is given by F (11.42) UG = G . Sρ G In the case of a single sphere, for instance, the velocity is that of the luid far from the particle. The literature on the heat and mass transfer coeficients is extensive [19–23], and several correlations can be found for each particular situation. For instance, for the case of particles in a gas percolating packed (ixed or moving) bed, the following are recommended [24]: • For 90 < NRe < 5,000: N Sh =

2.06 0.425 1/ 3 N Re N Sc ε

(11.43)

Auxiliary Equations and Basic Calculations

247

• For 5,000 < NRe < 10,300: N Sh =

20.4 0.185 1/ 3 N Re N Sc . ε

(11.44)

Here ε is the void fraction of the ixed or moving bed. It is important to remember that using the corresponding mass transfer parameters, the above relations can be applied to obtain the Sherwood number at each instance. For the speciic case of single spherical (or near spherical) particles in a luid low, in the range 0.6 < N Sc < 3200 (therefore valid for liquids as well) and 0.5 1.8 < N Re N Sc < 6 × 10 5, the following can be applied [25]: 0.31 N Sh = N Sh0 + 0.347 N 0Re.62 N Sc .

(11.45)

For N GrM N Sc > 108, the following holds true: 0.250 0.25, N Sh0 = 2.0 + 0.569 N Gr N Sc M

(11.46)

and for N GrM N Sc > 108, one should apply the following: 0.577 N Sh0 = 2.0 + 0.0254N 0Gr.333 N Sc . M

(11.47)

An alternative method, originally developed for convective heat transfer coeficients [26], can be used for cases of particles in a gas low. It is applicable to spherical and cylindrical particles and is given by n3 N Nu = n l n 22 N Re N1Pr/ 3 .

(11.48)

Here, N Re =

ρG U G d P 6n 2µ G (1 − ε)

(11.49)

and • • • •

n1 = 0.91, n3 = 0.49 for NRe < 50 n1 = 0.61, n3 = 0.59 for NRe ≥ 50 n2 = 1.0 for spherical particles n2 = 0.86 for cylindrical particles

Again, the above may be applied to obtain the Sherwood numbers by using the corresponding transfer parameters. As we have seen, the computations require calculations of several properties for individual gaseous species and their mixtures. These can be found in excellent publications [7, 8, 27–31]. Finally, it should be remembered that the above empirical and semiempirical formulas assume little or no interaction between heat and mass transfers. For situations

248

Solid Fuels Combustion and Gasification

where this happens, conjugate effects should be considered. This subject is beyond the scope of the present introductory text, but the interested reader can ind excellent material in the literature [19, 22].

11.8 11.8.1

ENERGY-RELATED PARAMETERS ENERGY PRODUCTION DUE TO GAS–GAS REACTIONS

It should be remembered that the nomenclature (Tables 8.1 through 8.3) reserved 500 positions for the gas components (1 ≤ j ≤ 500) and another 500 for the solidphase components (501 ≤ j ≤ 1000). Moreover, 40 positions were kept for the gas– solid reactions (1 ≤ i ≤ 40) (Table 8.4) and another 20 positions for the gas–gas reactions (41 ≤ i ≤ 60) (Table 8.5). Therefore, the rate of energy production or consumption due to gas–gas reactions was mentioned in the Equation 7.37 (Chapter 7) and is given by 500

R Q ,G = ∑ R hom , jh j,TG .

(11.50)

j=1

Here, the rate by homogeneous reactions is given by Equation 11.1. Note again that the enthalpy (hj) of a participating component j includes the formation value at 298 K and a sensible value computed between 298 K and its temperature.

11.8.2

ENERGY PRODUCTION DUE TO GAS–SOLID REACTIONS

The rate of energy production or consumption due to gas–solid reactions, indicated in Equation 7.38 (Chapter 7), is given by  dA S  1000 dz  R h R Q ,S =  het , j j, TS  dVS  ∑ j =1 dz  

(11.51)

where the rate of heterogeneous reactions is given by Equation 11.2. Note that summation includes gas and solid components because they are produced or consumed during these reactions. The correction for available area of reacting solid (Equation 11.7) against solid-phase available volume (Equation 11.8) is necessary for unit coherence with computations of RQ,S.

11.8.3

ENERGY TRANSFER DUE TO MASS TRANSFER

The rate of energy transfers associated with the mass transfer between the solid phase and the gas phase is given by

R h,G

 dA S  500 dz  R = het , j h j,TS − h j,TG .  dVG  ∑   j=1 dz

(

)

(11.52)

249

Auxiliary Equations and Basic Calculations

Since R h,G is given as function of volume of gas phase, whereas R het,j is given on the basis of area of active solid phase, it is necessary to multiply the summation by the ratio between the available area of external particles of the reacting solid and the volume of gas phase. Similarly, R h,S

 dVG   dA S  500   dz dz  R R het , j h j,TS − h j,TG = −  =−  dVS  h,G .  dVS  ∑ j =1     dz  dz 

)

(

(11.53)

It should be noted that the rate of energy transfer (R h,S) is based on volume occupied by the active solid phase.

11.8.4

ENERGY TRANSFER DUE TO CONVECTION

The rate of energy transfer by convection between gas and solid particles is given by R C,G = α SG ( TS − TG )

dA S dVG

dz .

(11.54)

dz

The convective heat transfer coeficient can be computed through the relations given in Section 11.7.

11.8.5

ENERGY TRANSFER DUE TO THERMAL RADIATION

Energy transfer due to thermal radiation is also called radiative transfer (RR) and is included in Equations 7.14, 7.15, 7.37, and 7.38 (Chapter 7). 11.8.5.1 Particle-Gas According to the irst approximation, the gas is assumed to be transparent to thermal radiation. Although this is not true for water and carbon dioxide (among others), nitrogen and oxygen are transparent for a signiicant range of thermal radiation frequencies. Therefore, no relations are presented here to account for those absorptions. The interested reader will ind plenty of material throughout the literature on radiative heat transfer. 11.8.5.2 Particle-Particle Intensive radiative and conductive heat transfers take place between particles at different temperatures. These transfers occur in all directions. In fact, the heat transfers in the axial (or vertical) direction are dissipative factors that contribute to smoothing the temperature proiles or avoiding sharp variations. However, the present introductory model considers that heat is transferred only in the radial direction and therefore between particles at the same height in the bed. Consequently, because of the basic hypothesis of the present one-dimensional model, no heat transfer occurs between particles of the same type. If the exposed-core model were the dominant process, part of the detached ash particles would be carried with the gas stream. The other part would probably ill the

250

Solid Fuels Combustion and Gasification

gaps between reacting carbonaceous particles. Nevertheless, those gaps are small, and heat transfer by conduction promptly equalizes their temperatures with those of the neighboring carbonaceous particles. On the other hand, if absorbent or inert particles are present in the bed, the radiative and conductive heat transfers (even for particles at the same height) can no longer be neglected. As this situation is more common in luidized beds, the correlations for such transfers will be presented in Chapter 15. Improvements on the present model for moving beds could include a similar treatment. 11.8.5.3 Particles and Walls Evidently, heat transfers between particles and reactor walls will take place. The model neglects this exchange because of considerations made in Chapter 7. Actually, because of wall insulation, the rate of energy lost by the bed to the walls (and from them to external environment) is usually negligible in most industrial moving-bed furnaces and gasiiers. Exceptions would be cases where water or gas jackets are installed between the external wall and the bed interior. In such situations, the heat lost by convection and radiation (and even conduction of particles in contact with the jacket) might be considerable. Therefore, an improvement is possible by adding terms on the energy balances (Equations 7.37 and 7.38). To simplify, one could account for all heat exchanges between the walls and bed via convection between gas and walls. This approach is taken for the model to be presented in Chapter 13. The interested researcher might include a similar treatment in the present model.

11.9

A FEW IMMEDIATE APPLICATIONS

With the objective of illustrating the use of some equations and methods shown so far, a few simple applications have been selected.

11.9.1

BURNOUT TIME

Take the case of a single fuel particle suspended in a gas stream containing oxygen. An estimation may be made of the time required to consume a given fraction of carbonaceous solid fuel by combustion. Of course, that generally constitutes a complex problem. Therefore, for the sake of a simpliied calculation, several assumptions can be made, such as the following: 1. Constant composition temperature and pressure of atmosphere around the particle. This is not what would be found by a particle during its travel through the furnace (no matter what type). The surrounding gas composition, temperature, and pressure would change from point to point. 2. Constant particle dimensions and approximate shape. This is not such a strong assumption, because the unexposed-core model can be acceptable as a model. 3. Constant particle temperature. This is another strong assumption because particle temperature would change from point to point. In addition, depending on its size and properties, the temperature might vary considerable throughout its radius (see comments on Biot number in Chapter 7).

251

Auxiliary Equations and Basic Calculations

4. Constant particle properties, such as densities (apparent and real), porosity, etc. Like the previous assumption, this is a strong assumption. Despite all those approximations, the following calculations might illustrate a few interesting points. The rates of gas–solid reactions, which consume components of the solid phase, have been deduced and are given in Chapter 9, by Equation 9.32 in the case of the unexposed-core model and by Equation 9.46 for the exposed-core model. On the other hand, the rate of consumption of a gas component j by a gas–solid reaction i is given by V dρ j . (11.55) ri = p A p dt Since the reaction rate is provided on the basis of its unit area, the correction should be made by multiplying the right side by the particle volume and dividing by its area. Now take the case of the oxidation of the carbon, which is here simpliied and written as C + O2 → CO2.

(11.56)

Therefore, if in this case j represents the oxygen, the same equation is valid for carbon. The concept of fractional carbon conversion, or simply carbon conversion, f, is given by ρ f = 1 − 514 (11.56) ρ 514,I where the denominator is the original concentration of carbon in the particle and the numerator is the concentration at a given instant or position. Therefore, Equation 11.55 can be written as V df ri = p ρ 514,I . (11.57) Ap dt Integration from the instant at which the carbonaceous particles are put into contact with the oxidant to a given instant θ leads to f

θ=

Vp df ρ 514, I ∫ . Ap r 0 i

(11.58)

For a gaseous atmosphere with constant conditions involving the spherical (or near spherical) particles, Equation 9.32 or 9.46 can be used to write θ=

3 f d 2p, Iρ 514, I ∑ U kdf . 12(ρ j,∞ − ρ j,eq ) k =1 ∫0

(11.59)

Here, the various resistance can be computed either using the exposed-core or the unexposed-core model. Of course, the exposed-core particles should lead to a smaller burning time than the unexposed-core particles.

252

Solid Fuels Combustion and Gasification

Let us integrate for the unexposed-core case. Combining Equations 9.33, 9.34, 9.35, and 11.59, one arrives at θ=

d 2p, I ρ 514 , I 12 ρ j,∞ − ρ j,eq

(

)

Φ   3 − 3(1 − f )2 / 3 − 2f f 2 db + +   (11.60) ∫ 2D j, A D j, N Φ aΦ b coth( b) − 1   N Sh D j,G

with parameter a given by Equation 9.56. The above integration can easily be performed. As expected, θ is 0 when f equals 0 (a = 1). Equation 11.60 can now be used to estimate the residence time required to achieve a desired level of carbon conversion. However, it should be remembered that the above computation can be applied only if the conditions set at the beginning of these deductions are satisied. Therefore, the above solution cannot be used for classical cases such as moving bed, luidized bed, or suspension combustion of carbonaceous solid particles. Nevertheless, and as shown below, it allows some preliminary estimates in particular situations.

11.9.2 BOILER WITH A CHAIN OR ROLLING GRATE The application of Equation 11.60 is illustrated for the case of a boiler where coal or biomass particles are burned on a rolling or chain grate. A simple representation of the grate of this equipment is shown in Chapter 3, Figure 3.7c. Once the power output of the boiler is decided, the mass low of coal (or other carbonaceous solid) can be set. For this, the eficiency of such systems should be known, as well as the usual level of carbon conversion. To show an example, let us assume some values and criteria. It should be stressed that the following values, criteria, or assumptions may not be the best or even the recommended ones for engineering design and are used here only to illustrate the application of few concepts. Suppose a boiler is consuming F (kg/s) of coal. If the intended carbon conversion is known,* the grate basic dimensions, its velocity, and the height of coal layer can be calculated. For this, the following steps can be used: 1. A boiler is designed to operate with a given oxygen (or air) excess ae. Low air excesses would require less power input to the compressor or fan, but they would lead to lower carbon conversions. On the other hand, higher air excesses would lead to greater power input but might provide higher carbon conversions. In addition, it should be remembered that air excesses that are too high would also lead to relatively low temperatures in the bed and therefore relatively low carbon conversions. Usual values for the air excess in cases of solid combustors are around 40%, or ae = 0.40. Therefore, if the excess level is given, the mass low of air (Fair) to be blown through the grate can be approximated by Fair = (1 + a e )Fst −air. *

(11.61)

The criterion for this is not within the scope of the present course, but usual values surpass 95%.

Auxiliary Equations and Basic Calculations

253

The total air low for complete (or stoichiometric) combustion of carbonaceous solid is approximately given by Fst −air = 11.47(1 − w 700 − wash )F.

(11.62)

Here, w700 and wash are the moisture and ash fractions, respectively, in the feeding coal or carbonaceous solid. 2. Assume a carbon conversion f. Usual values vary from 0.92 to 0.99 (or 92% to 99%). Depending on the requirements of the intended equipment or process, other levels might be used. 3. Assume an average temperature in the bed. Values around 1300 K are commonly found in the present type of combustor. However, care should be taken not to surpass the ash-softening temperature. The average temperature in the bed can be recalculated by an energy balance in the bed, assuming a well-mixed reactor. If the layer of particles is not too thick, the average temperature of the bed should fairly represent the temperatures of particles throughout the bed. 4. As during their travel on the grate, the particles should be maintained as in a ixed bed, and low stress between the particles is found in such cases. Hence, the unexposed-core model can be applied. In addition, the supericial velocity of the gas must not surpass the minimum luidization velocity. The diameter should be computed from the particle size distribution (see Chapter 4). It might be argued that it would be better for the supericial velocity not to surpass the luidization velocity of the smallest particle. Nevertheless, this last criterion may lead to air velocity that is too low, which would require larger grate areas. Consequently, a compromise must be established. Of course, some particles will be carried with the gas, and the entrainment rate is a variable of boiler design. The apparent density of particles at the end of their journey can be approximated by  w fix − carb  ρp,app,f =  wash ,I + (1 − f )  ρp,app,I . w fix − carb + w V  

(11.63)

Here, wash,I, wix-carb, and wV are the fractional ash, ixed carbon, and volatile content in the feeding coal, respectively. 5. Using the equations shown in Chapter 4, the maximum supericial velocity of the air can be set as the minimum luidization velocity of average particle size, with density given by Equation 11.63. It should be noted that the maximum velocity of the gas would be found near the top layer, where it reaches higher temperatures (around 1300 K). Therefore, the injection velocity of the air would be smaller than that. A fair method might assume the air injection velocity (UG) equal to the minimum luidization velocity multiplied by the ratio between the air injection temperature and the top layer temperature. 6. With the value of air mass low (Fair) and its maximum allowed supericial injection velocity UG, the area of the grate can be computed by

254

Solid Fuels Combustion and Gasification

Fair (11.64) U Gρair where the air density is computed at the average temperature in the bed. 7. The minimum and maximum residence times to achieve conversion f can be determined. The maximum would be required by the largest particles at the top of the coal layer. The minimum would be required by the smallest particles at the bottom of the layer. Usually, the most signiicant value is the required residence time for the average size particle, at the top layer, to achieve desired conversion level f. The average size should be computed from the size distribution of feeding particles. Equation 11.60 can now be used to calculate the residence time. Note that the periods for heating up particles, as well as for drying and devolatilization, are not included here. However, at the usual temperatures found for grate combustion, those periods are negligible compared with those necessary for almost complete particle consumption. For instance, total devolatilization of small coal* particles at 1000 K requires just 7 minutes [32], whereas high levels of ixed carbon conversion may take much longer. The example presented below arrives at necessary residence time around 1 hour. In any case, computations may be improved by considering heating, drying, and pyrolysis. 8. To use Equation 11.60, the following may be used: A grate =

• The approximate oxygen concentration in the air surrounding the particles at the top of the coal layer can be provided by ρ O2 ,∞ = 0.21

ae P .  1 + a e RT

(11.65)

• The equilibrium oxygen concentration ρ O2 ,eq equals 0. This is fairly true for carbon combustion. • Concentration of carbon in feeding particles is approximated by ρ 514,I =

• • • • *

ρp,I wfix − carb . 12

(11.66)

This is reasonable because all other solid-phase components—such as H, O, N, and S—would also be oxidized. Parameter a can be calculated from Equation 9.56 assuming nearly spherical particles, or p = 2. Dj,G equals DO2,G (here, oxygen is the j reacting component). As seen before, the diffusivity can be approximated by Equation 11.18. The effective diffusivity of the gas through porous ash and nucleus can be evaluated by the equations in Section 11.4.2. Depending on the low regime, Equation 11.43 or 11.44 can be used to calculate the Sherwood number and the mass transfer coeficient.

For a reference, Niksa [37] indicates that below 1 mm, pyrolysis yields and product distributions are independent of particle sizes.

255

Auxiliary Equations and Basic Calculations

• The Thiele modulus is given by Equation 9.8 (Chapter 9). As the reaction involved here is R.1, the reaction order n is 1, and the reaction coeficient calculated by  E  k1 = k 0 ,1 exp  − 1  .    RT

(11.67)

The pre-exponential and activation energies of reaction R.1 are given in Table 8.7 (Chapter 8). • For a desired conversion f, calculate θ from Equation 11.60. 9. Once the area of the grate has been computed by Equation 11.64 and its length/width ratio is known,* the required velocity for the grate is simply given by v grate =

L grate . θ

(11.68)

10. The height of the coal (or carbonaceous solid) layer over the grate can be computed by y grate =

F . v grate W(1 − ε)ρp, I

(11.69)

Here, W is the width of the grate. The void fraction of the bed can be set as approximately 0.5. 11. Finally, the grate load of the grate can be approximately calculated by grate load =

H HV F A grate

(11.70)

where HHV is the high heating value (J/kg) of the fuel. The reader should note that this parameter assumes total consumption of the carbonaceous fuel. To illustrate the above procedure, let us imagine that one intends to build a rolling grate boiler to consume 10 ton/hour (2.778 kg/s) of coal. Let the coal properties be those described in Problem 4.3 (Chapter 4), with the following addition information regarding the process: • • • • • • *

Air excess: 40% Desired ixed carbon conversion: 99% Average particle diameter: 10 mm Grate length/width ratio: 2 Air injected at ambient temperature Average bed temperature: 1300 K

Equipment manufacturers set standard or usual values. Sometimes, that ratio is dictated by the layout of the factory where it is going to be installed.

256

Solid Fuels Combustion and Gasification

Calculate the basic dimensions of the grate, its velocity, and the height of the solid fuel on the grate. The following values have been obtained: • • • • • • • • • • • • • • • • • • •

Mass low of air (Fair): 38.18 kg/s Supericial velocity of the air (at 1300 K): 0.911 m/s Area of the grate: 154.7 m2 Length of the grate: 17.59 m Width of the grate: 8.79 m Concentration of oxygen above the grate: 5.625 × 10 –4 kmol/m3 a = 0.215 Dj,G = 2.410 × 10 –4 m2/s Dj,A = 2.268 × 10 –4 m2/s Dj,N = 9.64 × 10 –5 m2/s k1 = 2.608 × 105 s–1 Φ = 164.5 First term between the brackets on the right side of Equation 11.60: 2893 s/m2 Second term between the brackets on the right side of Equation 11.60: 1942 s/m2 Third term between the brackets on the right side Equation 11.60: 195 s/m2 θ = 4138 s (or 1.15 hours) vgrate = 4.25 mm/s ygrate = 10 cm If the coal has a heating value around 31 MJ/kg (high heating value), the grate load would be around 560 kW/m2

Some comments follow: a) Despite a different approach from that usually taken by boiler designers, the above methodology leads to grate loads comparable to values recommended by furnace manufacturers. b) The calculation allows veriication of the relative inluence of the various resistances inluencing the rate of solid fuel consumption. Those are represented by the integrals in Equation 11.60. In the above particular case, the diffusion through the gas boundary layer surrounding the particles provided the greatest resistance. The one representing chemical kinetics combined with the diffusion through the core led to the smallest value. This is mainly because the carbon-oxygen reaction is very fast at 1300 K. The situation would be different for lower temperatures or if very low oxygen concentrations were available around the particles. In such cases, most attacks to the carbonaceous core would be carried by other gases, such as H2O or CO2. As we have seen, the above method cannot be applied, for instance, in cases of gasiication. If gasiication is intended, the residence time should consider the contribution of terms where each reaction involved must be considered at each point of the coal (or other carbonaceous) layer. A program to compute the gas and solid composition proiles through the bed at various positions of its horizontal and vertical coordinates could provide much better evaluations.

Auxiliary Equations and Basic Calculations

257

c) Of course, there are several ways of inluencing the residence periods for a desired conversion of solid fuels. For instance, the required residence time can be decreased if the average particle size is decreased. However (see Step 5), this would lead to reductions in the maximum gas supericial velocity UG allowed through the bed, leading, of course, to an increase in the grate area. In addition, the cost of grinding increases when smaller particles are desired. d) Another important variable for the design of equipment geometry and operational conditions is the pressure losses imposed on the gas stream passing through the grate and bed of particles. For instance, savings in capital costs due to smaller or more compact equipment could be jeopardized by increases in costs from the additional or more powerful compressors or fans needed to overcome pressure losses in grates and distributors.

11.10

PRESSURE LOSSES

The prediction of pressure losses imposed on luids passing through equipment or systems is extremely important for engineering design. It allows, among other details, the determination of pumping or blowing power requirements. In the case of packed beds, the total pressure loss in the passing luid is given by losses in the distributor or grate and the losses during low through the bed. The following sections present a few methods for computation of such losses, which have been adapted from several texts [27, 33–36]. Throughout the present book, the correlations have been developed strictly in SI (International System) units.

11.10.1

PRESSURE LOSS IN A BED OF PARTICLES

The methods by which to calculate pressure losses of gases lowing through ixed and luidized beds are shown below. 11.10.1.1 Fixed Beds As illustrated by Figure 3.2, the pressure loss in a gas passing through a ixed bed of particles is proportional to the velocity of the gas stream. This is valid in cases of constant temperature throughout the bed. The methods assume the following parameters or variables as known: • The average diameter (dp) of the particles in the bed • The bed height (zD) • The supericial velocity (U0), average temperature (T0), and pressure (P0) of the gas stream entering the bed (before injection) (Figure 11.1) • The average temperature (T1) of the gas stream leaving the bed The following steps describe the method to calculate the pressure drop in a ixed bed of particles:

258

Solid Fuels Combustion and Gasification U1, T1, P1

Bed of particles

zD

U0, T0, P0

FIGURE 11.1 Schematic showing the variables for the computation of pressure loss in a bed of particles.

1. If the average void fraction in the bed (ε) is not known, adopt it as 0.45. This value is easily determined by the following relation: ε = 1−

ρbulk . ρapp

(11.71)

The deinitions of various particle-related densities are presented in Chapter 4. 2. Average temperature in the bed is calculated from Tav =

T0 + T1 . 2

(11.72)

3. For the present calculation, the average velocity through the bed is deined as U av = U 0

Tav . T0

(11.73)

Note that for ixed or moving beds, the above velocity should be below the minimum luidization (see Chapter 4). 4. Gas density (ρG), viscosity (µG), and any other property are estimated at the average temperature using the methods presented in the literature [7, 8, 29, 30]. 5. The Reynolds number is given by N Re =

ρG U avd p . µG

(11.74)

µ 2G . d 3pρG

(11.75)

6. Parameter a1 is calculated by a1 = N Re

Auxiliary Equations and Basic Calculations

259

7. The coeficient a2 is given by a 2 = N Re

U avµ G . d 2p

(11.76)

8. Dimensionless parameter a3 is given as a3 =

P0 . P

(11.77)

where P is the average pressure in the bed. A reasonable irst approximation is to take a3 = 1. 9. The dimensionless parameter a4 by the Ergun equation [35], or  1− ε  (1 − ε)2 a 4 = a 3 150 a1 + 1.75 3 a 2  . 3 ε ε  

(11.78)*

10. Finally, the pressure loss in the bed is calculated by ∆PD = a 4z D .

(11.79)

As the SI system is used, the pressure loss will be given in Pascals (Pa). A few observations should be made regarding the above method: • It assumes a linear (or nearly linear) temperature proile in the bed. As shown by Figure 3.6 (Chapter 3), this is not the case in a moving-bed gasiier or combustor. In such equipment, the expansion near peaks of temperature leads to higher gas velocities than average considering just the entering and exiting values. Therefore, the actual pressure loss through the bed is larger than that calculated by the above method. However, the computation can be substantially improved if the temperature proile is known. The deviation may be decreased by dividing the bed into sections where the temperature variation might be assumed to be almost linear. For instance, using Figure 3.5, the section from the bottom to the peak of temperature might be used as one section and the rest as another. • It assumes that the composition of the gases through the bed does not vary. Actually, the composition in a combustor or gasiier changes dramatically as the gas stream crosses the bed. It is recommended to use an average composition of the gas. In this case, it is believed that a simple arithmetical average should allow reasonable values for computations of pressure losses. • Under laminar conditions (NRe < 1), the term a1 in the Ergun equation, Equation 11.78, is the most inluential, whereas the term a2 is the important one under turbulent conditions (NRe > 1000).

*

The general Ergun equation, Equation 4.9, should be used in cases of nonspherical particles.

260

Solid Fuels Combustion and Gasification

11.10.1.2 Fluidized Beds Equation 11.78 is also useful in computing gas pressure losses through the luidized bed. However, one should be careful because the void fraction ε should be the average found in the luidized bed. As seen in Figure 3.2 (Chapter 3), once the supericial velocity reaches the minimum luidization, the pressure loss in a luidized bed remains the same throughout the entire bubbling regime. Therefore, if the minimum luidization void fraction and minimum luidization velocity were used to substitute for ε and U0, respectively, the method above would provide a irst approximation for the pressure drop. The correct correlations for the void fractions under minimum luidization conditions, as well as the average in any other situation, are described in Chapter 14. Having in mind that the luidization is a process in which the bed of particles is lifted by the gas stream, the pressure loss can alternatively be estimated by the weight of the bed (particles in the bed) distributed on the distributor plate. Therefore, if the mass held by the bed is known, the computation is straightforward. In view of possible mistakes or approximations used in computations of void fractions, it is advisable to calculate the pressure loss in the gas phase through a luidized bed by this second alternative.

11.10.2

PRESSURE LOSS IN A DISTRIBUTOR

Several types of gas distributors can be used in pilot and industrial-scale packed beds. Among the most common are perforated plate, porous plate, and perforated tubes or lutes. Various other designs might be considered variations of these conigurations. 11.10.2.1 Perforated Plate A schematic of a perforated plate is shown in Figure 11.2. The computation of pressure loss in a gas passing through a perforated plate can be made by the procedure described below. Referring to Figure 11.2, there are two possibilities: Hplate/dorif ≤ 0.015 and Hplate/dorif > 0.015. These are subdivided into cases related to the Reynolds number. The following steps should be taken: 1. Calculate the Reynolds number as follows: N Re =

ρG U orif d orif µG

(11.80)

where Uorif is the average velocity of the gas trough the oriices. If the area of the plate through which the gas lows is called A0 and the total area of the plate A1, that velocity is given by U orif = U G

A1 . A0

(11.81)

2. Calculate the ratio between the lowing area (A0) and the total area of the plate (A1) given by A a1 = 0 . (11.82) A1

261

Auxiliary Equations and Basic Calculations Orifices

Bed

Hplate

Plenun

FIGURE 11.2 Schematic of a perforated plate as distributor.

3. If Hplate/dorif ≤ 0.015 and Nre > 105, then determine the coeficient a2 by the following relationship:

(

a 2 = 0.707 1 − a1 + 1 − a1

)

2

1 . a12

(11.83)

4. If Hplate/dorif ≤ 0.015 and Nre ≤ 105, then perform the following steps: 4.1. Use the Reynolds number, given by Equation 11.80, to obtain parameter a3 using the following equation: a 3 = 0.44178 b1 − 0.217159 b12

(11.84)

+ 0.0574618 b13 − 4.83879 × 10 −3 b14 . where b1 = log10 (N Re ).

(11.85)

4.2. Evaluate parameter a4 through the formula a 4 = 5.79038 − 0.735783 a 1 + 0.0369379 a 12 − 5.64686 b1 −2

(11.86)

+ 2.27944 b − 0.409812 b + 2.72761 × 10 b . 2 1

3 1

4 1

If the calculated value for a4 is negative, assume it as 0. 4.3. Calculate a5 from a 5 = 1 + 0.707 1 − a1 .

(11.87)

4.4. Obtain parameter a2 by a2 =

a 4 + a 3 (a 5 − a1 )2 . a12

(11.88)

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Solid Fuels Combustion and Gasification

5. If Hplate/dorif > 0.015 and Nre > 105, then perform the following steps: 5.1. Determine coeficient a6 by the simple following relation a6 =

H plate . d orifice

(11.89)

5.2. For a6 ≤ 2, calculate a7 by the formula a 7 = 1.4746 − 1.6131a 6 + 0.6326 a 62

(11.90)

− 0.114 a + 0.0096 a − 0.0003 a . If a6 > 2 or a7 < 0, take a7 = 0. 5.3. Estimate a8 using 3 6

4 6

5 6

 100  a 8 = 0.1  1.46a 9 + N Re  

0.25

(11.91)

where a9 is the relative roughness of the internal walls of the oriices of the perforated plate. In the absence of a value, take the roughness as 0.01. 5.4. Compute a10 by

(

)

a10 = 0.5 + a 7 1 − a1 (1 − a1 ) + (1 − a1 )2.

(11.92)

5.5. Determine a2 from a 2 = ( a10 + a 8a 6 )

1 . a12

(11.93)

6. If a6 > 0.015 and Nre < 105, then calculate a 2 = ( a 4 + a 3a10 + a 8a 6 )

1 . a12

(11.94)

7. Estimate the pressure loss in the distributor ΔPdist as ∆Pdist = a 2

ρG U 2G . 2

(11.95)

11.10.2.2 Porous Plate For computation of pressure loss in a gas stream through a porous plate, the following procedure may be followed: 1. The porosity of the plate ε is usually known and should vary between 0.15 and 0.50 m3 (of pores)/m3 (of plate). In the absence of a speciic value, use 0.430. 2. Given T0 and T1, or the gas temperature at positions before and after the porous plate, calculate the average temperature by Equation 11.72.

Auxiliary Equations and Basic Calculations

263

3. Given the gas velocity U0 at T0, the average velocity should be obtained using Equation 11.73. 4. Obtain the gas density (ρG) and viscosity (µG) at the average temperature. 5. Calculate the Reynolds number as N Re =

1 ρG U avd pore ε µG

(11.96)

where dpore is the average pore diameter. Typical diameters of pores vary from 2 × 10 –4 to 7 × 10 –6 m. That should be determined in the laboratory. If no speciic value is available, use 4 × 10 –5 m. 6. If NRe < 3, calculate the parameter a11 by the correlation with NRe as a11 =

180 . N Re

(11.97)

7. If NRe > 3, calculate the parameter a11 by the correlation with NRe as a 11 =

164 7.68 + . N Re N 0Re.11

(11.98)

8. The dimensionless parameter a12 is given by a 12 = 2

T1 − T0 . Tav

(11.99)

9. Evaluate the dimensionless parameter a2 by a 2 = a 11

H plate + a 12 . d pore

(11.100)

The calculation of pressure loss is very sensitive to that diameter. Therefore, a laboratory determination of the average pore diameter is very important. 10. Finally, using a2 given by Equation 11.100, estimate the pressure loss in the plate by Equation 11.95. 11.10.2.3 Perforated Tubes or Flutes A simpliied schematic of this kind of distributor is shown in Figure 11.3. This case is composed of two main pressure losses: • The one due to low inside the tubes of the lutes crossing the insulation (the calculation is similar to the perforated plates [Figure 11.2]). • The one due to low through the side oriices of the lute.

264

Solid Fuels Combustion and Gasification Bed

Orifices

Hdist

Flutes

Plenun

FIGURE 11.3

Schematic showing the lute tubes that compose the distributor.

The method for estimation of those pressure losses is as follows: 1. Compute the average gas velocity inside each lute tube (Ulute) by a simple relation between the supericial velocity for the empty bed and the area available for gas to cross the distributor plate through the lute tubes. 2. Estimate the pressure loss as a perforated plate. Use the method shown above, where Ulute would replace Uorif, and dlute (internal diameter of the lute) would replace dorif. Therefore, the Reynolds number should be given by N Re − flute =

ρG U fluted flute . µG

(11.101)

In the above, the coeficient a2 (to be used in Equation 11.95) for pressure loss at the lute has been obtained. The Reynolds number, as calculated by Equation 11.101, is now used for computations that would allow estimating the pressure loss in the gas through the oriices. 3. If NRe-lute < 4 × 104 3.1. Take a11 = 1 3.2. Determine a12 by  100  a12 = 0.1  1.46a 9 + N Re − flute  

0.25

(11.102)

where a9 is the relative roughness of the internal walls of the oriices of the perforated plate. In the absence of a value, take the roughness as 0.01. 3.3. The coeficient a13 is given by a13 = 45a12 4. If NRe-lute ≥ 4 × 104, take a11 = 1.5 and a13 = 1.1.

(11.103)

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Auxiliary Equations and Basic Calculations

5. Calculate a14 by a14 = 1.19a11a13.

(11.104)

6. Compute a15, which is the ratio between the area of oriices in a lute and the cross-section of the lute. 7. Evaluate the Reynolds number at oriice by N Re − orif =

N Re − flute . a15

(11.105)

8. Parameter a16 is given by a16 = 1 − a15.

(11.106)

If a16 less than 0, assume it as 0. 9. Calculate a17, given by a17 = 1 + 0.5a16 + 1.35 a16 .

(11.107)

2 3 4 a18 = 0.44178b18 − 0.217159b18 + 0.0574618b18 − 0.00483879b18

(11.108)

10. Obtain a18 by

where

b18 = log10 ( N Re − orif ).

(11.109)

If a18 acquires a negative value, take it as 0. 11. Compute 2 a 19 = 5.79038 − 0.735783a 15 + 0.0369379a 15 2 3 − 0.409812 b18 + 0.0272761b148 . − 5.64686 b18 + 2.27944 b18

(11.110)

If a19 acquires a negative value, take it as 0. 12. The pressure loss coeficient a20 should be obtained through a 20 = (1 + a18 )a17 + a19.

(11.111)

13. The total pressure loss coeficient is estimated using 2

2

U  U  a 2 = a 20 + a14  flute  + a17  orif  .  UG   UG 

(11.112)

14. Finally, Equation 11.95 should be used to determine the pressure loss in the gas through the lute duct and side oriices. Of course, it should be added to the one calculated from item 2 above.

266

Solid Fuels Combustion and Gasification

11.11 11.11.1

EXERCISES PROBLEM 11.1*

Estimate the rates of Reactions R.1 and R.2 (Chapter 8) by the two possible models (unexposed-core and exposed-core models) under the following conditions: • Composition of the gas that surrounds the particle (molar fractions): CO2 = 0.030; CO = 0.045; O2 = 0.100; H2O = 0.230; H2 = 0.35 × 10–4; N2 = 0.595 • Solid fuel composition (wood), immediate analysis (% mass, wet basis): moisture = 5%, volatile = 38%, ixed carbon = 47.6%, ash = 9.4%; ultimate analysis (mass fraction, dry basis): C = 0.732, H = 0.0510, O = 0.0790, N = 0.0090, S = 0.0300, ash = 0.0990 • Pressure = 100 kPa • Temperature = 1200 K • Bed bulk density = 700 kg m–3 • Particle apparent density = 1200 kg m–3 • Particle real density = 2100 kg m–3 • Original particle diameter = 10 mm • Solid fuel conversion = 50% • Gas velocity = 0.28 m s–1

11.11.2

PROBLEM 11.2**

For the case of the unexposed-core model, calculate the respective effectiveness factor.

11.11.3

PROBLEM 11.3*

Repeat the computations for combustor grate design as shown in Section 11.9.2, but for a particle diameter twice that used there. Compare the results for burning time and grate velocity.

11.11.4

PROBLEM 11.4**

Write computer routines to determine pressure losses in gas streams passing through perforated plates. Be as general as possible and assume the minimum number of variables.

11.11.5

PROBLEM 11.5**

Repeat the last problem for the case of porous plates. Compute the values for a case where ambient air is passing through a 2-cm-thick plate. Assume a plate porosity of 0.3, a grain diameter of 6 × 10 –5 m, and a supericial velocity of air of 1 m/s.

11.11.6

PROBLEM 11.6**

Repeat Problem 11.4 for a distributor composed of lutes. Propose a design for the lutes and their distribution in the gas distributor. Be sure to avoid impossible situations such as using more than the available area for oriices in each lute

Auxiliary Equations and Basic Calculations

267

or more than the available distributor area to accommodate the lutes. Sensible design should also allow spaces between lutes for proper cleaning and maintenance. As a rule of thumb, the distance between neighboring lutes should be at least twice the diameter of lutes.

11.11.7

PROBLEM 11.7*

Using the routine developed in Problem 11.5 and for a case where ambient air is passing through the 2-cm-thick plate (with 0.3 porosity), plot a graph of pressure losses against air velocity within the range of 0.1–10 m/s.

11.11.8

PROBLEM 11.8***

Try to: a) b)

11.11.9

Write the radiative heat transfer terms between particles and particles if at different temperatures. Consult Chapter 15. Include those into the energy balances (Chapter 7, Equations 7.37 and 7.38) of moving-bed combustors and gasiiers.

PROBLEM 11.9***

Follow the suggestion made for improvement in Section 11.8.5.3 and include terms for heat transfers between bed and walls in Equations 7.37 and 7.38.

11.11.10

PROBLEM 11.10****

Try to: a) b) c)

Rewrite the basic energy equations presented by Chapter 7 by including the terms for axial heat transfers. Using a simple treatment for radiative heat transfer (see Chapter 15), add the terms of radial and axial radiative heat transfers between particles in those equations. Insert the heat transferred by conduction between particles in the treatment.

REFERENCES 1. de Souza-Santos, M.L., Development of a Simulation Model and Optimization of Gasiiers for Various Fuels (Desenvolvimento de Modelo de Simulação e Otimização de Gaseiicadores com Diversos Tipos de Combustíveis), IPT-Inst. Pesq. Tec. Est. São Paulo, SCTDE-SP, Report No. 20.689, DEM/AET, São Paulo, Brazil, 1985. 2. de Souza-Santos, M.L., and Jen, L.C., Study of Energy Alternative Sources; Use of Biomass and Crop Residues as Energy Source: Part B: Development of a Mathematical Models and Simulation Programs for Up-Stream and Down-Stream Moving Bed Gasiiers. (Estudo de Fontes Alternativa de Energia. Parte B), IPT-Institute for Technological Research of São Paulo, Report No. 16.223-B/DEM/AET, São Paulo, Brazil, 1982. 3. Jen, L.C., and de Souza-Santos, M.L., Modeling and simulation of ixed-bed gasiiers for charcoal (Modelagem e simulação de gaseiicador de leito ixo para carvão), Brazilian Journal of Chemical Engineering (Revista Brasileira de Engenharia Química), 7(3–4), 18–23, 1984.

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4. Yu, Q., Brage, C., Chen, G., Sjöström, K., Temperature impact on the formation of tar from biomass pyrolysis in a free-fall reactor, J. Analytical and Applied Pyrolysis, 40–41, 481–489, 1997. 5. Brage, C., Yu, Q., and Sjöström, K., Characteristics of evolution of tar from wood pyrolysis in a ixed-bed reactor, Fuel, 75(2), 213–219, 1996. 6. Field, M.A., Gill, D.W., Morgan, B.B., and Hawksley, P.G.W., Combustion of pulverized coal, Brit. Coal Utiliz. Res. Assoc. Mon. Bull., 31(6),285–345, 1967. 7. Poling, B.E., Prausnitz, J.M., and O’Connell, J.P., The Properties of Gases and Liquids, 5th Ed., McGraw-Hill, New York, 2000. 8. Reid, C.R., Prausnitz, J.M., and Poling, B.E., The Properties of Gases and Liquids, 4th Ed., McGraw-Hill, New York, 1987. 9. Walker, P.L., Jr., Rusinko, F., Jr., and Austin, L.G., Gas reactions of carbon, in Advances in Catalysis, Vol. XI, Academic Press, New York, 1959, 133–221. 10. de Souza-Santos, M.L., Modelling and Simulation of Fluidized-Bed Boilers and Gasiiers for Carbonaceous Solids, PhD thesis, University of Shefield, Shefield, United Kingdom, 1987. 11. de Souza-Santos, M.L., Comprehensive modelling and simulation of luidized-bed boilers and gasiiers, Fuel, 68, 1507–1521, 1989. 12. de Souza-Santos, M.L., Application of comprehensive simulation to pressurized bed hydroretorting of shale, Fuel, 73, 1459–1465, 1994. 13. de Souza-Santos, M.L., Application of comprehensive simulation of luidized-bed reactors to the pressurized gasiication of biomass, Journal of the Brazilian Society of Mechanical Sciences, 16(4), 376–383, 1994. 14. DeSai, P.R., and Wen, C.Y., Computer Modeling of the MERC Fixed Bed Gasiier, U.S. Department of Energy Report, MERC/CR-78/3, Morgantown, WV, March 1978. 15. Yoon, H., Wei, J., and Denn, M.M., A model for moving-bed coal gasiication reactors, AIChE J., 24(5), 885–903, 1978. 16. Davini, P., DeMichele, G., and Bertacchi, S., Reaction between calcium-based sorbents and sulphur dioxide: a thermogravimetric investigation, Fuel, 70, 201–204, 1991. 17. Elias-Kohav, T., Sheintuch, M., and Avnir, D., Steady-state diffusion and reactions in catalytic fractal porous media, Chem. Eng. Sci., 46(11), 2787–2798, 1991. 18. Kasaoka, S., Sakata, Y., and Tong, C., Kinetics evaluation of reactivity of various coal char for gasiication with carbon dioxide in comparison with steam, Intern. Chem. Eng., 25(1), 160–175, 1985. 19. Bird, R.B., Stewart, W.E., and Lightfoot, E.N., Transport Phenomena, John Wiley, New York, 1960. 20. Slattery, J.C., Momentum, Energy, and Mass Transfer in Continua, Robert E. Krieger, New York, 1978. 21. Incropera, F.P., and DeWitt, D.P., Fundamentals of Heat and Mass Transfer, 4th Ed., John Wiley, New York, 1996. 22. Luikov, A.V., Heat and Mass Transfer, Mir, Moscow, 1980. 23. Treybal, R.E., Mass Transfer Operations, 2nd Ed., McGraw-Hill, New York, 1968. 24. Gupta, A.S., and Thodos, G., Direct analogy between mass and heat transfer to beds of spheres, AIChE J., 9(6), 751–754, 1963. 25. Steinberger, R.L., and Treybal, R.E., Mass transfer from a solid soluble sphere to a lowing liquid system, AIChE J., 6, 227, 1960. 26. Kothari, A.K., MSc thesis, Illinois Institute of Technology, Chicago, 1967. 27. Perry, J.H., Green, D.W., and Maloney, J.O., in J.H. Perry, ed., Perry’s Chemical Engineers Handbook, 7th Ed., McGraw-Hill, New York, 1997, 12-1–12-90. 28. Alemasov, V.E., Dregalin, A.F., Tishin, A.P., and Khudyakov, V.A., Computation methods, in Thermodynamic and Thermophysical Properties of Combustion Products, Glushko, V.P., Ed., Vol. I, Viniti, Moscow, 1971.

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269

29. Yaws, C.L., Hood, L.D., Gorin, C., Thakore, S., and Miller, J.W., Jr., Correlation constants for chemical compounds, Chemical Engineering, 16, 79–87, 1976. 30. Yaws, C.L., Schorr, G.R., Shah, P.N., and Miller, J.W., Jr. Correlation constants for chemical compounds, Chemical Engineering, 22, 153–161, 1976. 31. Yaws, C.L., Schorr, G.R., Shah, P.N., McGinley, J.J., and Miller, J.W., Jr., Correlation constants for liquids, Chemical Engineering, 25, 127–135, 1976. 32. Smoot, L.D., and Pratt, D.T., Pulverized-Coal Combustion and Gasiication, Plenum Press, New York, 1979. 33. Kunii, D., and Levenspiel, O., Fluidization Engineering, 2nd Ed., John Wiley, New York, 1991. 34. Geldart, D., Gas Fluidization Technology, John Wiley, Chichester, United Kingdom, 1986. 35. Ergun, S., Fluid low through packed columns, Chemical Engineering Progress, 48, 91–94, 1952. 36. Idel’cik, I.E., Memento de Pertes de Charge, Eyrolles, Paris, 1960. 37. Niksa, S., Rapid coal devolatilization as an equilibrium lash distillation, AIChE J., 34, 790–802, 1988. 38. Adánez, J., Miranda, J.L., and Gavilán, J.M., Kinetics of a lignite-char gasiication by CO2, Fuel, 64, 801–804, 1985.

Simulation 12 Moving-Bed Programs and Results CONTENTS 12.1 Introduction ................................................................................................. 271 12.2 From Model to Simulation Program ........................................................... 271 12.2.1 Simulation Strategy ....................................................................... 274 12.2.2 Programming Language ................................................................ 274 12.2.3 Program Writing ............................................................................ 275 12.3 Updraft Moving Bed ................................................................................... 275 12.3.1 Program Chart ............................................................................... 276 12.3.2 Simulation Results ......................................................................... 278 12.3.3 Comments ...................................................................................... 279 12.4 Downdraft Moving Bed .............................................................................. 286 12.4.1 Program Chart ............................................................................... 288 12.4.2 Simulation Results and Discussion................................................ 289 12.5 Exercises ..................................................................................................... 295 12.5.1 Problem 12.1 .................................................................................. 295 12.5.2 Problem 12.2 .................................................................................. 295 12.5.3 Problem 12.3 .................................................................................. 296 12.5.4 Problem 12.4 .................................................................................. 296 12.5.5 Problem 12.5 .................................................................................. 296 12.5.6 Problem 12.6 .................................................................................. 296 12.5.7 Problem 12.7 .................................................................................. 296 12.5.8 Problem 12.8 .................................................................................. 296 12.5.9 Problem 12.9 .................................................................................. 296 References .............................................................................................................. 297

12.1

INTRODUCTION

This chapter presents considerations on how to organize a model to build a simulation program. The example of a moving-bed combustor or gasiier is used to illustrate this project. In addition, results from simulations and comparisons against real operations are shown.

12.2

FROM MODEL TO SIMULATION PROGRAM

The basic and auxiliary equations and correlation necessary for modeling a movingbed gasiier have been described. The next step is to build the simulation program, in which these equations and correlation should be arranged in an orderly way. 271

272

Solid Fuels Combustion and Gasification

They should also work together without mathematical or logical contradictions; otherwise, the computational program would not be completed or would lead to false or absurd results. From a general point of view, the stage of development of computer software, particularly for engineering applications, is still in its infancy compared with the level of industrial production. Even with advances in coding performed by other programs, many programming techniques are comparable to the industrial equivalent of musket production [1]. Most codes are produced by experienced researchers one at a time; therefore, it is still a handicraft job. Although this may sound romantic, it leads to serious consequences when eficiency, safety, and reliability are concerned. For example, the opening of a large international airport in the United States was delayed for more than a year because of bugs in the software developed to control its luggage distribution system. In Brazil, some time ago, even after three computational systems were applied, a sizable ethylene production unit had to be completely dismantled after several unsuccessful attempts to make the controlling system work as intended. There are numerous other examples: controlling systems that led to the loss of satellites, air trafic controls that ran more than $100 million over budget and had to be abandoned after years of investment, and so forth. Luckily, the task proposed here is not as serious or complex as those examples, but every precaution should be taken to avoid many problems. One should be aware that the development of a computer simulation program is a time-consuming task. The period spent on modeling is almost insigniicant compared with the time consumed in collecting reliable published information, building up a workable computer program, and testing the simulator against real operational data. Moreover, every model and simulation program must be maintained; that is, it needs to be improved by expanding its range of application and including new and more accurate auxiliary equations published in the literature. This ensures that the program will be kept up to date and competitive. According to the discussion in Chapter 1, the development of a simulation program is by no means a straightforward job. First, most of the time, building a coherent mathematical structure is not easy and requires repetitive revisiting of the code. Second, the validation procedure, which is carried out by comparing the computed results with actual experimental results, demands a good amount of effort and time. This last task is absolutely necessary. No matter how sophisticated and mathematically coherent a model is, it is worthless if it is unable to reproduce the basic characteristics of the process or equipment operation within an acceptable range of deviation. That acceptable deviation varies from case to case of the process or equipment to be simulated. There are no simple rules to determine that satisfactory level, but it is possible to recognize several factors inluencing comparisons between simulations and real operations, such as the following: • Conidentiality. Usually, pilots or industrial units are under some sort of commercial agreement preventing the publication of several details regarding geometry or operational conditions. • Unreliability and inaccuracy. After careful examination, it is common to ind laws in reported data. Most are due to careless measurements,

Moving-Bed Simulation Programs and Results

•

• •

•

273

mistakes, and misinterpretations of the operational data. The data seldom state the conditions under which they have been measured. In addition, even a supposedly steady-state operation is subject to local luctuations in its parameters. Take, for example, the case of the operation of a moving-bed gasiier to which coal is continuously fed. Several questions can arise: • What are the deviations between actual and average reported physical characteristics—such as composition, density, particle size distribution, etc.—found in the feeding coal? • What is the luctuation of mass rate of coal feeding into the reactor? • What are the luctuations of other input streams to the reactor, such as mass low of air and steam, their temperatures, and their pressures? Other questions related to measurements under unstable conditions may also arise. For instance, during the operation of a moving-bed gasiier, the position of temperature peak obtained at the combustion zone can easily vary 5% around an average. The same sort of variations can be expected for concentrations and pressures. Added to that, the intrinsic precision of the instruments used during the measurements should be considered. Factors such as reliability and reproducibility of measurements have to be accounted for. For most situations, the intrinsic uncertainties of the instruments are masked by the luctuations of the measured variable around a supposedly steady operational value. Incomplete information. Most published reports do not reveal all inputs required to run the simulation program. Precision of the correlations. As we have seen, even for models based on fundamental equations, such those presented here, several empirical and semiempirical correlations are needed to provide values for various parameters during computations. Those are taken from the literature. Nonetheless, as in any mathematical model, the relationships are simple representations of the physical reality and therefore already affected by error. For example, most correlations for physical-chemical properties lead to deviations of around a few percent. The problem increases in cases of chemical reaction kinetics, where correlations or parameters might be affected by deviations around 30% or even higher. The intended application. Of course, the simulation of a satellite operation would require much smaller deviations than those acceptable for a coal gasiier. On the other hand, basic and auxiliary equations describing the former problem are inluenced by lower uncertainty than for the latter. As we have seen, deviation requirements are in line with the precision of available knowledge on the subject. Other situations can be found in which small deviations between simulation and reality should be guaranteed. One such case is when decisions regarding huge inancial investments are involved. For example, in selecting the best strategy for an electric power generation unit, deviations above 1% are normally unacceptable because this unit represents such a large investment of inancial resources.

274

Solid Fuels Combustion and Gasification

By this point, the reader should be eager for a suggestion concerning the maximum deviation value allowed in our present case of moving-bed gasiiers. Experience shows an average acceptable number around 5% for most parameters. These parameters include temperatures, pressures, compositions, residual particle size distributions, and mass lows. However, this should not be taken as an absolute rule.

12.2.1

SIMULATION STRATEGY

Like the strategy adopted during the mathematical modeling, the development of a simulation program should follow few basic steps: 1. Choose a programming language. 2. Write a block diagram for the program strategy. This is usually called the program chart. 3. Start writing the program. 4. Test the program, having in mind its robustness, which includes the following points: a) Whether it runs, without interruption, within the range of operation that it is intended to simulate. b) Whether the existing convergence procedures work properly, that is, whether the convergence of variables are always achieved. 5. Certify whether the simulation reproduces (within the expected deviation) the experimental or real operational data.

12.2.2

PROGRAMMING LANGUAGE

The choice of a language in which the program would be written should consider the following points: • How acquainted is the programmer with the various available languages usually used in engineering and scientiic programs? • Is the language universal enough, and is it available in most mainframes and personal computers? • Is the language compatible with those used in auxiliary mathematical and other commercially available packages? Usually, the simulation codes use routines or procedures to solve systems of linear and nonlinear equations, systems of ordinary or partial differential equations, convergence procedures, etc. There are several available commercial computational libraries installed in most mainframes and available in personal computer versions. These libraries are written in languages such as FORTRAN, PASCAL, and C. To avoid incompatibilities or the necessity of translating compilers, it is convenient to use the same language as the one used by the commercial mathematical package. The present chapter presents the cases of updraft and downdraft moving-bed gasiier simulations, and those should be useful to illustrate how a program can be organized. Let us start with the updraft version.

Moving-Bed Simulation Programs and Results

12.2.3

275

PROGRAM WRITING

Once the strategy is set, the developer may start the actual writing of the program. The fundamental requirements to achieve a robust and reliable program are clarity and organization. Before and during the task of writing a computer simulation program, one should put herself or himself in the position of someone who is trying to understand the code only by reading its lines. It has been veriied that time invested in making detailed comments at each point, as well as notations, references, and data feeding instructions, pays huge dividends, such as saving a lot of time and effort during the development of improvements in the program. Moreover, the analysis of possible laws in a computer program is among the most time-consuming tasks. This task is greatly facilitated by attention to clarity and order. The irst step includes listing the following items as part of the comments at the top of the main program: • The objective of the program • A description of its basic logic structure • Fundamental remarks on the assumptions, equations, and principles that were applied • If possible, a list of the references for each aspect of the program • A list of the nomenclature and the logic behind it • A list of the input data with their meaning and units • A similar list for the outputs As a second step, one should add comments at each point of key calculations inside the main program, as well as inside subroutines. Each subroutine must include the same items as described above for the main program. The developer should maintain a uniform nomenclature throughout the program. All this could sound too troublesome, but commercial and successful programs use more than half their listings for comments. Finally, a complete user manual must be written. It should include all the above features presented in a formal manner, as required by a book. In addition, the manual should contain examples of simulation results and comparisons against real operations.

12.3

UPDRAFT MOVING BED

As mentioned before, the model presented in Chapter 7 has been tested and used to improve existing operations, as well in designing new ones. It has been applied for processes consuming various types of woods, sugar cane bagasse, coals, charcoals, and charcoal pellets. The present section shows a few results and comparisons against real operations.* The simulation program derived from the model shown in previous chapters is known in the literature as Comprehensive Simulator for Fluidized and Moving Beds *

Unfortunately, several other operational data cannot be presented due to contractual restrictions between the author and clients. Those presented here have been published in open literature. However, the same degree of precision has been obtained in all cases.

276

Solid Fuels Combustion and Gasification

equipment (CSFMB).* As demonstrated below, it is capable of simulating movingbed reactors, and details regarding modeling and simulation of luidized bed equipment are presented throughout the remaining chapters.

12.3.1

PROGRAM CHART

Developing the program chart for the programming task is the equivalent of developing the model chart during the modeling phase. It consists of a block diagram in which the sequential and reiterative steps of the program are shown. That is very helpful during the actual writing of the program. For example, take the program chart for the case of updraft moving-bed simulation, as illustrated in Figure 12.1. It presents the computational procedure or block diagram, where parameter A is the guessed value for the fractional conversion of carbon, and B is the computed value for that conversion. The irst step of any simulation program is the data reading.† In our case, the simulation inputs are related to the following: 1. The physical characteristics of injected lows into the equipment, such as: 1.1. The mass low, temperature, composition (proximate and ultimate analysis), density, porosity, heating value, and particle size distribution of carbonaceous particulate solid feeding. 1.2. The mass low, temperature, and composition of the gas streams injected into the gasiier or combustor. In the case of the downdraft concept, those injections occur at a point usually above the bed, whereas they occur at the base of the bed in the case of the updraft or countercurrent type. Normally, one of those streams contains the oxidant. In the case of a gasiier, steam may be added as an independent stream or mixed to the stream with oxidant. As shown, steam may be necessary to control the peak of temperature, as well as to promote the production of hydrogen. Of course, air is the cheapest oxidant mixture, but some processes use pure oxygen. In the case of gasiication, this latter alternative is aimed at increasing the combustion enthalpy of the produced gas. Although it is not very common, some processes use other gases, such as carbon dioxide, as agents. 1.3. If intermediate gas injections are present, the above variables should be described and related to the height at which each injection is made. 2. A description of the equipment geometry, such as: 2.1. The reactor diameter. The diameter may be constant or vary throughout the bed height. In the latter case, some equations described in Chapter 7 should be reviewed because the cross-section (S) of the reactor is no longer constant. Moreover, a method to describe the diameter against bed height should be set. 2.2. The bed height (in case of updraft) or depth (in case of downdraft) (zD). *

†

Also known as CeSFaMB; submitted to the U.S. Trademark Ofice and copyright by U.S. Copyright Ofice; previous versions also known as CSFB. Despite concentrating here on updraft equipment, a few comments regarding the downdraft version are also made.

277

Moving-Bed Simulation Programs and Results

Data reading

Assume guess for carbon conversion = A

Preliminary calculations A

Solve system of differential equations

Computed carbon conversion = B

Is “A” near or equal to “B”?

Yes

Perform final calculations

Set boundary conditions

Print results

No

A

End

FIGURE 12.1 Program chart for the case of an updraft combustor or gasiier.

2.3. Description of the external and internal insulation, i.e., thickness, average thermal conductivity, and emissivity. These data are required to compute the heat losses to the environment. On the other hand, the reactor may be considered adiabatic, at least as a irst reasonable approximation. In several cases of good insulation, heat losses around and below 2% of the total energy input are common. 2.4. Other internal characteristics. The equipment of such moving-bed combustors or gasiiers may include features such as water jackets surrounding the reactor, mixing devices with refrigerated paddles, etc. Steam or hot water might be produced inside the jacket, and the work delivered by mixers might not be negligible regarding energy balances. Therefore, their effects should be added to the model equations. 3. Control variables. These are the following: 3.1. Printing control. Apart from printing data and results, it could be necessary to register the values of several intermediary computed variables. This may be important to follow the progress of the computation

278

Solid Fuels Combustion and Gasification

and to allow diagnosis of faults and problems, mainly during the phase of program development. A device for printing priorities may be set. It can be composed of a series of parameters that, for instance, instruct the program to print (or not print) some key variables. Another useful and eficient way to follow the evolution of variables is the debugging procedures provided by several compilation programs. 3.2. Parameters to control convergences and solutions of differential equations, such as maximum allowed deviation, maximum number of iterations, etc. A single block, as illustrated in Figure 12.1, represents the data reading step. According to the chart, the second step is composed of preliminary calculations. Information found in publications or provided by laboratories is presented in several different forms and systems of units. For instance, the ultimate analyses of the carbonaceous feeding are usually given on a dry basis. However, water must be considered to describe the actual composition. The required transformations from published values to the desired unit system could be made automatically by simple routines. More sophisticated routines can be set to recognize a range of possible units and transform to the required system. It is advisable to use SI (International System) units in all computations because this system has been adopted as the oficial one in most countries, and it is required by almost all technical and scientiic publications. The next steps are the core of the computational strategy. In the present case, the computation begins with a guessed value for the fractional conversion of carbon, marked as A in the diagram. B indicates the computed value for that conversion. The iterative procedure involves the computations using the equations described in Chapter 7, followed by those auxiliary aspects (Chapters 8 through 11). Once the convergence is achieved, the simulation results are printed.

12.3.2

SIMULATION RESULTS

To exemplify the validity of the model, comparisons between simulation outputs and parameters measured during operations of a moving-bed gasiier [2–4] are presented. The set below refers to charcoal updraft moving-bed gasiication with fundamental characteristics shown in Table 12.1. A cylindrical gasiier with a 0.5-m internal diameter, 4 m high, was used. Its internal surface was insulated with 0.15-m-thick refractory bricks with thermal conductivity around 0.7 W m–1 K–1. Air and saturated steam were injected into the gasiier. Table 12.2 shows the conditions set during periods of steady-state operation. The actual operational and simulation values for concentrations for the exiting gas are shown in Table 12.3. Figure 12.2 presents the temperature proiles in the bed (height measured from the bases) obtained from simulation compared against measured values (marked with black stars) for Test 1 [4]. Figures 12.3 through 12.7 illustrate important process characteristics, such as concentration proiles of several gases and tar throughout the bed.

279

Moving-Bed Simulation Programs and Results

TABLE 12.1 Characteristics of the Charcoal Used for Gasification Tests Characteristic Proximate analysis (%, wet basis) Moisture Volatile Fixed carbon Ash

4.60 13.93 79.66 1.81

Ultimate analysis (%, dry basis) C H O N S Ash Low heat value (MJ kg–1)

86.9 3.3 6.5 1.3 0.1 1.9 31.96

Density Particle, apparent (kg m–3) Particle, real (kg m–3)

12.3.3

Value

550 1400

COMMENTS

As we have seen, reasonable predictions of real operations are possible even with a relatively simple model like that described in Chapter 7. Good agreement has been obtained regarding temperature, as demonstrated by Figure 12.2. Simulation has been able to reasonably reproduce the peak of temperature and the overall trend. TABLE 12.2 Basic Operational Conditions Applied in Various Updraft Moving-Bed Gasification Tests Operational Condition Bed height (m) Charcoal feeding temperature (K) Injected air temperature (K) Injected steam temperature (K) Charcoal feeding mass low (kg s–1) Air feeding rate (kg s–1) Steam feeding rate (kg s–1) Average reactor interior pressure (kPa) Average charcoal particle diameter (mm)

Test 1

Test 2

3.5 298 318.1 431.4 1.18 × 10–2 3.87 × 10–2 4.70 × 10–3 101.6 27.9

3.5 298 316.9 444.0 1.26 × 10–2 4.17 × 10–2 6.33 × 10–3 103.0 12.7

280

Solid Fuels Combustion and Gasification

TABLE 12.3 Experimental and Simulation Compositions of Gas from Updraft MovingBed Gasification of Charcoal (Mol %, Dry Basis, Tar-Free) Run 1 Chemical Species

Experimental

Simulation

Experimental

Simulation

13.2 n.d.a n.d. n.d. n.d. 51.2 n.d. 0.4 n.d. 30.0 4.6 n.d. 0.6 n.d. n.d. n.d. n.d. n.d.

13.2118 0.0075 0.2142 0.0000 0.0000 54.8146 0.0000 0.0000 0.0060 24.4331 6.9376 0.0043 0.3234 0.0036 0.0416 0.0000 0.0000 0.0021

14.0 n.d. n.d. n.d. n.d. 49.8 n.d. 1.2 n.d. 28.6 4.6 n.d. 1.8 n.d. n.d. n.d. n.d. n.d.

13.9317 0.0062 0.1805 0.0000 0.0000 54.3603 0.0000 0.0000 0.0076 22.6089 8.3598 0.0028 0.2276 0.0034 0.4092 0.0001 0.0000 0.0020

H2 H2S NH3 NO NO2 N2b N2O O2 SO2 CO CO2 HCN CH4 C2H4 C2H6 C3H6 C3H8 C6H6 a b

Run 2

n.d., not determined. Obtained by difference.

In Table 12.3, note the reasonable agreement between simulations and experimentations regarding the concentration of hydrogen in the exiting gas stream. On the other hand, deviations between experimental and simulation results have been observed for carbon monoxide and dioxide concentrations. However, it is important to mention that during those early tests at IPT [2–4], a simple Orsat gas analyzer was used for measurements of concentrations in the exiting gas. Another possible explanation for the deviations could be found in the catalyst effect of biomass affecting main gasiication reactions, such as Reactions R.2 through R.5 (see Chapter 8, Table 8.4). As mentioned before, ashes of given fuels may have catalyst or poisoning effects on gasiication reactions. In such cases, special kinetics can be obtained through determinations at laboratories. CSFMB also allows insertion of speciic parameters related to kinetics. Such a process is legitimate, and many simulation programs require proper calibration to account for deviations of kinetics from published parameters. Temperature proiles somewhat similar to those presented in Figure 12.2 were obtained by several researchers [5–11]. As mentioned before, the development of models is a continuous task. Although moving-bed technology has lost ground to luidized beds (bubbling and circulating),

281

Moving-Bed Simulation Programs and Results 1600 1400

Temperature (K)

1200 1000 800 600 400 Emuls.gas 200

0

0.5

1

Carbonac.

Inert

1.5

2

Average 2.5

3

3.5

Height (m)

FIGURE 12.2 Experimental and simulation values of temperatures (K) against height (m) in the bed for a case of charcoal gasiication in an updraft moving bed (Test 1).

0.2

Molar fraction

0.15

0.1

0.05

CO2 0 0.E1

5.E-1

1.E0

1.5E0

CO 2.E0

O2 2.5E0

3.E0

3.5E0

Height (m)

FIGURE 12.3 Proiles of CO2, CO, and O2 concentrations in the bed for a case of charcoal gasiication in an updraft moving bed (Test 1).

282

Solid Fuels Combustion and Gasification 0.18 H2O

H2

CH4

0.16 0.14

Molar fraction

0.12 0.1 0.08 0.06 0.04 0.02 0

0

0.5

1

1.5

2

2.5

3

3.5

Height (m)

FIGURE 12.4 Proiles of H2O, H2, and CH4 concentrations in the bed for a case of charcoal gasiication in an updraft moving bed (Test 1).

1.E0 1.E-1

Molar fraction

1.E-2 1.E-3 1.E-4 1.E-5 1.E-6 1.E-7 H2 O 1.E-8

0

0.5

1

1.5

2

2.5

H2

CH4 3

3.5

Height (m)

FIGURE 12.5 Proiles of H2O, H2, and CH4 concentrations in the bed for a case of charcoal gasiication in an updraft moving bed (Test 1) (logarithmic scale of Figure 12.4).

283

Moving-Bed Simulation Programs and Results 6.E-3 H2S

NH3

Tar/Oil

5.E-3

Molar fraction

4.E-3

3.E-3

2.E-3

1.E-3

0.E1

0

0.5

1

1.5

2

2.5

3

3.5

Height (m)

FIGURE 12.6 Proile of tar concentration in a case of charcoal gasiication in an updraft moving bed (Test 1).

1.E0 SO2

1.E-1

NO

C2H6

1.E-2 1.E-3 Molar fraction

1.E-4 1.E-5 1.E-6 1.E-7 1.E-8 1.E-9 1.E-10 1.E-11 1.E-12

0

0.5

1

1.5

2

2.5

3

3.5

Height (m)

FIGURE 12.7 Proiles of SO2, NO, and C2H6 concentrations in the bed for a case of charcoal gasiication in an updraft moving bed (Test 1).

284

Solid Fuels Combustion and Gasification 1.E0 C  O2

1.E-1

C  H2O

C  CO2

C  H2

C  NO

1.E-2

Rates (kmol/m2/s)

1.E-3 1.E-4 1.E-5 1.E-6 1.E-7 1.E-8 1.E-9 1.E-10 1.E-11 1.E-12 1.E-13

0

0.5

1

1.5

2

2.5

3

3.5

Height (m)

FIGURE 12.8 Rates of the main heterogeneous reactions throughout the bed for a case of charcoal gasiication in an updraft moving bed (Test 1).

new models and improvements continue to appear in the literature. For instance, a model including dissipation terms (diffusion and conduction) in the axial direction has been developed by Cooper and Hallett [12]. It seems that deviations regarding temperature proiles between simulation and actual operation have been decreased. This should encourage even newer developments. From Figures 12.3 through 12.12, it is possible to observe the following: 1. Figure 12.3 shows the fast decrease in oxygen concentration near the combustion peak. Of course, it coincides with the surges in temperature (Figure 12.2). 2. The peaks of CO2 concentration (Figure 12.3) and temperature coincide. After that event, CO2 concentration decreases, mainly because of reaction with carbon (Reaction R.3, Table 8.4, Chapter 8). Nonetheless, one should be careful to interpret the graphs because these are relative concentrations that may appear to be declining when other gases are being produced. For instance, at regions near the top of the bed, devolatilization introduces several gases in the mixture, leading to reductions in other gas concentrations. The rate proile of Reaction R.3 can be observed in Figure 12.8. 3. Obviously, the concentrations of fuel gases, such as CO and H2, increase after almost complete consumption of oxygen. For instance, the rate and equilibrium of hydrogen oxidation is such that no oxygen can be detected in the reducing region of the bed with the usual precision of on-line analyzers, let alone Orsat apparatus. The presence of oxygen in the exiting gas (Table 12.3) increases the suspicion that determinations of gas compositions during the experiments had some laws.

285

Moving-Bed Simulation Programs and Results

4. The concentration of water vapor diminishes, while that of hydrogen increases (Figure 12.4), mainly because of Reaction R.2 (Table 8.4, Chapter 8). Its rate is illustrated in Figure 12.8, which demonstrates the importance of that reaction for gasiication processes. The small increase in water is observed around the peak of combustion due to oxidation of hydrogen in the fuel. 5. The methane concentration proile can be observed in Figure 12.4 or its logarithmic version Figure 12.5. Although devolatilization and tar decomposition reactions (Chapter 8, Reactions R.7 through R.9, R.50, and R.54) are responsible for most methane production, methane is also produced by Reaction R.4. Among others, the rate of Reaction R.4 can be observed in Figure 12.8, which shows its signiicance in methane production. That justiies high pressures for hydrocarbon-seeking processes because they favor the equilibrium toward higher concentrations of methane. 6. Figure 12.7 shows how the nitrogen oxide is produced rapidly around the peak of combustion and consequently the peak of temperature (Figures 12.2 and 12.3). The subsequent decrease is due to reduction of NO due to Reaction R.5 (Table 8.4) and the augmentation of the mass of other chemical species added to the gas phase. The rates of NO production by Reaction R.1 and reduction by R.5 can be observed in Figure 12.8. 7. Figure 12.8 also shows how the maximum rate of fuel combustion (Chapter 8, Reaction R.1) supplants all other rates by orders of magnitude. This explains the peak of temperature (Figure 2.2) even in the presence of endothermic reactions (Reactions R.2, R.3, etc.). 8. Figure 12.9 illustrates the rates of a few more heterogeneous reactions, emphasizing the main devolatilization and tar coking. C  N2O

V  Tar + Gas

Tar  Char

V  H2

C-drying

1.E0 1.E-1 1.E-2

Rates (kmol/m2/s)

1.E-3 1.E-4 1.E-5 1.E-6 1.E-7 1.E-8 1.E-9 1.E-10 1.E-11 1.E-12

0

0.5

1

1.5

2

2.5

3

3.5

Height (m)

FIGURE 12.9 Rates of a few more heterogeneous reactions throughout the bed for a case of charcoal gasiication in an updraft moving bed (Test 1).

286

Solid Fuels Combustion and Gasification 1.E0 CO  H2O

1.E-1

CO  O2

H2  O2

CH4  O2

C2H6  O2

1.E-2 1.E-3 Rates (kmol/m3/s)

1.E-4 1.E-5 1.E-6 1.E-7 1.E-8 1.E-9 1.E-10 1.E-11 1.E-12 1.E-13

0

0.5

1

1.5

2

2.5

3

3.5

Height (m)

FIGURE 12.10 Rates of the main homogeneous reactions throughout the bed for a case of charcoal gasiication in an updraft moving bed (Test 1).

9. Figure 12.10 presents rates of the main homogeneous reactions, demonstrating that the shift reaction (Reaction R.41, Table 8.5, Chapter 8) remains active throughout the reactor. Thus, assumptions of chemical equilibrium at the gas exiting position—as used by zero-dimensional models—are invalid. 10. Figure 12.11 shows other homogeneous reaction rates. The tar cracking inds its peak near the position where devolatilization starts (Figure 12.9). That occurs at positions not too far from the bed top, where the downtraveling solid fuel meets the rising gas stream at temperatures high enough to enhance pyrolysis (Figure 12.2). 11. Finally, Figure 12.12 demonstrates the importance of tar hydrogenation (Reaction R.54, Table 8.5, Chapter 8) in the conversion of tar into light chemical species. However, the decrease in tar concentration (Figure 12.6) is not too fast because of the decreasing temperatures near the top of the bed.

12.4

DOWNDRAFT MOVING BED

Chapter 3 describes why downdraft gasiiers usually lead to lower tar concentrations in the gas produced compared with the equivalent updraft models. However, it has also been seen that downdraft versions have severe limitations on size. That is why no large commercial gasiication process applies to downdraft reactors. This also explains the dificulty of inding reliable experimental data for that type of gasiier in the available literature. Nonetheless, the technique is applied to small units, usually within power outputs below 1 MW. CSFMB has been able to reproduce operations of both updraft and downdraft gasiiers; however, the data and results available to this author regarding the latter

287

Moving-Bed Simulation Programs and Results 1.E0 NH3  O2

1.E-1

H2S  O2

N2  O2

Tar  O2

Tar  Gas

1.E-2 1.E-3 Rates (kmol/m3/s)

1.E-4 1.E-5 1.E-6 1.E-7 1.E-8 1.E-9 1.E-10 1.E-11 1.E-12 1.E-13

0

0.5

1

1.5

2

2.5

3

3.5

Height (m)

FIGURE 12.11 Rates of more homogeneous reactions throughout the bed for a case of charcoal gasiication in an updraft moving bed (Test 1).

3.5E-5 Tar  H2 C3H8  O2

3.E-5

C2H4  O2 C6H6  O2

C3H6  O2 HCN  O2

Rates (kmol/m3/s)

2.5E-5 2.E-5 1.5E-5 1.E-5 5.E-6 0.E1

0

0.5

1

1.5

2

2.5

3

3.5

Height (m)

FIGURE 12.12 Rates of other homogeneous reactions throughout the bed for a case of charcoal gasiication in an updraft moving bed (Test 1).

288

Solid Fuels Combustion and Gasification

Data reading

Preliminary calculations

Carbon conversion is zero or at feeding value

Computed carbon conversion

Solve system of differential equations

Set boundary conditions

Perform final calculations

Print results

End

FIGURE 12.13 Program chart for the case of downdraft combustor or gasiier.

cannot be published because of the conidentiality clauses of a contract. On the other hand, for the sake of completeness, simulation outputs are presented regarding a case with input conditions similar to those described above for updraft moving-bed gasiication.

12.4.1

PROGRAM CHART

Similarly to the updraft case, a chart for the downdraft has been developed and is presented in Figure 12.13. One should keep in mind that the coordinate system used here (Chapter 7, Figure 7.1) differs from that applied to the updraft model (Figure 7.5). According to the description in Chapter 7, the downdraft version does not require convergence to determine boundary value conditions. Those are described by the mass lows (or compositions) and temperatures of both gas and solid phases z = 0, and they are given by the conditions of gas injection and solid fuel feeding. This allows much faster computations compared with similar cases of updraft reactors.

289

Moving-Bed Simulation Programs and Results

12.4.2

SIMULATION RESULTS AND DISCUSSION

The same geometry and input conditions as for the irst case of the updraft gasiier presented above have been used. Figure 12.14 shows the temperature proile generated by CSFMB. The following can be noticed compared with the proile shown in Figure 12.2: • The maximum temperatures are similar for both cases. • Higher average temperatures throughout the bed are observed for downdraft operations. The composition of gas produced during the downdraft gasiication is described in Table 12.4 and can be compared with Test 1, reported in Table 12.3. The simulations predicted a higher concentration of carbon monoxide in the gas from the downdraft reactor than from the updraft version. The main reason for that is the higher average temperature achieved in the downdraft process in relation to the value obtained in the updraft version. As mentioned before, the great advantage of downdraft version is the lower rate of tar released with the exiting produced gas stream. This can be observed in Table 12.5, which compares two operations under the same conditions. It is also noticeable that higher average temperatures throughout the bed led to larger eficiencies and carbon conversion in the case of downdraft version operating under the same inputs and geometry as the updraft gasiier. A few graphs related to concentration and reaction rate proiles throughout the bed of the present example of a downdraft gasiier are useful for comparison against the equivalent updraft version. 1400 Emuls.Gas

Carbonac.

Inert

Average

1200

Temperature (K)

1000

800

600

400

200 0

0.5

1

1.5

2

2.5

3

3.5

Depth (m)

FIGURE 12.14 Temperature (K) proiles against bed depth (m) for a case of charcoal gasiication in a downdraft moving bed (conditions similar to those of updraft Test 1).

290

Solid Fuels Combustion and Gasification

TABLE 12.4 Composition of Gas from Downdraft Moving-Bed Gasification of Charcoal (Simulation Result) Chemical Species H2 H2S NH3 NO NO2 N2 N2O O2 SO2 CO CO2 HCN CH4 C2H4 C2H6 C3H6 C3H8 C6H6

Molar % (tar free, dry basis) 14.2618 0.0097 0.2120 0.0000 0.0000 51.9623 0.0000 0.0000 0.0051 24.7614 7.5508 0.0124 1.2246 0.0000 0.0000 0.0000 0.0000 0.0000

Figure 12.15 presents the composition proiles of important gaseous species. It can be compared with the proiles at the updraft process given in Figure 12.3. The higher average bed temperatures led to a higher concentration of carbon monoxide during the downdraft operation. The effect on hydrogen production is not so different from both operations; however, methane concentration in the exiting gas is improved by downdraft operations, TABLE 12.5 Main Parameters of Equivalent Moving-Bed Downdraft and Updraft Operations (Based on Input for Test 1 of Charcoal Gasification) Condition Mass low of exiting gas stream (kg/s) Mass low of tar exiting with the produced gas (kg/s) Mass low of solids discharged from the bed (kg/s) Carbon conversion (%) Hot eficiency (%) Cold eficiency (%) Exergetic eficiency (%)

Updraft

Downdraft

5.188 × 10–2 1.502 × 10–3 1.977 × 10–3 84.00 65.54 53.69 80.02

5.369 × 10–2 2.817 × 10–6 1.573 × 10–3 87.66 82.81 68.83 85.45

291

Moving-Bed Simulation Programs and Results

as seen by comparing Figures 12.4 and 12.16. This can be explained by two main factors: a) Tar decomposition increases the production of methane, and this destruction is almost complete during downdraft operations. b) Particles are kept at high temperatures longer in downdraft reactor compared with the equivalent downdraft operation. Therefore, high rates of methane production by Reaction R.4 (Table 8.4, Chapter 8) are maintained longer than in the updraft operation, as veriiable by comparing Figures 12.8 and 12.17. The rates of important homogeneous reactions can be observed in Figure 12.18 and compared with the corresponding rates for the updraft operation given in Figure 12.10. Of course, all main processes in a downdraft gasiier occur near the top of the bed. Nevertheless, the rate of shift reaction remains relatively high throughout the bed, and that conirms the importance of that reaction in both downdraft and updraft processes. The effectiveness of tar destruction in a well-designed and well-operated downdraft gasiier can also be appreciated in Figure 12.19. Since pyrolysis occurs near the top of the bed (z = 0), the released tar is dragged by the gas stream through the combustion region below. On the other hand, that may not be the case when: a) The bed area is too large, which allows for the formation of preferential channels. These channels provide escape routes where temperatures that 0.25

Molar fraction

0.2

0.15

0.1

0.05 CO2 0 0.E1

5.E-1

1.E0

1.5E0

2.E0

2.5E0

CO 3.E0

O2 3.5E0

Depth (m)

FIGURE 12.15 Proiles of CO2, CO, and O2 concentrations in the bed for a case of charcoal a downdraft gasiication (conditions equivalent to those of updraft Test 1).

292

Solid Fuels Combustion and Gasification 0.2 H2O

H2

CH4

Molar fraction

0.15

0.1

0.05

0

0

0.5

1

1.5

2

2.5

3

3.5

Depth (m)

FIGURE 12.16 Proiles of H2O, H2, and CH4 concentrations in the bed for a case of charcoal gasiication in a downdraft moving bed (conditions equivalent to those of updraft Test 1).

1.E0 C  O2

1.E-1

C  H2 O

C  CO2

C  H2

C  NO

1.E-2 1.E-3 Rates (kmol/m2/s)

1.E-4 1.E-5 1.E-6 1.E-7 1.E-8 1.E-9 1.E-10 1.E-11 1.E-12 1.E-13 0

0.5

1

1.5

2

2.5

3

3.5

Depth (m)

FIGURE 12.17 Rates of the main heterogeneous reactions throughout the bed for a case of charcoal gasiication in a downdraft moving bed (conditions equivalent to those of updraft Test 1).

293

Moving-Bed Simulation Programs and Results 1.E0 CO  H2O

1.E-1

CO  O2

H2  O2

CH4  O2

C2H6  O2

1.E-2

Rates (kmol/m3/s)

1.E-3 1.E-4 1.E-5 1.E-6 1.E-7 1.E-8 1.E-9 1.E-10 1.E-11 1.E-12 1.E-13

0

0.5

1

1.5

2

2.5

3

3.5

Depth (m)

FIGURE 12.18 Rates of the main homogeneous reactions throughout the bed for a case of charcoal gasiication in a downdraft moving bed (conditions equivalent to those of updraft Test 1).

2.5E-3 H2S

NH3

Tar/Oil

Molar fraction

2.E-3

1.5E-3

1.E-3

5.E-4

0.E1

0

0.5

1

1.5

2

2.5

3

3.5

Depth (m)

FIGURE 12.19 Proile of tar concentration in the case of charcoal gasiication in a downdraft moving bed (conditions equivalent to those of updraft Test 1).

294

Solid Fuels Combustion and Gasification 1.E0 SO2

1.E-1

NO

C2H6

1.E-2 1.E-3 Molar fraction

1.E-4 1.E-5 1.E-6 1.E-7 1.E-8 1.E-9 1.E-10 1.E-11 1.E-12

0

0.5

1

1.5

2

2.5

3

3.5

Depth (m)

FIGURE 12.20 Proiles of SO2, NO, and C2H6 concentrations in the bed for a case of charcoal gasiication in a downdraft moving bed (conditions equivalent to those of updraft Test 1).

1.E0 1.E-1

SO2

NO

C2H6

1.E-2 1.E-3 Molar fraction

1.E-4 1.E-5 1.E-6 1.E-7 1.E-8 1.E-9 1.E-10 1.E-11 1.E-12 0.000001

0.00001

0.0001

0.001

0.01

0.1

1

10

Depth (m)

FIGURE 12.21 Proiles of SO2, NO, and C2H6 concentrations in the bed for a case of charcoal gasiication in a downdraft moving bed (conditions equivalent to those of updraft Test 1) (log-log scales).

Moving-Bed Simulation Programs and Results

295

are lower than the average in the combustion region are found and tar is not completely cracked or cooked. b) The operational conditions are not correctly set. For instance, if the oxygen ratio is too low, a relatively low temperature would be observed in the combustion zone. This may deprive the process of conditions for the complete destruction of tar. Finally, the concentration proiles of a few other gaseous species are presented in Figure 12.20 (logarithm scale for concentration) and Figure 12.21 (log-log). From those, it is possible to verify the production of SO2 and NO at the combustion region and, after that, the reduction of nitrogen oxide, mainly due to Reaction R.5.

12.5

EXERCISES

12.5.1

PROBLEM 12.1*

Having in mind the discussion in Chapter 7 related to ignition temperature in the case of updraft moving beds, the reader is asked to describe how the ignition temperature can be computed in cases of downdraft gasiiers.

12.5.2

PROBLEM 12.2****

Write a simulation program for an updraft moving-bed gasiier that is supposed to operate under the following conditions: • There is a steady-state regime. • The gasiier is continuously fed with porous spherical particles of pure graphite. • Air is continuously fed through the distributor or bottom of the bed. • Assume, for instance, the following data: • Mass low of feeding particles, 1.0 × 10 –2 kg s–1. • Mass low of air, 3.0 × 10 –2 kg s–1. • Gasiier internal diameter, 0.5 m. • Bed height, 2.0 m. • Apparent density of feeding particles, 1000 kg m–3. • Real density of feeding particles, 2000 kg m–3. • Global bed density, 700 kg m–3. • Operating pressure, 100 kPa. • Average diameter of feeding particles, 1.0 × 10 –2 m. • Feeding particles are dry and volatile free; therefore, there is no need to introduce routines or calculations regarding those processes. • As a irst approximation, assume the temperature of solid phase as 700 K at the basis of the bed. In a second model, the student may reine the search for this boundary condition. • In order to allow the combustion of CO, assume that the concentration of water in the entering gas is 1/100 of the concentration of oxygen. • Assume adiabatic rector, i.e., neglect the heat transfer to environment. • Use the routines developed in previous chapters or use the available literature [13–16] and Appendix B to set the calculations for the physical-chemical properties.

296

Solid Fuels Combustion and Gasification Guiding steps: 1. 2. 3. 4. 5. 6. 7. 8.

12.5.3

Write the basic equations or the mass and energy differential equations. Write the boundary conditions. Develop a simpliied block diagram for the computation. Write the auxiliary equations for the chemical kinetics, equilibrium, physical and chemical properties, enthalpy and heat transfer calculations, etc. Develop more detailed block diagrams for each subroutine or part of your model. Consult the literature [17–20] or commercially available packages (such as IMSL®) for the convergence routines and for methods of solving the systems of differential equations. Write the program. Obtain the numerical results for the concentrations and temperature proiles throughout the bed.

PROBLEM 12.3****

Improve the previous program by adding a routine for the computation of rates and composition of pyrolysis yields. For a irst approximation, just use the Loison and Chauvin equations (Equations 10.23–10.28, Chapter 10). Integrate them into the program and run it for the case of charcoal. Use the input described in Section 12.3.

12.5.4

PROBLEM 12.4****

Repeat the previous problem using the functional group combined with DISKIN model.

12.5.5

PROBLEM 12.5****

Improve your program even further by applying the model developed in Problem 10.4.

12.5.6

PROBLEM 12.6****

Repeat Problem 12.2 for the case of a downdraft moving-bed gasiier.

12.5.7

PROBLEM 12.7****

Apply the improvements developed in Problem 12.3 to the program developed in for Problem 12.6 for the case of a downdraft combustor or gasiier.

12.5.8

PROBLEM 12.8****

Repeat Problem 12.5 for the case of a downdraft reactor.

12.5.9

PROBLEM 12.9****

Use the program developed to optimize the operations of charcoal gasiier presented in Section 12.3. Set the objective of maximizing the cold eficiency (see Chapter 5). Keep all conditions constant, varying only the mass lows of air and steam injected into the reactor.

Moving-Bed Simulation Programs and Results

297

REFERENCES 1. Gibbs, W.W., Software’s chronic crisis, Scientiic American, 86–95, September 1994. 2. de Souza-Santos, M.L., Development of a Simulation Model and Optimization of Gasiiers for Various Fuels (Desenvolvimento de Modelo de Simulação e Otimização de Gaseiicadores com Diversos Tipos de Combustíveis), IPT-Inst. Pesq. Tec. Est. São Paulo, SCTDE-SP, Report No. 20.689, DEM/AET, São Paulo, Brazil, 1985. 3. de Souza-Santos, M.L., and Jen, L.C., Study of Energy Alternative Sources; Use of Biomass and Crop Residues as Energy Source: Part B: Development of a Mathematical Models and Simulation Programs for Up-Stream and Down-Stream Moving Bed Gasiiers. (Estudo de Fontes Alternativa de Energia. Parte B), IPT-Institute for Technological Research of São Paulo, Report No. 16.223-B/DEM/AET, São Paulo, Brazil, 1982. 4. Jen, L.C., and de Souza-Santos, M.L., Modeling and simulation of ixed-bed gasiiers for charcoal (Modelagem e simulação de gaseiicador de leito ixo para carvão), Brazilian Journal of Chemical Engineering (Revista Brasileira de Engenharia Química), 7(3–4), 18–23, 1984. 5. Hobbs, M.L., Radulovic, P.T., and Smoot, L.D., Combustion and gasiication of coals in ixed-beds, Prog. Energy Combust. Sci., 19, 505–556, 1993. 6. DeSai, P.R., and Wen, C.Y., Computer Modeling of the MERC Fixed Bed Gasiier, U.S. Department of Energy Report MERC/CR-78/3, Morgantown, WV, March 1978. 7. Yoon, H., Wei, J., and Denn, M.M., A model for moving-bed coal gasiication reactors, AIChE J., 24(5), 885–903, 1978. 8. Stillman, R., Simulation of a moving bed gasiier of a western coal, IBM J. Res. Dev., 23, 240–252, 1979. 9. Hobbs, M.L., Radulovic, P.T., and Smoot, L.D., Modeling ixed-bed coal gasiiers, AIChE J., 38, 681–702, 1992. 10. Radulovic, P.T., Ghani, M.U., and Smoot, L.D., An improved model for ixed-bed combustion and gasiication, Fuel, 74(4), 582–594, 1995. 11. Ghani, M.U., Radulovic, P.T., and Smoot, L.D., An improved model for ixed-bed combustion and gasiication: sensitivity analysis and applications, Fuel, 75(10), 1213–1226, 1996. 12. Cooper, J., and Hallett, W.L.H., A numerical model for packed-bed combustion of char particles, Chemical Engineering Science, 55, 4451–4460, 2000. 13. Perry, J.H., Green, D.W., and Maloney, J.O., in Perry’s Chemical Engineers Handbook, 7th Ed., McGraw-Hill, New York, 1997, 12-1–12-90. 14. Poling, B.E., Prausnitz, J.M., and O’Connell, J.P., The Properties of Gases and Liquids, 5th Ed., McGraw-Hill, New York, 2000. 15. Reid, C.R., Prausnitz, J.M., and Poling, B.E., The Properties of Gases and Liquids, 4th Ed., McGraw-Hill, New York, 1987. 16. Alemasov, V.E., Dregalin, A.F., Tishin, A.P., and Khudyakov, V.A., Computation methods, in Thermodynamic and Thermophysical Properties of Combustion Products, Vol. I, Glushko, V.P., Ed., Viniti, Moscow, 1971. 17. Carnahan, B., Lutter, H.A., and Wilkes, J.O., Applied Numerical Methods, John Wiley, New York, 1969. 18. Bakhvalov, N.S., Numerical Methods, Mir, Moscow, 1977. 19. Kopachenova, N.V., and Maron, I.A., Computational Mathematics, Mir, Moscow, 1975. 20. Lapidus, L., Digital Computation for Chemical Engineers, McGraw-Hill, New York, 1962.

Fluidized13 Bubbling Bed Combustion and Gasification Model CONTENTS 13.1 Introduction ................................................................................................. 299 13.2 Mathematical Model ................................................................................... 299 13.2.1 Basic Assumptions ........................................................................300 13.2.2 Basic Equations .............................................................................304 13.3 Boundary Conditions ..................................................................................309 13.3.1 Boundary Conditions for Gases ....................................................309 13.3.2 Boundary Conditions for Solids .................................................... 310 13.4 Exercises ..................................................................................................... 313 13.4.1 Problem 13.1 .................................................................................. 313 13.4.2 Problem 13.2 .................................................................................. 313 13.4.3 Problem 13.3 .................................................................................. 314 13.4.4 Problem 13.4 .................................................................................. 314 13.4.5 Problem 13.5 .................................................................................. 314 13.4.6 Problem 13.6 .................................................................................. 314 13.4.7 Problem 13.7 .................................................................................. 314 13.4.8 Problem 13.8 .................................................................................. 314 13.4.9 Problem 13.9 .................................................................................. 314 References .............................................................................................................. 314

13.1

INTRODUCTION

There is an impressive body of work on modeling and simulation of bubbling luidized beds. Although these works include several aspects of the process, differences among models can be better appreciated after some experience in the subject. Appendix F presents the main modeling characteristics and related references on the subject. In the present chapter, the basic aspects of a mathematical model for bubbling luidized-bed equipment are shown. This model can be used to simulate a wide range of pilot and industrial-scale luidized-bed reactors, including boilers and gasiiers.

13.2

MATHEMATICAL MODEL

The superiority of a one-dimensional model over a zero-dimensional model is obvious. For cases of packed beds—such as moving beds—the convenience and 299

300

Solid Fuels Combustion and Gasification

usefulness of a one-dimensional approach in relation to higher ones, such as twoand three-dimensional, have already been demonstrated. The success of the present simulation for packed beds at least provides a strong indication that modeling of luidized-bed processes can be accomplished at the same level as used for moving beds. As illustrated in Chapter 16, the present level of modeling provides good predictions of operational conditions of bubbling luidized beds from bench to large industrial scales.

13.2.1

BASIC ASSUMPTIONS

Figure 13.1 illustrates the model chart that includes the system of coordinates used to set the system of fundamental equations governing the processes inside bubbling luidized-bed equipment. The basic assumptions of the model are as follows: A. There are two main phases in the bed: emulsion and bubbles. The bubbles are free of particles. This is a very good approximation, which has been conirmed by experimental veriication. Some exceptions may occur in cases of turbulent luidization, as already explained in Chapter 3. However, most of the industrial units using the bubbling luidization technique operate far from turbulent regime. As an immediate consequence of the present assumptions, all solid species taking part in the process remain in the emulsion, Cyclones Freeboard Carbonaceous PF

Gas Tubes or jackets Entrained particles

PF

Absorbent

Gas-gas reactions

Inert

PF

Tubes or jackets

PF Gas-solid reactions

PF

Bed Bubble phase

Solids

Returning particles

Emulsion phase

Tubes or jackets

Gas-gas reactions

Emulsion gas

Carbonaceous UC

PF

Absorbent

Gas-gas reactions

Solids

Inert

Gas-solid UC reactions

Heat transfer Gas inlet

UC

Tubes or jackets Recycling to bed

Heat & mass transfers

PF = Plug-flow UC = Uniform composition

FIGURE 13.1 Simpliied diagram of the basic structure of the model.

Bubbling Fluidized-Bed Combustion and Gasification Model

301

which is therefore composed of the interstitial gas and solid species. Here, three possible solid particles will be considered: carbonaceous (coal, biomass, etc.), inert (sand, etc.), and sulfur-absorbent (limestone, dolomite, etc.). It is important to note that some authors [1, 2] refer to other phases, called cloud and bubble-wake. The cloud is a thin layer of gas with particles surrounding the bubbles. The bubble-wake is similar to the cloud but is a region that stays below the ascending bubbles. The average size of particles in the cloud and in the wake is smaller than the average found in the emulsion. This is relatively easy to understand and has already been noted in Chapter 3. In the present treatment, the clouds and wakes are considered part of the emulsion. B. The equipment operates in a steady-state regime. Consequently, feeding and withdrawing rates of all streams (gases and solids) are constant. As noted before, no real process operates under a perfect steady-state regime. Fluctuations of almost all variables are inevitable. Nonetheless, in many cases, it is possible to assume average constant values for inputs and outputs, thus preserving the steady-state approximation. For instance, as described in Chapter 3, the bubbles travel through the bed and explode at its top, thus severely disturbing that region. For that reason, there is no clear delimitation of bed surface. Nonetheless, the present model assumes an average value for the bed level without any loss in simulation applicability or precision. Another variable that brings some interpretation problems is the average temperature in the bed. Usually, temperature measurements are made using a series of thermocouples inserted in that region. However, the tip of a thermocouple may be immersed in the emulsion and the next instant a bubble traveling upward may pass through that position. Since the temperatures of emulsion and bubbles usually differ considerably, sharp variations in the values registered by that thermocouple should be expected. The reader should be aware of the dynamic nature of the phases present in a luidized bed. When the text refers to temperature in the bubble at a given height in the bed, it means the horizontal cross-sectional average temperature inside the bubbles that at a given instant are traveling through that particular height. The same is valid for the emulsion and for the solid particles. The average temperature measured by thermocouples in a luidized bed is a particular value between temperatures of gas in the emulsion, gas in the bubbles, and particles near the thermocouple position. That average is not the common mixed-cup temperature but a sort of statistical mean. C. Gas streams passing through the emulsion, bubble phase, and freeboard follow a plug-low regime. For the case of the emulsion, this assumption is justiied similarly to the case of moving beds. As shown in Figure 13.2, particles circulate in both directions, upward and downward. Therefore, it is possible to imagine, at least as a good irst approximation, that the overall inluence of their movement on the gas lowing regime would be negligible. The approximation of plug-low for the gas in the bubble phase is also reasonable. This is easily understood if the bubbles are thought to be lowing in a separate vertical chamber. Mass and energy transfers occur between the

302

Solid Fuels Combustion and Gasification Upward global movement (FH = S GH)

Downward global movement (FH = S GH)

FIGURE 13.2 Typical circulation paths of particles in a bubbling luidized bed.

emulsion and bubbles through an imaginary continuous interface. The total area of that interface equals the total external area of bubbles present in the bed at a given instant. D. The low of particles in the freeboard also follows a plug-low regime. Actually, an upward plug-low of particles with a decreasing mass rate is assumed. This approximation becomes increasingly cruder for positions far above the bed surface. Since a smaller upward gas velocity is found at regions near the walls, heavier particles returning to the bed do so through paths nearer them. Luckily, the number of particles found in the bed is much higher than in the freeboard. Thus it is easy to recognize that most of the processes and transformations occur in the bed. Consequently, it is reasonable to take a more relaxed approach for the freeboard because that would exert little inluence in the total deviation between simulation and measured values. E. One-dimensional model or variations occur in the vertical or axial z direction. This is somewhat justiied by assumptions C and D. In addition, this

Bubbling Fluidized-Bed Combustion and Gasification Model

F.

G.

H.

I. J.

K.

303

approach is proved very reasonable by the comparisons of simulation and operational results illustrated in Chapter 16. Second-order transport terms are negligible. Therefore, it is assumed that the mass transfer by diffusion and the energy transfer by conduction in the axial direction are overwhelmed by the convective transfers. This is a reasonable approximation because of the high circulation rates of particles and high mass lows of gases in that direction. Homogeneous composition of the solid particles throughout the bed is assumed. This is justiiable because of the high circulation rate in the bed. Hence, particles with low concentration, as in carbon, are quickly replaced by others with higher average concentrations. The same reasoning is valid for other components in the solid phase, thus ensuring the present approximation. Despite uniform composition for each solid phase in the bed, the same is not imposed for the temperatures. This may seem to be a contradiction. Nonetheless, the temperature of a particle exerts a strong inluence on the heterogeneous reaction rates taking place inside it, as well as on the homogeneous reactions of the gas layer around it. The situation is more dramatic for the cases of combustion and gasiication near oxygen-rich regions of the bed base. There, the fast exothermic combustion reactions tend to increase the temperature of the particles faster than they can the replaced by cooler ones. For instance, increases in the temperature of solid carbonaceous particles near the distributor have been experimentally veriied in combustors and gasiiers. The effects of drying and devolatilization of carbonaceous solids are felt only near feeding positions. Kinetics and processes involved during the devolatilization and drying combined with circulation rates of particles are used to determine the region of the bed where these processes are considered. The same is applied to the cases of limestone and inert drying. The gas–solid reactions are described by one of two possible basic models: unexposed and exposed core. The gas phases are transparent concerning radiative heat transfer. Heat transfers between all phases, as well as between these phases and internal surfaces (walls and tubes), involve convection, conduction, and radiation. However, the present approximation does not introduce any major deviation due to thin gas layers between the particles. On the other hand, radiative heat transfers between particles are accounted for. Usually, luidized beds operate with sand (or another inert substance) and/or sulfur-absorbent (limestone or dolomite) particles. At certain axial positions in the bed or freeboard, it is possible to ind considerable differences between temperatures of carbonaceous and the above solid particles. Thus, intense radiative heat transfer would occur. In addition, radiative heat transfer takes place between solid particles and immersed surfaces in the bed or in the freeboard. This is the case in boilers, where banks of tubes for steam generation are usually inserted into both these regions. Instead of using the momentum transfer equations, the luidization dynamics are described by empirical and semiempirical correlations. Those are

304

Solid Fuels Combustion and Gasification

presented in Chapter 14. Such phenomena include, for instance, the rate of particle circulations in the bed, entrainment, bubble behavior, etc. This approach is assumed because luidization dynamics are complex and impossible to properly treat under one-dimensional modeling.

13.2.2

BASIC EQUATIONS

The basic differential equations describing mass and energy balances for components and phases are described below. They can also be found elsewhere [3, 4], although in different forms. The fundamental continuity equations and the balances are similar to those presented in Chapter 7 for the gas at each phase (emulsion and bubbles). Thus, it is possible to write the following: • The mass balance in the emulsion gas phase can be described by 3 dFGE, j = ∑ ( R het ,SE,m , jγ m ,ESE ) + R hom,GE, jε ESE + G MGEGB,jγ BS, 1 ≤ j ≤ 500. (13.1) dz m =1

This equation is similar to Equation 11.14; however, now several possible solid phases and another gas contained in the bubbles are involved. The index m refers to the particular solid phase (m = 1 for the carbonaceous solid, m = 2 for limestone, and m = 3 for the inert substance). The irst term of the right side (Rhet,SE,m,j) is the contribution to the variation on the mass low of gas component j due to the production (or consumption) by gas–solid or heterogeneous reactions. The second term (R hom,GE,j) is due to the gas– gas or homogeneous reactions, and the third (GMGEGB,j) represents the mass transfer of component j between bubble and emulsion phase. Equations and correlation to allow computations of the parameters involved are described in Chapters 14 and 15. The absence of second-order derivatives of mass low should be noted. This is due to simpliication F, described above. • The mass balance in the bubble phase is described by dFGB, j = R hom ,GB, jε BS − G MGEGB, jγ BS, 1 ≤ j ≤ 500. dz

(13.2)

This equation is similar to the previous one. Nonetheless, because of assumption A, there is no term involving the production or consumption of component j due to the heterogeneous reactions. • Because of simpliication G, the mass balance should refer to the total conversion of solid, as in a well-stirred reactor, or Λ D, j = 1 −

FLD, j , 501 ≤ j ≤ 1000 FID, j

(13.3)

where the mass low (FLD,j) of component j leaving the bed is given by an average computed throughout the entire bed, or

Bubbling Fluidized-Bed Combustion and Gasification Model 3

FLD, j = FID, j − ∑

zD

∫R

γ

S dz, 501 ≤ j ≤ 1000.

het ,SE , m , j m , E E

m =1 z = 0

305

(13.4)

• Energy balance for the emulsion phase is described by FGE c GE

3  dTGE = ε ESE  −R QGE + ∑ ( R CSEGE,m + R hSEGE,m ) dz m =1 

+ ( R CGBGE + R MGBGE ) where

(13.5)

 εB − R CGETD − R GEWD  εE 

500

FGE = ∑ FGE, j.

(13.6)

j =1

RQGE represents the rate of energy generation (or consumption, if negative) per unit of reactor height due to gas–gas chemical reactions. RCSEGE,m is the rate of convective heat transfer between each solid species m and the emulsion gas. Gaseous species are exchanged between the solid phase m and the emulsion. Since these phases are not necessarily at the same temperature, the mass transfers result in energy transfers as well and are symbolized by the rate R hSEGE,m. The term RCGBGE represents the rate of heat transfer by convection between the emulsion gas and the bubbles. R MGBGE is the rate of energy exchanged between bubbles and emulsion due to the reciprocal mass transfers. In view of the particular forms of heat transfers between emulsion and bubbles, RCGBGE and R MGBGE appear multiplied by the volume ratios between bubble and emulsion. More details on these aspects are presented in Chapters 14 and 15. If tubes are immersed in the bed, RCGETD represents the heat transfer by convection between the emulsion gas and those tubes. RGEWD symbolizes the rate of heat transfer between the emulsion gas and the bed wall. Equation 13.5 can be compared with Equation 7.37 or 7.38 (Chapter 7). It is easy to see that the term multiplying the whole right side of Equation 13.5 is equivalent to the void fraction (of emulsion gas in the reactor) times the horizontal cross-sectional area occupied by the emulsion. This will be even clearer below, after the deductions of derivatives of volumes and areas of phases against the reactor axial coordinate. • Energy balance for the bubble phase is governed by FGBc GB

dTGB = ε BS  −R QGB − R CGBGE − R hGBGE − R CGBTD  dz

(13.7)

where 500

FGB = ∑ FGB, j. j =1

(13.8)

306

Solid Fuels Combustion and Gasification

RQGB represents the rate of energy generated or consumed because of homogeneous reactions occurring in the bubble phase. The meanings of RCGBGE and R hGBGE have already been explained. RCGBTD stands for the rate of heat transfer by convection between bubbles and tubes, if any are immersed in the bed. • Energy balances for the solids are found. Rigorously, those balances cannot be described by a differential equation in the same way as is possible in the case of the moving bed (Chapter 7). This is due to assumption G, which models the motion of particles as random. Nonetheless, a global or overall energy balance is feasible and would be used to determine the average temperatures of each solid in the bed. As detailed below, those averages can be used to set boundary conditions for the solution of the differential equations. As discussed in assumption G, the model assumes a well-stirred process that allows taking advantage of the fact that a pluglow model for the gas in emulsion provides its composition and temperature at each height (or axial position). This is so because the particles moving either up or down in the bed exchange heat and mass with the gas phase around them. Therefore, those transfers may be quantiied at each point. However, as described above, and to maintain the mathematical and physical coherence of energy conservation laws, the average temperatures of particles in the bed are dictated by overall energy balances (Section 13.3.2). Consequently, if the temperature proiles of solids could be found, their averages should match the ones computed by overall balances as well. That would be ensured by an iterative procedure, as explained in Chapter 16. On the other hand, as detailed in Chapter 14, Section 14.4, if higher supericial velocities for the injected gas into the bed are used, the circulation rate of particles increases. That promotes intense stirring of solids, and the temperature of each particulate phase would approach homogeneity, or would experience little variation throughout the bed. Hence, the derivative of temperature for each solid phase should be inversely proportional to the circulation rate (FH,m) of that solid (m) in the luidized bed. To reconcile or combine all above considerations, the following equation has been proposed [3, 4]: FH ,m cSE ,m

 dTSE ,m εE = γ m ,E SE  − R QSE ,m − R CSEGE ,m + R hSEGE ,m 1 εE − dz 

)

(

3

(

− R RSETD ,m − ∑ R RSESE ,m ,n + R CSESE ,m ,n n =1

)

(13.9)

  , 1 ≤ m ≤ 3. 

For each solid species m (1 = carbonaceous; 2 = limestone; 3 = inert) present in the bed, it is possible to verify the following: • The derivative of temperature on the left side would decrease for increases in the circulation rate of the solid (FH,m) and vice-versa. • Equation 13.9 is the counterpart of Equation 13.5; hence, the energy balance regarding the various effects of exchanges with emulsion is

Bubbling Fluidized-Bed Combustion and Gasification Model

307

respected, and one should follow the explanations given there to understand the meaning of each term. Nevertheless, unlike the gas phase, where only homogeneous reactions take place, the irst term on the right side of Equation 13.9 (RQSE,m) represents the rate of energy generation or consumption due to gas–solid or heterogeneous chemical reactions. In addition, R RSETD,m symbolizes the rate of heat transfer due to radiation between the solid m and the surfaces of tubes, if any are immersed in the bed. The radiation and conduction heat transfers between different solid species are accounted for by RRSESE,m,n and RCSESE,m,n, respectively. • The integrity of energy conservation is preserved. Figure 3.11 helps to illustrate the coherence of computing a temperature proile by Equation 13.9. If the bed operates with relatively low circulation rates, the particles near the bottom (height = 0) would reach higher temperatures. This is mainly because they would have more time in contact with the oxygen-rich gas stream that is being injected through the bed distributor. At the same time, their convective heat transfer to the gas is lower because of relatively low velocities between particles and gases. To compensate or maintain the same average, the temperature of others particles, located at higher positions, would be lower than the average in the bed. If the circulation increases, the temperature of coal particles at the bottom (z = 0) would decrease, and that of the others— above the average height where the average is located—would increase. The totally lat temperature proile would be reached for an ininite circulation rate. However, no matter what the situation, the energy conservation law is not violated since the overall balance is respected. Actually, the present approach allows reasonable considerations regarding the temperature proiles of individual solid species [3, 4]. Most onedimensional models for luidized beds assume constant temperatures or lat temperature proiles (see Appendix F). That assumption is based on experimental measurements of temperatures in bubbling luidized combustors or gasiiers. This happens because any thermocouple inserted at any point in the bed would be able to measure an average. The present model shows considerable differences between various phases at various points in the bed, whereas the computed average (Figure 3.11) remains almost constant, thus agreeing with experiments as well. Of course, Equation 13.9 is a proposition or model. Nonetheless, it permits explanation of several aspects of pilot and industrial operations, such as the following: a) Average temperatures are nearly constant, even at points very near the distributor surface. This occurs despite the injection of a cold gas stream through the distributor. b) Technical staff from National Coal Board Research Center* reported that agglutination of particles started near the distributor. This is an *

Personal communication with the author.

308

Solid Fuels Combustion and Gasification

indication that if the temperature of solid fuel were to surpass the ash-softening temperature, it would occur at points where higher temperatures are usually found. c) As will be shown in Chapter 16, the present model is able to reproduce experimental data reasonably well, not just for the average temperatures but also for low and compositions of gas from the equipment. This is a good indication of reasonable modeling because chemical kinetics strongly depends on temperatures. As will be demonstrated, most chemical reactions between fuel and gases (mainly oxygen) occur at points near the distributor. Actually, only a two-dimensional (or three-dimensional) approach, where the momentum transfer equations are included, allows rigorous differential energy balances for the solid phases. That would provide an approximation for the path of particles in the bed and allow estimation of temperature proiles of each particle with more precision. As shown in Chapter 14, the two-dimensional approach for particle circulation is not too dificult. However, when all mass and energy balances are taken into account, a very complex system of nonlinear and highly coupled partial differential equations would emerge. This leads to signiicant augmentations in convergence problems, loss of robustness, and consumption of computing time. Resolutions for those systems have eluded many researchers. Hence, the increase in mathematical complexity seems not to be required or even justiiable. Besides, as demonstrated in Chapter 16, the present one-dimensional approach leads to good reproduction of the conditions in real equipment. Finally, it is important to remember that, like the model for moving beds, the present model also assumes that ash particles leaving the carbonaceous solid are incorporated into the inert solid phase (m = 3). Such particles would be generated during the consumption of carbonaceous solid if the exposed-core model were applied for the heterogeneous reactions. • Mass balances for the components in the freeboard are described by 3 dFF , j = ∑ (R het ,SF ,m , jγ mS) + R hom ,GF , jεS, 1 ≤ j ≤ 1000. dz m =1

(13.10)

It should be noted that in Equation 13.10, a chemical species j belongs to solid kind m. Assumption D above allows Equation 13.10 to be written. The last term (R hom,GF,j) concerns only the gas phase. It should be remembered that just one gas phase lows though the freeboard space. Equation 13.10 is similar to Equation 13.1, but it also includes chemical species from solid phases. The term F F indicates the net low in the upward direction. As noted in Chapter 3, the mass lows of solids tend to decrease for higher positions because of the disengaging process. The equations and correlation, which provide for the computation of the decay of particle mass low, are shown in the next chapter.

Bubbling Fluidized-Bed Combustion and Gasification Model

309

• Energy balances for gases in the freeboard are described by FGF c GF

3   dTGF = εS  − R QGF + ∑ (R CSFGF ,m + R hSFGF ,m ) − R CGFTF − R GFWF  dz m =1  

(13.11)

where 500

FGF = ∑ FF , j.

(13.12)

j =1

Equation 13.11 is similar to Equation 13.5. • Energy balances for the solids in the freeboard are described by FSF ,m c SF ,m

ε dTSF ,m  = (1 − ε)S  −R QSF ,m − (R CSFGF ,m + R hSFGF ,m ) 1− ε dz 

(13.13)

 −R RSFTF ,m − ∑ (R RSFSF ,m , n )  , 1 ≤ m ≤ 3 n =1  3

where FSF ,m =

1000

∑F

F, j

.

(13.14)

j = 500

Again, in Equation 13.14, chemical species j belongs to solid m. Equation 13.13 is similar to Equation 13.9.

13.3

BOUNDARY CONDITIONS

The sets of differential equations concerning the bed section (Equations 13.1 through 13.9) should be solved from the surface of distributor (z = 0) to the top of the bed (z = zD), whereas Equations 13.10 through 13.14 should be solved from that last position to the top of the freeboard (z = zF).

13.3.1

BOUNDARY CONDITIONS FOR GASES

As the conditions of the gas stream injected through the distributor (z = 0) are known, it is easy to set the total gas low rate (FG), its composition (wG,j), and temperature (TG) as: wGE, j,z = 0 = wGB, j,z = 0 = wG, j,z = 0 , 1 ≤ j ≤ 500

(13.15)

TGE,z = 0 = TGB,z = 0 = TG,z = 0 .

(13.16)

and

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Solid Fuels Combustion and Gasification

It should be noted that an approximation is made here because the temperature of gas mixture (TG,z=0) is assumed to be equal to the average temperature of gas in the plenum. Actually, the temperature of the gas increases a little when it passes through the distributor plate or system (Figure 13.3). The mass low rates of gas components injected into the emulsion and bubble phases are given by FGE, j,z = 0 = FGE,z = 0 wG, j,z = 0 , 1 ≤ j ≤ 500

(13.17)

FGB, j,z = 0 = FG , j,z = 0 − FGE, j,z = 0 , 1 ≤ j ≤ 500.

(13.18)

and

The fraction of the injected gas (FG,z=0) that is diverted to the emulsion phase (FGE,z=0) can be determined by the luidization dynamics. This topic is covered in Chapter 14. Naturally, for the freeboard, the boundary conditions are those at the top of the bed (z = zD). Consequently, once the system of differential equations related to bed section is solved, the conditions for the freeboard would be set by the following: FGF , j,z = zD = FGE, j,z = zD + FGB, j,z = zD , 1 ≤ j ≤ 500.

(13.19)

The temperature of the gas phase entering the freeboard is given by a simple mixedcup temperature of the gas leaving the bed from the emulsion and from the bubble phase, or TGF ,z = zD = T* +

FGBc GB (TGB,z = zD − T* ) + FGE c GE (TGE,z = zD − T* ) . FGBc GB + FGE c GE

(13.20)

The following points should be observed: • This model (Equation 13.15) allows a gas stream injected into the reactor to have any composition. Usually, air is used in combustors and mixtures of air and steam in gasiiers. However, there are several processes using pure oxygen, oxygen and steam, carbon dioxide, etc. • Also, a preheated mixture of the injected gas (Equation 13.16) can be used. • The total mass low (FG,z=0) at the given temperature and composition would determine the supericial velocity of gases at the distributor surface. Of course, that should be equal to or greater than the minimum luidization value (see Chapter 4). The average characteristics of particles in the bed would depend on the whole simulation. They would be computed after each iteration or solution of the differential system through the bed. This is better described in Chapter 16.

13.3.2

BOUNDARY CONDITIONS FOR SOLIDS

The most dificult conditions to set are those concerning the temperature of the various solids at the base of the bed. A solution is described on page 311.

311

Bubbling Fluidized-Bed Combustion and Gasification Model

TD,av

z

TG,z=0

Bed

TD,z=0

z=0 Distributor insulation or plate

xdist

Gas Plenum

TG,plenum

FIGURE 13.3 Scheme to illustrate the method used to set the boundary conditions for the temperatures of solid phases.

Consider the region of the distributor through which the mixture of gases is injected into the bed (Figure 13.3). After being injected into the plenum, the gas mixture passes through the distributor plate (see conigurations in Section 11.10.2) and enters the bed. For boilers and gasiiers, this stream is usually at a lower temperature than the bed average. This is also true between the distributor surface and the bed average. Thus, heat is transferred between the bed and the distributor surface with a rate equal to the rate transferred by conduction through the insulation plate. Therefore, it is possible to write λ dist ,z = 0 ( TD,z =0 − TG,plenum ) = α D −dist ( TDav − TD,z =0 ) . x dist

(13.21)

The various involved parameters are described below: 1. λdist is the average thermal conductivity for the distributor block between the temperatures at the distributor surface (TD,z=0) and for the average of the gas phase in the plenum (TG,plenum). Note that TG,plenum is not the average temperature of gas at the surface of distributor (TG,z=0) because the gas usually (at least in most cases of combustors and gasiiers) heats up when traveling through the distributor plate or any other device. 2. The heat transfer coeficient between the bed and the distributor surface (αD-dist) can be found in the literature [5] and is detailed in Chapter 15. In fact, it is assumed that this coeficient also includes the contribution due to the heat transfer by radiation. 3. The average bed temperature (TD,av) is given from a zero-dimensional energy balance using Equation 5.4 without the power production term. It should be performed after each iteration or solution of differential system (Equations

312

Solid Fuels Combustion and Gasification

13.1 through 13.9) throughout the bed. The energy balance should take into account the enthalpies (sensible plus formation) of all bed entering and leaving streams, including: 3.1. Gases injected through the distributor or at any other point of the bed. 3.2. Solids fed into the bed. In the cases of solid fuels, the formation enthalpy is not known. A good approximation is to apply the concept of representative formation enthalpy (based on LHV) as shown in Section B.7 (Appendix B) to compute the enthalpy of injected fuel. If the fuel is preheated, the sensible enthalpy should be added. 3.3. Gas and solid streams leaving at the top of the bed. Note that the withdrawal of energy due to low of solids entrained to the freeboard may be neglected. This is because in normal operations, almost all of those particles return to the bed. The temperature of gas leaving the bed and entering the freeboard can be assumed to be equal to the average in the bed. This is justiiable because of the lat proiles of temperatures in the majority of the bed (see Figure 3.11). 3.4. Solid withdrawn from overlow pipes or through the bottom of the bed. Like the temperature of the gas stream, the temperatures of those streams can be approximated by the average in the bed. The energy carried by the exiting low of unconverted solid fuel should be accounted for. The simulation would provide the conversion not just for carbon, but also for each component of the fuel. Hence, the composition of residue can be calculated, and its H HV (and LHV; see Equation B.41, Appendix B) can be calculated by formulas listed in Appendix B. In any case, the following approximation gives reasonable values: (L HV )residue = (L HV )original fuel (1 − f514 ). This would allow application of representative formation enthalpy for the residue (Equation B.45). 3.5. Fluids entering and leaving tubes (if any) immersed in the bed. Heat . (Q) would be lost to or gained from jackets, electrical heating devices, or others to walls. Similarly to the case discussed above for setting energy balances for solids in the bed, the above balance provides the closing condition to ensure that overall energy conservation is respected. 4. The temperature of the distributor surface (TD,z=0) can be computed from Equation 13.21. However, it should also match the average that accounts for the contribution from all phases at the bed base, or 3

TD,z = 0 = T* +

(TG ,z = 0 − T*)FG ,z = 0c G + ∑ (Tm ,z = 0 − T*)FH ,m c m m =1 3

FG ,z = 0c G + ∑ FH,m c m m =1

.

(13.22)

313

Bubbling Fluidized-Bed Combustion and Gasification Model

In the above, TG,z=0 is assumed to be equal to TG,plenum. Usually, such an approximation does not introduce signiicant deviations in the overall simulation. Equation 13.22 allows estimation of the temperature of the carbonaceous particles at the bed base (z = 0). Here, FH,m is the circulation low of particle type m (1 = carbonaceous, 2 = limestone [if any], 3 = inert [if any]). The correlations for computation of circulation rates are shown below. Equations 13.21 and 13.22 provide the parameters for an iterative convergence procedure until the computed values of the distributor surface (T D,z=0) are found. Then, the temperature for each solid phase at the bed base (z = 0) can be estimated. As inert solid (sand or other) and sulfur-absorbent (limestone, dolomite, or any other) particles do not introduce signiicant energy generation or consumption, their temperature at bed basis (z = 0) is assumed to be equal to that of the distributor surface. Usually, two opposing extreme temperatures at z = 0 are those of carbonaceous particles and the injected gas stream. The temperature of the inert (or sulfurabsorbent, if present) particles should be an intermediate value between those two. Therefore, this allows the temperature of carbonaceous fuel at the bed base to be set (Tm=1,z=0). Finally, it is necessary to set the mass low rate of particles at the top of the bed or at freeboard basis. The rate is given by lm

FSF , j,z = zD = ∑ FY,m , l,z = zD w PLD, j , 1 ≤ m ≤ 3, 501 ≤ j ≤ 1000.

(13.23)

l =1

The irst term in the summation represents the upward mass low of particle type m measured at the freeboard basis (or bed surface). This is called the entrainment low at that position. The phenomenon of entrainment is explained in Chapter 3, and methods of computing it, as well other parameters, are shown in Chapters 14 and 15.

13.4

EXERCISES

13.4.1

PROBLEM 13.1*

Indicate where and how the following factors could be included in the overall energy balance for the bed: a) A water jacket—where liquid water enters and steam or hot water leaves— surrounds the bed. b) Steam, generated at the water jacket, is injected with the gas mixture into the plenum and then into the bed. c) Endothermic reactions taking place inside the tubes of a bank immersed in the bed.

13.4.2

PROBLEM 13.2*

Develop a block diagram to compute the boundary conditions for solid phases (Tm,z=0).

314

Solid Fuels Combustion and Gasification

13.4.3

PROBLEM 13.3**

Write a routine to compute the boundary conditions. Assume the following inputs as known: • Mass lows, temperatures, and compositions of all injected gases and feeding solids into the bed • Carbon conversion in the bed • Mass lows, temperatures, and compositions of the gas leaving the bed, as well as solids leaving the bed through overlow tube

13.4.4

PROBLEM 13.4**

Include terms of diffusion and conduction transfers in the axial (vertical) direction in the equations in Section 13.2.2.

13.4.5

PROBLEM 13.5***

After solving Problem 13.4, set the boundary conditions for the resulting system of differential equations.

13.4.6

PROBLEM 13.6**

Write the differential equations of momentum transfer in the bed. Do not include a dissipative term (or viscosity).

13.4.7

PROBLEM 13.7****

Write the differential mass and energy equations if a two-dimensional approach is used for modeling a luidized-bed combustor or gasiier.

13.4.8

PROBLEM 13.8****

Include the equations for momentum transfer in the above (two-dimensional) approach. Neglect the interference of tube banks.

13.4.9

PROBLEM 13.9****

Set the boundary conditions for a complete two-dimensional model of a bubbling luidized bed.

REFERENCES 1. Kunii, D., and Levenspiel, O., Fluidization Engineering, 2nd Ed., John Wiley, New York, 1991. 2. Davidson, J.F., and Harrison, D., Fluidized Particles, Cambridge University Press, Cambridge, United Kingdom, 1963. 3. de Souza-Santos, M.L., Modelling and Simulation of Fluidized-Bed Boilers and Gasiiers for Carbonaceous Solids, PhD thesis, University of Shefield, Shefield, United Kingdom, 1987. 4. de Souza-Santos, M.L., Comprehensive modelling and simulation of luidized-bed boilers and gasiiers, Fuel, 68, 1507–1521, 1989. 5. Zhang, G.T., and Ouyang, F., Heat transfer between the luidized bed and the distributor plate, Ind. Eng. Chem. Process Des. Dev., 24(2), 430–433, 1985.

14 Fluidization Dynamics CONTENTS 14.1 Introduction ............................................................................................... 315 14.2 Splitting of Gas Injected into a Bed .......................................................... 316 14.2.1 Quantitative Description..............................................................317 14.2.2 Void Fractions.............................................................................. 318 14.2.3 Volumes and Areas ...................................................................... 320 14.2.3.1 Volume Ratios ............................................................. 320 14.2.3.2 Bubble Area-Volume Ratio ......................................... 320 14.2.3.3 Area Ratios.................................................................. 320 14.3 Bubble Characteristics and Behavior ........................................................ 321 14.4 Circulation of Particles.............................................................................. 323 14.4.1 Background.................................................................................. 324 14.4.2 A Review of Soo’s Work.............................................................. 326 14.5 Entrainment and Elutriation ...................................................................... 331 14.5.1 TDH ............................................................................................. 333 14.6 Particle Size Distribution .......................................................................... 333 14.7 Recycling of Particles................................................................................ 335 14.8 Segregation ................................................................................................ 337 14.9 Areas and Volumes at Freeboard .............................................................. 338 14.10 Mass and Volume Fractions of Solids ....................................................... 338 14.11 Further Study ............................................................................................ 339 14.12 Exercises ................................................................................................... 339 14.12.1 Problem 14.1 .............................................................................. 339 14.12.2 Problem 14.2.............................................................................. 339 14.12.3 Problem 14.3.............................................................................. 339 14.12.4 Problem 14.3..............................................................................340 References ..............................................................................................................340

14.1

INTRODUCTION

Any model of luidized-bed equipment requires information regarding its dynamics. As the present model does not include the momentum equations, such information should be given through empirical or semiempirical correlations. They would provide methods or equations to describe the following: • Dynamics related to the bubble phase. This item includes the rate of growth of the bubbles in the bed, their pattern, their shape, the velocity of their ascent, maximum or critical size, etc. Another important question is, What is the fraction of the gas injected through the distributor that goes to the bubble 315

316

Solid Fuels Combustion and Gasification

phase? In addition, there are factors that may affect the behavior of the bubbles in the regions of tube banks that may be immersed in the bed. The equations that determine the bubble size are particularly important because predictions of slugging-low should be made to avoid operational problems [1]. • Dynamics of particles. This item includes, among many other factors, the path and velocity of particles in the bed and in the freeboard, the rate of circulation (or turnover) in the bed, rate of ines generation, mass lows of particles in the freeboard, etc. Moreover, some operational situations may lead to segregation of the lighter particles from the rest of the bed. These lighter particles tend to loat on the surface of the bed. The correlations to recognize this problem should be implemented into the program. If not, simulations assuming perfect mixing of particles in the bed might render false all predictions. Actually, segregations usually cause signiicant operational problems. As an example, let us imagine a bed with wood chips and sand. If the luidization is not vigorous enough, the wood chips could loat at the bed surface. Consequently, since the fuel would not be present inside the bed, its combustion would occur near the bed surface. The temperature there might increase above the ash-softening one, and agglutination of particles may start, resulting in bed collapse. The reverse is also dangerous, i.e., the inert or sulfur-absorbent portion segregated at the bed top, leaving the bed with too high a concentration of carbon, which may also lead to temperatures surpassing ash-softening values. Despite the complexity of above phenomena, the correlations presented below are relatively simple.

14.2

SPLITTING OF GAS INJECTED INTO A BED

As described in the previous chapter, the gas coming from the plenum passes through the distributor and enters the bed. At that point, a fraction would be diverted to the emulsion phase and the remaining to bubbles. The splitting ratio between the emulsion and bubble phases is among the important pieces of information needed for any model of a bubbling luidized bed. A simpliied model assumes the two-phase theory [2], by which minimum luidization conditions characterize the emulsion phase. Any excess of gas injected through the distributor would be diverted to the bubble phase. Following that theory, other researchers [3, 4] veriied that for positions above the distributor, the emulsion progressively departs from the minimum luidization condition. Therefore, the void fraction in the emulsion at points far from the distributor was higher than the values found at positions near the base of the bed. These researchers created a model called the three-phase theory, which assumes that most of the emulsion is under minimum luidization conditions, whereas the layers coating the bubbles have void fractions higher than the minimum luidization conditions. Such layers are known as clouds. Actually, the presence of clouds can be veriied in experiments. Nevertheless, the present model uses the following simpliication: 1. The clouds are part of the emulsion. 2. At the distributor (z = 0), the two-phase theory is adopted.

317

Fluidization Dynamics

3. The two-phase theory is abandoned for positions above the distributor, and the mass transfers between these phases dictate the lows of gas through the emulsion and bubble. Therefore, the above method preserves the experimental veriication of two-phase theory for points near the distributor, while maintaining the integrity of mass and energy balances, shown in the last chapter. The two-phase theory is used only for setting the boundary conditions. This approach proved to be very reasonable, and computations showed how the void fraction of the emulsion deviated from the minimum luidization value [5–7].

14.2.1

QUANTITATIVE DESCRIPTION

The quantitative description of luidization dynamics starts with the deinition of emulsion gas velocity as UE =

FGE . ρGESE

(14.1)

UE is the average velocity in the emulsion at a given axial position z. That velocity may vary considerably throughout the bed. This is even more noticeable in cases of combustors and gasiiers because of sharp variations in gas temperature. At a given height in the bed, the total mass low of gas is given by the contributions from emulsion and the bubbles, or FG = FGE + FGB .

(14.2)

Similarly, the bed cross-sectional area S is divided into the portion occupied by the emulsion (SE) and that occupied by the bubble (SB) phases, or S = SE + SB.

(14.3)

Now, if the two-phase theory is applied at the base of the bed, it is possible to write U E,z = 0 = U mf ,z = 0 .

(14.4)

It should be remembered that, mainly in reacting systems, the gas properties, such as temperature, pressure, and composition, change with the height (z). Thus, changes occur even for the minimum luidization velocity, which can be computed by the equations presented in Chapter 4. It is actually impossible to maintain a minimum luidization regime throughout the bed for reactors where the gas temperature increases throughout the bed height. This is true even if the injected low is exactly at the minimum luidization conditions. The reduction in gas density results in increases of supericial gas velocity, departing from the minimum luidization velocity. This is the case for all combustors and most gasiiers.

318

Solid Fuels Combustion and Gasification

At each height z, the fractions SE and SB of reactor cross-sectional area are deduced by considering that the volume occupied by the bubble phase is due to the bed expansion from the static condition, as follows:

Hence,

SBz D = S ( z D − z Dst ) .

(14.5)

zD    1  SB = S  1 − st  = S  1 − . zD  fexp   

(14.6)

The bed expansion factor indicates the ratio between the bed in a given situation and the respective volume at minimum luidization. A correlation [8] that is valid within a wide range of conditions is given by fb exp = 1 +

.083 1.032( U − U mf )0.57 ρ0GA 0.166 0.063 0.445 ρP U mf d D

(14.7)

to be used when dD < 0.0635 m and fb exp = 1 +

14.314( U − U mf )0.738 d1P.006ρ0P.376 0.126 0.937 ρGA U mf

(14.8)

for dD ≥ 0.0635 m. Since the parameters involved in those correlations are functions of axial positions z, so is factor f bexp. Usually, SB/SE increases with height because of increases in the difference between local gas supericial velocity and minimum luidization velocity (U – Umf ). Since the bed is modeled as separated into bubble and emulsion phases (Figure 13.1), the velocity of the gases through the bubble section (U B) and through the emulsion (UE) is computed by the ratio between the respective volume lows and cross-sectional areas (SB and SE). These values can be calculated at each point during the solution of the differential mass and energy balances.

14.2.2 VOID FRACTIONS Void fraction is deined as the fraction of the total volume that is occupied by gas; therefore the following hold true: • The total void fraction in the bed is given by ε=

VG . V

(14.9)

Here, V is the total volume of the luidized bed, and VG is the volume occupied by gas. • The void fraction caused by the presence of bubbles in the bed is given by εB =

VB . V

(14.10)

319

Fluidization Dynamics

As the bubbles are assumed to be free of particles, VB is the total volume of bubbles, as well as the volume occupied by gas in the bubbles. • The void fraction in the emulsion is given by εE =

VGE VE

(14.11)

where VE is the volume of the bed taken up by the emulsion gas, and VE is the volume occupied by the emulsion phase (gas plus solids). Since the following holds true: VG = VB + VGE ,

(14.12)

it is possible to write ε=

VB VGE V V V + = ε B + GE E = ε B + ε E E V V VE V V

= εB + εE or

V − VB = ε B + ε E (1 − ε B ) V εB = 1 −

1− ε . 1 − εE

(14.13)

(14.14)

The two-phase theory is adopted at the base of the bed; therefore, ε E,z = 0 = ε mf ,z = 0 .

(14.15)

An empirical correlation for void fraction at minimum luidization is described [9] by 1/ 3

 1  εmf =   .  14φA 

(14.16)

It should be stressed that this is an empirical relationship, and no universal relationship really exists between void fraction and sphericity. The determination of average sphericity of particles is a standard laboratory procedure. For lack of better value, the number 0.7 can be adopted for most ground materials, such as sand, coal, and limestone. One should be aware that unless it is at the basis of the bed (z = 0), the void fraction in the emulsion εE is different from the respective value at minimum luidization (εmf ). The computation of the void fraction in the emulsion can be made by the following correlation [10]: 1

 U  6.7 ε E = εmf  E  .  U mf 

(14.17)

320

Solid Fuels Combustion and Gasification

Finally, from the deinition of bed expansion factor, it is easy to see that ε = 1−

1 − ε mf . fb exp

(14.18)

Relations to estimate void fractions in the freeboard section are presented in Section 14.9.

14.2.3

VOLUMES AND AREAS

The mass and energy transfer parameters require computations of interface areas and volumes of involved phases. Their differential areas and volumes are deined below. 14.2.3.1 Volume Ratios The differential volume of the reactor is simply given by dV = S. dz

(14.19)

The variation of volume occupied by the emulsion gas against bed height is given by dVGE dVGE dVE dV = = ε E (1 − ε B )S, (14.20) dz dVE dV dz and the equivalent for the bubble phase is given by dVGB dVB dVB dV = = = ε BS. dz dz dV dz

(14.21)

The differential volume of particle species m per unit of bed height can be computed using dVPE,m dVPE,m dVPE dVE = = fm′′′(1 − ε E )(1 − ε B )S. dz dVPE dVE dz

(14.22)

14.2.3.2 Bubble Area-Volume Ratio The ratio between the surface area and bubble volume is simply given by dA B

2 dz = dA B = A B = πd B = 6 . 3 dVB dB dVB VB π d B dz 6

(14.23)

14.2.3.3 Area Ratios Using the above relations, the available surface area of bubbles per unit of the bed height is provided by

321

Fluidization Dynamics

dA B dA B dVB 6 = = ε BS. dz dVB dz dB

(14.24)

The available external surface area of a particle kind m per unit of bed height is deduced as follows: dA PE,m dA PE,m dVPE,m dVPE dVE 6 fm′′′(1 − ε E )(1− ε B )S. = = dz dVPE,m dVPE dVE dz dm

14.3

(14.25)

BUBBLE CHARACTERISTICS AND BEHAVIOR

While bubbles rise through the bed, their diameters increase because of decreases in the static pressure and coalescence of neighboring bubbles. Various correlations describing the bubble diameter according the vertical position z are found in the literature. Mori and Wen [11] proposed the following:  z  d B = d Bmax − (d Bmax − d Bmin ) exp  −0.3  d D  

(14.26)

where the maximum attainable diameter is provided by 0.4

d Bmax = 2.59g −0.2 S( U − U mf )  .

(14.27)

The minimum diameter (or diameter at z = 0) depends on the type of distributor. For perforated plates, it is given by d Bmin = 1.38g

−0.2

 S(U − U mf )    n orif  

0.4

(14.28a)

and for porous plates, by d Bmin = 3.77

(U − U mf )2 . g

(14.28b)

Other correlations [12], such as d B = 0.430(U − U mf )0.4 (z + 0.1272)0.8 g −0.2 ,

(14.29)

can be applied as well. For most cases, this equation produces results similar to those from Mori and Wen [11]. However, Equation 14.29 does not consider the maximum size of a stable bubble or options for distributor designing. An improvement was introduced by Horio and Nonaka [13]. Despite using the same equations for the maximum (Equation 14.27) and minimum (Equations 14.28a and 14.28b) bubble diameters, they added a series of considerations of the frequency

322

Solid Fuels Combustion and Gasification

of bubble formations combined with the rate of bubble coalescence to obtain the following correlation:  d B − d Beq   d Bmin − d Beq 

1−

   

c1 c2

 d B + C3   d B + C3 min 

1+

   

c1 c2

 z − z0 = exp  −0.3 dD 

 . 

(14.30)

Here, z0 is the reference position at which the bubbles are generated, and the parameters denoted by C can be computed from the following:  dD   g  C1 = 2.56 × 10 −2 U mf  4d  C2 =  C12 + Bmax  dD  

0.5

(14.31)

0.5

(14.32)

C3 = 0.25 d D (C1 + C2 )2

(14.33) 2

d Beq

0.5     d = 0.25d D  −C1 +  C12 + 4 Bmax   . d D    

(14.34)

Compared with previous works, the computations using the above correlations lead to smaller bubbles and gentler increases of diameter against bed height. They also agree with experimental data better. The velocity of bubble rise can be computed by the following correlation [4]: U B = U − U mf + 0.711(gd B )1/ 2 .

(14.35)

The most important conclusions drawn from the above equations are as follows: • The bubble size is a strong function of vertical distance traveled by the bubble. • The bubble tends to a maximum stable size and breaks into smaller ones when surpassing that threshold. • The bubble size is also a strong function of the difference between the actual gas supericial velocity and the minimum luidization value. Since in cases of combustors or gasiiers, the injected cold gas experiences fast expansion just above the distributor (z = 0), so does bubble dimensions. Hence, if small average bubble sizes are intended throughout the bed, a good strategy is to preheat the gas injected through the distributor. This is particularly important in cases of gasiiers in which the amount of oxygen transferred from the bubbles to the emulsion increases with the available speciic area of interface between these phases. This leads to even higher mass transfers

Fluidization Dynamics

323

of oxygen between bubbles and emulsion. Thus, oxygen can be completely consumed at points nearer the distributor, leaving the remaining at reducing conditions, which allows for the accumulation of fuel gases. Moreover, larger residence times are provided for bubbles in the bed, because their ascending velocity decreases with size.

14.4

CIRCULATION OF PARTICLES

Particles circulate at fast rates inside luidized beds. Figure 13.2 (Chapter 13) illustrates a possible situation, where the paths of two basic streams are shown. Of course, this is a very simpliied scheme, because actual circulations are composed of many toroidal paths. With an increase in the supericial velocity of the gas in the bed, these toroids can split in two, four, eight, etc., as illustrated by Figure 14.1. The circulation low of particles is given by the contribution of all upward lows. A similar mass low is obtained by adding all downward movements. The parameter that describes the low rate of a particulate solid species m (m = 1–3) in the bed is represented by the symbol FH,m. If this value is divided by the bed cross-sectional area, the circulation lux GH,m is obtained. Both FH,m and GH,m are functions of the axial position in the bed. Usual values for the circulation lux are in the range of 102 to 103 kg m–2 s–1. Therefore, it is easy to understand the homogeneity of each solid-phase composition, as well as the average temperature throughout a bubbling luidized-bed. As could be seen in Chapter 13, the rate of particle turnover or circulation inside a luidized bubbling bed is an extremely important factor. It strongly inluences the

FIGURE 14.1 Splitting of circulation routes due to increases in gas velocity through a luidized bed.

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Solid Fuels Combustion and Gasification

mass and heat transfer rates between the various phases, as well as between these phases and immersed tubes and walls. Low circulation rates could lead to the development of high-temperature spots, which might be above the particle ash-softening point. In such situations, agglomeration of particles may take place, and the process could follow two routes: a) Formation of lumps with a stable maximum size. These drop to the bottom of the bed because their minimum luidization velocities are above the gas velocity in the bed. Indeed, some industrial systems provide continuous or automatic removal of lumps from the distributor. b) Continuous and fast growth of lumps, leading to the total collapse of the bed. This is called bed melting, and it is among the most equipment-damaging problems of luidized-beds. Usually, the melting solids adhere to the reactor internal refractory walls, which may force the replacement of that refractory lining. If tubes are immersed in the bed, they would also be damaged. Another important aspect of the circulation rate of particles is its strong inluence on the erosion and corrosion rates of tubes and walls in the bed [14, 15]. As we have seen, precise correlation of the circulation rates of individual particle solid phases improves the precision and extends the range of applicability of a simulation program.

14.4.1

BACKGROUND

Among the pioneers in the area, Talmor and Benenati [16] proposed the average circulation rate of particles to be represented by the following equation: G H = 785 (U − U mf ) exp( −6630 d P ).

(14.36)

Therefore, they found a strong inluence of the difference between the actual luidization velocity and the minimum value on the average circulation rate. The dependence on the average particle diameter gives some insight into the problem as well. If two beds operate with the same luidized material and similar differences between actual and minimum luidization velocities, the one with larger particles would present a signiicantly smaller circulation rate. However, the expression above is empirical and is useful only within a relatively small range of operational conditions. For instance, the exponential dependence on the average particle diameter imposes strong limits for extrapolations or applications to cases outside the range of experimental determinations (67–660 µm). An attempt [5, 6] to apply the above equation in order to predict the rate of circulation of an individual solid species m in a multiple-particle bed has led to the following relationship: G H,m = ρm (1 − ε mf )( U − U mf ) exp(−6630 d m )fm .

(14.37)

Of course, this is a strong simpliication of the problem. However, it responds well in cases in which the average diameters of each solid particle species m were not too

Fluidization Dynamics

325

different from each other. On the other hand, Equation 14.37 has also led to poor results, including failures in the solution of the differential equation system, when used for conditions outside the experimental range of Talmor and Benenati’s [16] work. This occurs in cases when particles with very different diameters or densities were present in the same bed, such as, for example, sand and wood particles. In addition, the exponential dependence on the diameter led to very low circulation rates for relatively large particles, therefore predicting very high temperature derivatives (Equation 13.9, Chapter 13) at some critical points in the bed. Geldart [17] wrote an excellent review of the subject and proposed the following semiempirical relationship: G H = ρP (1 − ε mf )(U − U mf )(C1 + 0.38C2 ) B3 .

(14.38)

The above relationship can be applied to a broader range of average particle diameters than Talmor and Benenati’s equation [16]. The coeficients C1, C2, and C3 depend on the luidization parameters and characteristics of luidized solid particles. Like the previous adaptation, a possible one has been tried for the above work, and it is described by the following equation: G H,m = ρm (1 − ε mf )( U − U mf )(C1,m + 0.38C2,m )C3,m .

(14.39)

Unfortunately, applications of the above equation to beds of multiple particle species led to large deviations between simulation and experimentation regarding operational parameters. Gidaspow and coworkers [18–23] developed solutions for systems of partial differential equations describing the momentum conservation in isothermal luidized beds. Their solutions for the velocity ields compared well against measurements. On the other hand, for a mixture with many different kinds of particles, the system of differential equations increases considerably. The application of such an approach in a comprehensive simulation would consume a great deal of computational time. Moreover, it must be remembered that in order to apply the model to furnaces and other reactors, it should be rewritten for nonisothermal and reacting beds. The additional coupling of mass and energy balances would greatly increase the complexity of the numerical problem, and using such a system as an appendix of any comprehensive simulation program would be cumbersome. Soo [24–26] reduced the number of momentum equations by assuming reasonable simpliications, allowing fast computations. At the same time, he proposed an ingenious set of solutions for the velocity ields that implicitly satisfy the continuity equations. His results brought new insights into the hydrodynamics of the bed, as well as the interplay between particles and bubble movements. In most cases, model and experiment results agree. Nevertheless, it seems that under certain conditions, the lack of convective terms in his momentum equations could lead to deviations against experimental measurements [27]. In addition, his approach considered just one solid species. The works of Soo were critically reviewed [28, 29], as described below.

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Solid Fuels Combustion and Gasification

14.4.2 A REVIEW OF SOO’S WORK Assume a luidized bed, with radius r0 and height zD. The basic equations for the mass and momentum continuity as put forth by Soo [25] are as follows:  1   ∂(ru G, r )   ∂u G,z   r   ∂r  +  ∂z  = 0     

(14.40)

 1   ∂(ru p, r )   ∂u p,z  r   ∂r  +  ∂z    

(14.41)

 =0 

 1 ∂  ∂u p,z  ∂ 2 u p,z  0 = ρp α p ( u G ,z − u p,z ) + µ p  r  + ∂z 2  − ρp g   r ∂r  ∂r  0=−

∂P − ρpα p u G,z,z = 0 − (ρp + ρG )g. ∂z

(14.42)

(14.43)

The boundary necessary and suficient boundary conditions are as follows: • One for uG,r in the radial direction: u G, r (r0 , z) = 0, 0 ≤ z ≤ z D

(14.44)

• One for uG,z in the axial direction: u G,z (r, 0) = u G,z,z = 0 , 0 ≤ r ≤ r0

(14.45)

• One for up,r in the radial direction: u p, r (r0 , z) = 0, 0 ≤ z ≤ z D

(14.46)

• Two for up,z in the radial direction: u p,z (r0 , z) = u p,z, W , 0 ≤ z ≤ z D ∂u p,z ∂r

= 0, 0 ≤ z ≤ z D

(14.47) (14.48)

r =0

• Two for up,z in the axial direction: u p,z (r, 0) = 0, 0 ≤ r ≤ r0

(14.49)

u p,z (r, z D ) = 0, 0 ≤ r ≤ r0

(14.50)

It should be noted that no entrainment of particles to the freeboard is allowed here.

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Fluidization Dynamics

In addition to those, Soo [25] imposed the following conditions for uG,z in the radial direction: u G,z (r0 , z) = u G,z, W , 0 ≤ z ≤ z D (14.51) ∂u G,z ∂r

= 0, 0 ≤ z ≤ z D .

(14.52)

r =0

Although they are not necessary for a solution of the continuity and momentum differential equations, the above additional conditions are acceptable from a physical point of view. Moreover, they permit the inclusion of more terms in the series proposed by Soo as approximated solutions and given by u G ,z = u G ,z ,z = 0

r  πz  ∞ + sin   ∑ Cn   z  D  n = 0  r0 

n

(14.53)

 πr   πz  ∞  C   r  u G ,r = −  0  cos   ∑  n     zD   z D  n = 0  n + 2   r0   πz  ∞  r  u p,z = sin   ∑ a n    z D  n = 0  r0 

n +1

(14.54)

n

 πr   πz  ∞  a   r  u p, r = −  0  cos   ∑  n     zD   z D  n = 0  n + 2   r0 

(14.55) n +1

.

(14.56)

The great advantage of these forms is that they intrinsically satisfy the continuity equations 14.40 and 14.41. In addition, they allow simple interpretations of the movement of particles in the bed. Figure 14.2 is intended to be simply a schematic representation of the axial velocity ield of particles given by Equation 14.55 and does not show the possibility of sliding conditions (14.47 and 14.51). When the proposed solutions are substituted into Equation 14.42, the following equation is obtained: n n  r  πz  ∞  r    πz  ∞ ρpα p  u G,z,z = 0 + sin   ∑ C n   − sin   ∑ a n     z D  n = 0  r0    z D  n = 0  r0  

  πz  ∞ a r + µ p sin   ∑ n 2 2n   z r0  r0  D = n 0   

n −2

−

(14.57)

 πz  ∞  r   π sin   ∑ a n    − ρpg = 0. 2 zD  z D  n = 0  r0   2

n

The inverse relaxation time αp is used by Soo, among other authors [18–23], to simplify the problem of momentum transfer between the gas and particulate phases. That relaxation factor can be better understood as the linear momentum transfer coeficient from the luid to the particle. Nevertheless, the use of such simpliied

328

Solid Fuels Combustion and Gasification Upward global movement (GH)

Downward global movement (GH)

FIGURE 14.2 Possible axial velocity ield of particles in a bed, as predicted by Soo [25].

treatment brings incoherence to the originally proposed solution (Equations 14.53 through 14.56) because it prevents coeficients an and Cn from being set as constants. Soo [25] smoothed the deviations by assuming the following approximation: αp =

g , u G ,z ,z = 0 + C 0 − a 0

(14.58)

which represents an application of Equation 14.57 for inviscid low at the center of the bed (z = z0/2 , r = 0). Following this, he assumed equality or at least closeness between the values of maximum velocities of particulate and the gas phases, which is symbolized by C0 − a 0 2.38/dD. Here, N Re,m ,1 =

ρG ( U G − u T,m ,1 )z d m ,1 µG

.

(14.77)

Several discrepancies have been observed in the reported correlation and values for the parameter aY in Equation 14.70. Wen and Chen [32] could not ind a correlation to determine this parameter. Their experimental work, as well as the published studies, showed that its value might vary from 3.5 to 6.4 m–1. They maintained that “since the value is not very sensitive in the determination of the entrainment rate, it is recommended that a value of 4.0 m–1 be used for a system which no information on entrainment rate is available” [32]. On the other hand, Lewis et al. [33] reported values from 0.4 to 0.8 s–1 for the product between aY and UG. Walsh et al. [34] found the same values to be between 2.7 and 3.7 s–1. These igures contradict a graph presented by Wen and Chen [32], in which the scattering of the characteristic length for the decay of particle low aY against the gas supericial velocity UG, cannot ensure the validity of the above correlation. In view of that and on the basis of the number of experimental tests carried out by Wen and Chen [32], it is recommended that above suggestion by them should be followed.

14.5.1

TDH

The concept of TDH is an approximation because from a rigorous point of view, there is always a decrease in the upward low of particles, no matter how high the freeboard is. The gas velocity is not uniform throughout the freeboard cross-section, and layers near the vertical walls present lower velocities, which allow even ine particles to slide down. Nevertheless, from a practical point of view, it is possible to ind an approximate deinition [32] and set the TDH as the height at which the entrainment rate is within 1% of the elutriation rate; it is therefore given by the following equation: z TDH = z D +

14.6

1  FY ,z = z D  ln . a Y  0.01FX 

(14.78)

PARTICLE SIZE DISTRIBUTION

A very important part of any model dealing with particulate solids is the determination of the size distribution under steady-state conditions. Several phenomena contribute to the modiication of the original particle size distribution of a solid species fed into a luidized bed. These effects are as follows: • Reduction of size due to chemical reactions, mainly for the carbonaceous particles when the exposed-core process is the mandatory mode

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Solid Fuels Combustion and Gasification

• Increase in the average size due to the entrainment, which carries the ine particles to the freeboard • Decrease in the average size due to generation of ines by attrition between particles and between particles and internal or immersed surfaces (tubes, walls, etc.) To provide a method by which to estimate the mass fraction wm,l of particles of kind m at level l (from now on, called m,l class) under steady-state conditions in the bed, a mass balance for that class should be made in the bed region. This balance should account for the combined effects of all interfering factors as mentioned above. For a bed in which no heterogeneous chemical reactions take place, the mass balance for particle class m,l is given by wm ,1FLD,m + Γ m ,1 = w I,m ,1FI,m + Γ m ,1+1 + (FY,m ,1,z = zD − FY ,m ,1,z = zF ) + FK ,m ,1. (14.79) The left side of the equation represents the losses of the bed for particle class m,l, and the right side represents the gains. The irst term on the left is the loss of class m,l due to the streams leaving the bed for the following reasons: • Forced withdrawals, such as overlow, which may be necessary to keep the bed height constant • Flow to the freeboard The second term on the left side of Equation 14.79 represents the rate of loss of class m,l to an inferior level m,l-1 due to attrition. The irst term on the right side represents the gain of particles of class m,l due to feeding into the bed. The second represents the gain of particles of class m,l due to the attrition of particles from the size level m,l+1, which is immediately above m,l. The difference of terms inside the parentheses represents the net return of particles from the freeboard, in other words, those particles of class m,l that enter the freeboard but are not withdrawn at its top. The last term represents the eventual, forced recycling to the bed of particles collected by cyclones or cleaning systems. When chemical consumption affects the particles of type m, the irst term on the right side of Equation 14.79 should consider that as losses. This is the typical case when the exposed core describes the heterogeneous reactions involving carbonaceous solid particles. In this situation, one may just replace the feeding rate FI,m with the leaving rate of particle class m,l from the system, FL,m. A good approximation is to multiply FI,m by the fraction of carbon conversion in order to obtain F L,m. The mass low of particles belonging to class m,l leaving the bed can be computed by (14.80) FLD ,m = FL ,m + FK ,m + FY ,m ,z =z − FY ,m ,z =z . D

F

The mass low of particles recovered by the cyclone system and recycled to the bed can be computed by FK ,m ,1 = FY,m ,z = zF ηcy ,m ,1fK .

(14.81)

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Fluidization Dynamics

A method for estimating the cyclone eficiency (for each class m,l) is presented in the next section. The factor f K represents the imposed recycling ratio, or the mass fraction of the particles collected by the cyclone, which are actually reinjected into the bed. The rate of production of ines due to attrition can be computed [35] by Γ m ,1 = Θm M D ( U G − U mf )fm wm ,1wsm ,1.

(14.82)

The parameter wsm ,1 is the result of contributions from all particles with diameters below dm,l, or wsm ,1 = 1 −

lmax

∑f

m

wm, k .

(14.83)

k = l +1

The upper value lmax is the maximum size found among particles of kind m. MD is the total mass of solids held by the bed. Once again, all these values correspond to the steady-state condition. For the sake of simplicity, the solid friability coeficients (Θm) have been taken as constants. However, other works [36–38] show that they are not truly constants. In particular, Vaux and Schruben [38] provide a relationship between the rate of production of ines and the time from the start of operation. It is known that the production rate of ines declines asymptotically with time and approaches a constant value. That limiting situation characterizes a steady-state condition. For bituminous coal and limestone, the friability constants are found in the literature [35] to be 9.11 × 10 –6 and 2.73 × 10 –6 m–1, respectively. From the above discussion, it is very likely that these values correspond to a steady-state condition. The friability factor for the inert portion (sand) can be assumed as 7.30 × 10 –6 m–1. As described elsewhere [5], this igure was calculated on the basis of the abrasion coeficient for that solid. Usually, a simple linear rule should be enough to estimate the value. Equation 14.79 implies that a recurrence or iteration procedure should be used to compute the particle size distributions for each solid m in the bed. The following method can be used: 1. The mass fraction of the largest particles, lmax, in the distribution cannot increase because of the generation of ines. Therefore, Γm,lmax+1 is equal to 0. 2. Apply Equations 14.79 through 14.82 to calculate the new wm,lmax. 3. Use Equation 14.82 to compute Γm,lmax. 4. As described above, use Equation 14.79 to compute wm,lmax-1 and Γm,lmax-1. 5. Repeat the process until l = 1.

14.7

RECYCLING OF PARTICLES

The entrainment and elutriation processes wash ine particles from the system (bed plus freeboard). Therefore, the residence times in the system for larger particles tend to be higher than for smaller ones. For carbonaceous solid fuel, that mechanism usually leads to lower carbon conversions for smaller particles compared with bigger

336

Solid Fuels Combustion and Gasification

ones. This may not be desirable, and recycling a portion or all of the particles collected by the cyclone can be used to increase the residence time of smaller ones. Despite that, recycling of ines is not always convenient because it leads to a decrease in the average size of particles in the bed. If the same rate of gas injection into the bed is maintained, increases in the difference between actual gas velocity and the minimum luidization will occur. Large bubbles will result, and even slugging low might be reached. In addition, increases the void fraction in the bed will be observed. If the bed height is maintained, the overall residence time of particles in the bed will decrease. Another consequence of recycling is to decrease the temperature in the bed, because particles usually cool during travel through the freeboard. If the low of recycling is high enough, the process can lead to a gradual decline of the temperature in the bed. The results can vary from a decrease in process eficiency to a complete shutdown because combustion or gasiication can no longer sustained. Therefore, recycling should be applied with caution, and comprehensive simulation may prevent low eficiencies or even operational embarrassments. One method of controlling undesired effects is to select partial recycling. The balance of particle size described above includes the term representing the low of particles recycled from the collecting system. However, the collection eficiency for each particle class m,l should be known. There are several methods for the computation of eficiencies, and a simple but fairly precise approach is suggested by Leith and Mehta [39], which describes the cyclone eficiency as  1  ηcy,m ,1 = 1 − exp  −2 ( a cy Bcy,m ,1 ) . 2n cy + 2  

(14.84)

Here, the parameter acy is a function of the cyclone geometry, which luctuates around 50 for a wide range of standard commercial designs. The parameter Bcy,m,l is related to the terminal velocity of the corresponding particle by Bcy ,m ,1 =

u T,m ,1u cy,G (n cy + 1). gd cy

(14.85)

The gas inlet velocity, ucy,G, is calculated using the fact that standard cyclones have a square cross-section entrance with area given by 2 Scy,entry = 0.125d cy .

(14.86)

Finally, the coeficient ncy is given from  ( 39.4d cy )0.14   T 0.3   cy  . n cy = 1 − 1 − 2.5    283 

(14.87)

Here, the value dcy is the representative diameter of the cyclone (internal in the cylindrical section), and the temperature Tcy is the average temperature throughout the cyclone and can be assumed to be the average value at the top of freeboard.

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Fluidization Dynamics

14.8 SEGREGATION As mentioned before, good operations of bubbling luidized combustors or gasiiers should avoid the phenomenon of particle segregation. A set of convenient relations that allows the prediction of these processes has been published by Nienow et al. [40]. Based on that work, a procedure to determine the various parameters and to incorporate the calculation into a comprehensive simulation program is described as follows: 1. After solution of the system presented in Chapter 13, determine the complete conditions of the steady-state regime, including the average diameter and density of each solid species m (carbonaceous = 1, sulfur absorbent = 2, inert = 3). The values computed for the middle of the bed would be acceptable. 2. Compute the minimum luidization velocity for each of those particulate species, Umf,m. For that, the composition, temperature, and pressure of the emulsion gas in the middle of the bed can be used. 3. Use the term mH (or heavy) for the solid species with the highest minimum luidization velocity, and use m L for the one with the lowest. 4. Calculate the mixing takeover velocity, i.e., supericial velocity of gas in the bed at which the segregation would be destroyed, as follows: 0.7 1.2 −1.1 1.4  u  ρm H   ϕ m H d m H    z D     mf ,m H  0.5 f . exp 2 2 1 u to = u   + 0 . 9 − − − mH   ρ  ϕ d   d    . (14.88)  mL   mL mL  D      u mf ,m L 

Here, fmH is the average mass fraction of the heaviest species in the bed. Similarly, all other values concerning densities, sphericities, diameters, and minimum luidization velocities are the respective average values in the bed. A good approximation is to take the values at the middle of the bed. The term in the exponent (bed height over bed diameter) is called the aspect ratio of the bed. 5. Compute the reduced gas velocity by u reduc =

 u  u − u to exp   . u − u mf  u to 

(14.89)

6. Finally, calculate the mixing ratio by m ratio =

1 . 1 − e − ureduc

(14.90)

The mixing ratio is also the ratio between the mass fractions of the heaviest species at the top and the bottom of the bed. In well-mixed beds, that ratio tends to be close to 1. Segregation is likely to occur if the ratio is less than or equal to 0.5.

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Solid Fuels Combustion and Gasification

As seen, the relative distance between the actual gas supericial velocity and that value at the minimum luidization condition is an important factor in the segregation. Larger values lead to higher circulation rates of solids, which should prevent segregation. Therefore, segregation may be avoided by the following steps: 1. Increasing the gas supericial velocity, either by decreasing the bed diameter or by augmenting the low of gas injected into the bed 2. Reducing the particle size of solids in the bed

14.9

AREAS AND VOLUMES AT FREEBOARD

Similarly to the bed section, relations concerning areas and volumes of various phases are fundamental for computations regarding the freeboard section. From the solution of differential equations for the freeboard (Chapter 13, Equations 13.10 through 13.14), the mass low of gas and each solid species (m = 1–3) is computed. Therefore, the void fraction at each point would be simply given by FGF

εF =

ρGF .  FSF ,m  FGF +  ρGF m∑ ρp,m  =1  3

(14.91)

The available surface area of each solid species can be calculated by dA PF ,m 6 fm′′′(1 − ε F )SF . = dz d p,m

(14.92)

Note that the cross-section area at the freeboard might differ from the area at the bed section. Actually, many designs allow larger diameters at the freeboard section to facilitate disengaging of particles. The average particle diameter (dp,m) and volume fraction of each solid ( fm′′′ ) would vary according to the position in the freeboard, as well. Those can be estimated through the relations related to entrainment, as seen before. Clearly, dVF (14.93) = ε FSF . dz

14.10

MASS AND VOLUME FRACTIONS OF SOLIDS

The mass fraction of particle kind m (fm) among all other solid species (m = 1–3) is easily calculated after each iteration through the bed using the computed values of remaining mass, or F fm = 3 LD,m (14.94) ∑ FLD,m m =1

339

Fluidization Dynamics

where FLD,m can be obtained by adding all mass lows of component j (F LD,j) belonging to the solid species m. Those can be computed by Equation 13.4 (Chapter 13), and the volume fractions can be computed by FLD,m fm′′′ =

ρp,m

3

∑  F

LD , m

m =1

 ρp,m 

.

(14.95)

For the freeboard, those can be calculated at each position as fm =

FSF ,m 3

∑F

(14.96)

SF , m

m =1

and

FSF ,m

fm′′′ =

14.11

ρp,m .  FSF ,m  ∑  ρp,m  m =1  3

(14.97)

FURTHER STUDY

The area of luidization dynamics is among the areas most studied. Major works and reviews [1, 3, 4, 17, 41–45] are rich sources of information for those interested in deeper and more detailed aspects of this ield.

14.12

EXERCISES

14.12.1 PROBLEM 14.1* Compare the proiles of bubble diameter throughout the bed height by applying correlations from the various authors mentioned in this chapter. Use the following data: • • • • • • •

14.12.2

Bed internal diameter: 0.5 m Bed height: 2 m Average temperature in the bed: 1000 K Distributor is a porous plate Average particle diameter: 0.1 mm Average particle apparent density: 2000 kg/m3 Air is passing through the bed

PROBLEM 14.2*

Referring to the model based on Soo’s work [24, 25] and described in Section 14.4.2, make a suggestion for the boundary condition of particle velocities at the top of the bed.

14.12.3 PROBLEM 14.3* Show that SB = ε BS.

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Solid Fuels Combustion and Gasification

14.12.4 PROBLEM 14.3** Develop a computational routine to estimate the mixing ratio in a bubbling luidized bed by following the route described in Section 14.8. Assume as known the equipment geometry and average values of the following parameters at the middle of the bed: particle densities, sphericities, mass fractions, gas composition, temperature, pressure, and supericial velocity.

REFERENCES 1. Davidson, J.F., and Harrison, D., Fluidization, Academic Press, New York, 1985. 2. Toomey, R.D., and Johnstone, H.F., Gaseous luidization of solid particles, Chem. Eng. Progress, 48(5), 220–226, 1952. 3. Kunii, D., and Levenspiel, O., Fluidization Engineering, 2nd Ed., John Wiley, New York, 1991. 4. Davidson, J.F., and Harrison, D., Fluidized Particles, Cambridge University Press, Cambridge, United Kingdom, 1963. 5. de Souza-Santos, M.L., Modelling and Simulation of Fluidized-Bed Boilers and Gasiiers for Carbonaceous Solids, PhD thesis, University of Shefield, Shefield, United Kingdom, 1987. 6. de Souza-Santos, M.L., Comprehensive modelling and simulation of luidized-bed boilers and gasiiers, Fuel, 68, 1507–1521, 1989. 7. de Souza-Santos, M.L., Application of comprehensive simulation of luidized-bed reactors to the pressurized gasiication of biomass, Journal of the Brazilian Society of Mechanical Sciences, 16(4), 376–383, 1994. 8. Babu, S.P., Shah, B., and Talwalkar, A., Fluidization correlations for coal gasiication materials—minimum luidization velocity and luidization bed expansion ratio, AIChE Symp. Ser., 176(74), 176–186, 1978. 9. Wen, C.Y., and Yu., Y.H., A generalized method for predicting the minimum luidization velocity, AIChE J., 12, 610–612, 1966. 10. Delvosalle, C., and Vanderschuren, J., Gas-to-particles and particle-to-particle heat transfer in luidized beds of large particles, Chem. Eng. Sci., 40(5), 769–779, 1985. 11. Mori, S., and Wen, C.Y., Estimation of bubble diameter in gaseous luidized beds, AIChE J., 21, 109, 1975. 12. Stubington, J.F., Barret, D., and Lowry, G., Bubble size measurements and correlation in a luidized bed at high temperature, Chem. Eng. Res. Des., 62, 173–178, 1984. 13. Horio, H., and Nonaka, A., A generalized bubble diameter correlation for gas–solid luidized beds, AIChE J., 33(11), 1865–1872, 1987. 14. Bouillard, J.X., Lyczkowski, R.W., Folga, S., Gidaspow, D., and Berry, G.F., Hydrodynamics of erosion of heat exchanger tubes in luidized bed combustors, Canadian Journal of Chemical Engineering, 67, 218–229, 1989. 15. Bouillard, J.X., and Lyczkowski, R.M., On the erosion of heat exchanger tube banks in luidized-bed combustors, Powder Technology, 68, 37–51, 1991. 16. Talmor, E., and Benenati, D., Solids mixing and circulation in gas luidized beds, AIChE J., 9, 536–540, 1963. 17. Geldart, D., Gas Fluidization Technology, John Wiley, Chichester, United Kingdom, 1986. 18. Gidaspow, D., Multiphase Flow and Fluidization, Academic Press, San Diego, CA, 1994. 19. Liu, Y., and Gidaspow, D., Solids mixing in luidized beds—a hydrodynamic approach, Chem. Eng. Sci., 36, 539–547, 1981. 20. Gidaspow, D., and Ettehadleh, B., Fluidization in two-dimensional beds with a jet. 2. Hydrodynamic modeling, Ind. Eng. Chem. Fundam., 22(2), 193–201, 1983.

Fluidization Dynamics

341

21. Gidaspow, D., Hydrodynamics of luidization and heat transfer: supercomputed modeling, Appl. Mech. Rev., 39(1), 1–23, 1986. 22. Gidaspow, D., Lin, C., Seo, Y.C., Fluidization in two-dimensional beds with a jet. 1. Experimental porosity distributions, Ind. Eng. Chem. Fundam., 22(2), 187–193, 1983. 23. Gidaspow, D., Tsuo, Y.P., and Ding, J., Hydrodynamics of circulating and bubbling luidized beds, in Proc. of the Materials Issues in Circulating Fluidized-bed Combustors, Argonne National Laboratory, June 19–23, 1989. 24. Soo. S.L., Note on motions of phases in a luidized bed, Powder Technology, 45, 169– 172, 1986. 25. Soo, S.L., Average circulatory motion of particles in luidized beds, Powder Technology, 57, 107–117, 1989. 26. Soo, S.L., Particulates and Continuum, Multiphase Fluid Dynamics, Hemisphere, New York, 1989. 27. Lin, J.S., Chen, M.M., and Chao, B.T., A novel radioactive particle tracking facility for measurement of solid motion in gas luidized beds, AIChE J., 31(3), 465–473, 1985. 28. Costa, M.A.S., and de Souza-Santos, M.L., Studies on the mathematical modeling of circulation rates of particles in bubbling luidized beds, Power Technology, 103, 110– 116, 1999. 29. Costa, A.M.S., Improvements on the Mathematical Modeling and Simulation of Circulation Rates of Solid Particles in Bubbling Fluidized Beds (in Portuguese), MSc thesis, Faculty of Mechanical Engineering, University of Campinas, São Paulo, Brazil, 1998. 30. Ergun, S., Fluid low through packed columns, Chemical Engineering Progress, 48, 91–94, 1952. 31. Saxton, J.A., Fitton, J.B., and Vermeulen, T., Cell model theory of homogeneous luidization: density and viscosity behavior, AIChE J, 16(1), 120–130, 1970. 32. Wen, C.Y., and Chen, L.H., Fluidized bed freeboard phenomena: entrainment and elutriation, AIChE J., 28, 117, 1982. 33. Lewis, W.K., Gilliland, E.R., and Lang, P.M., Entrainment from luidized beds, Chem. Eng. Prog. Symp. Ser., 58(38), 65–78, 1962. 34. Walsh, P.M., Mayo, J.E., and Beer, J.M., Reluxing particles in the freeboard of a luidized bed, AIChE Symp. Ser., 80(234), 119–128, 1984. 35. Merrick, D., and Highley, J., Particle size reduction and elutriation in a luidized bed process, AIChE Symp. Ser., 70(137), 366–378, 1974. 36. Chirone, R., D’Amore, M., and Massimila, L., Carbon attrition in luidized combustion of petroleum coke, Proc. 20th Symp. (Int.) on Combustion, Combustion Institute, Ann Arbor, MI, 1984. 37. Salatino, P., and Massimila, L., A descriptive model of carbon attrition in the luidized combustion of coal char, Chem. Eng. Sci., 40(10), 1905–1916, 1985. 38. Vaux, W.G., and Schruben, J.S., Kinetics of attrition in the bubbling zone of a luidized bed, AIChE Symp. Ser., 79(222), 97–102, 1983. 39. Leith, D., and Mehta, D., Cyclone performance and design, Atmospheric Environment, 7, 527–549, 1973. 40. Nienow, A.W., Rowe, P.N., and Cheung, L.Y.L., The mixing/segregation behaviour of a dense powder with two sizes of a lighter one in a gas luidised bed, in Fluidization, Davidson, J.F., and Keairns, D.L., Eds., Cambridge University Press, Cambridge, United Kingdom, 1978, p. 146. 41. Yang, W.C., Fluidization Solids Handling and Processing, Noyes, Westwood, NJ, 1999. 42. Lim, K.S., Zhu, J.X., and Grace, J.R., Hydrodynamics of gas–solid luidization, Int. J. Multiphase Flow, 21, 141–193, 1995. 43. Davidson, J.F., and Keairns, D.L., Fluidization, Cambridge University Press, Cambridge, United Kingdom, 1978. 44. Kwauk, M., and Li, J., Fluidization regimes, Powder Technology, 87, 193–202, 1996.

Parameters 15 Auxiliary Related to FluidizedBed Processes CONTENTS 15.1 Introduction ................................................................................................. 343 15.2 Mass Transfers ............................................................................................344 15.2.1 Bubbles and Gas in the Emulsion ..................................................344 15.2.2 Solids and Gas in the Emulsion ..................................................... 345 15.2.3 Solids and Gas in the Freeboard....................................................346 15.3 Heat Transfers .............................................................................................346 15.3.1 Bubbles and Gas in the Emulsion ..................................................346 15.3.2 Solids and Gas in the Emulsion ..................................................... 347 15.3.3 Solids and Solids ........................................................................... 348 15.3.3.1 Heat Transfer by Radiation ...........................................348 15.3.3.2 Heat Transfer by Convection ........................................348 15.3.4 Tubes and the Bed ......................................................................... 349 15.3.4.1 Convection between Tubes and the Bed ....................... 349 15.3.4.2 Convection between Internal Tube Wall and Fluid ...... 351 15.3.4.3 Radiation between Tubes and the Solids ...................... 353 15.3.5 Tubes and the Freeboard................................................................ 353 15.3.5.1 Convection between Tubes and the Gas Flowing through the Freeboard .................................................. 353 15.3.5.2 Radiation between Tubes and the Solids ...................... 354 15.3.6 Reactor and Ambient Atmosphere ................................................ 354 15.3.7 Bed and Distributor Surface .......................................................... 354 15.4 Parameters Related to Reaction Rates ........................................................ 355 15.5 Transition to Turbulent Fluidization ........................................................... 355 References .............................................................................................................. 357

15.1

INTRODUCTION

As we have seen, luidized-bed reactors involve a great number of phenomena. Among those are the transfers of energy and mass in almost all possible forms. The present chapter introduces the basic methods to allow computations of parameters related to those and other phenomena.

343

344

Solid Fuels Combustion and Gasification

15.2

MASS TRANSFERS

These transfer processes take place between the following phases: 1. Bubbles and gas in the emulsion 2. Solid particles and gas in the emulsion 3. Solid particles and gas in the freeboard As can be seen, no direct mass transfer is considered between the particles and the gas in the bubbles. This is due to the assumption that bubbles are free of particles. However, the effects of such eventual exchanges are accounted for because such transferences are indirectly computed through exchanges 1 and 2 listed above.

15.2.1

BUBBLES AND GAS IN THE EMULSION

The mass lux of each chemical species j between bubbles and emulsion can be computed by G MGEGB, j = ψ BEρ G M j ( y GB, j − y GE, j )

VB . AB

(15.1)

Among the various available correlations for the coeficient for the mass transfer are the following: a) The one presented in Kobayashi and Arai [1], given by ψ BE =

0.11 . dB

(15.2)

This correlation was applied in several works [2–4]. b) The one by Kunii and Levenspiel [5], which is an improvement on the above correlation and accounts for the effects of the bed dynamics and mass transfer properties, described as ψ BE =

1 . 1 1 + ψ BC ψ CE

(15.3)

The above equation is based on the three-phase model. The mass transfer between the bubble and emulsion (BE) is given by a mechanism of resistances, which considers the mass transfer between bubble and cloud (BC) and between cloud and emulsion (CE). These individual mass transfer coeficients are given by ψ BC = 4.5

U mf D0.5g 0.25 + 5.85 G l.25 dB dB

(15.5)

and ε D U  ψ CE = 6.78  mf 3G B  . dB  

(15.6)

Auxiliary Parameters Related to Fluidized-Bed Processes

345

Most of the parameters involved in these relationships have been introduced in Chapter 9. The above formulation is used in several mathematical models [6–9]. c) Sit and Grace [10] veriied that the Kunii and Levenspiel correlation underestimated the mass transfer in regions near the distributor where intense bubble interaction or coalescence takes place. Strong evidence for this comes from other works [8–11] as well. As is assumed in the present model, Sit and Grace [10] included the clouds around the bubbles as part of the emulsion. Their correlation is recommended because it reproduces the experimental data without any additional correction, and it is given by 1/ 2

ψ BE =

15.2.2

2 U mf 12  DG εmf U B  + 3/ 2   . dB dB  π 

(15.7)

SOLIDS AND GAS IN THE EMULSION

As shown before, the mass transfer of gaseous components from and to reacting solid particles depends on three basic resistances. Among those is the one offered by the gas boundary layer surrounding the particle. The coeficient of mass transfer between phase formed by particle kind m and the emulsion gas is provided by ψ SEGE,m = N Sh,m

DGρ GE . dm

(15.8)

Therefore, the determination of the Sherwood number for each particle is an important point during computations. However, several mathematical models set strong simpliications regarding this parameter. Overturf and Reklaitis [8], for instance, assume a constant Sherwood number of 2. Other works do not even mention the adopted value or correlation. Among the various works presented in the literature, the one by La Nause et al. [12] seems to be the most appropriate because of its relatively easy application to models of luidized-bed equipment. It can be used to predict the mass transfer rates for a wide range of temperature and luidization conditions and is given by 1/ 2

N Sh,m

 4d ε U  = 2εmf +  m mf B  .  πDG 

(15.9)

It is valid for particles smaller than the average, or (dm /dP,av) ≤ 1, whereas for particles larger than three times the average in the process, the number can be computed by 1/ 2

 4d ( U + εmf U B )  N Sh,m = 2εmf +  m mf  . πDG  

(15.10)

For the intermediate region, the following formula may be adopted: N Sh ,m = aN Sh ,m ,eq.15.9 + (1 − a ) N Sh ,m ,eq.15.10

(15.11)

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Solid Fuels Combustion and Gasification

where a=

15.2.3

(

3 − d m d p,av 2

).

(15.12)

SOLIDS AND GAS IN THE FREEBOARD

The correlations shown in Chapter 11, Section 11.7, can be applied to computations of the Sherwood number related to the mass transfer between gas and solid particles in the freeboards. As always, the average properties for the gas phase should be computed at each vertical position in the freeboard. The composition and temperature are dictated by the differential mass and energy balances.

15.3

HEAT TRANSFERS

As described previously, the luidized-bed processes involve a wide range of possible heat transfers among the various phases, as well as between those and surfaces immersed in the bed or in the freeboard. The most important heat transfers occurring in the bed are those between the following: • • • • • • • • • • • • • • • •

Gas in the bubbles and gas in emulsion Gas and solids in the emulsion Solids and solids in the emulsion Solids and solids in the freeboard Solids and gas in the freeboard Gas in the bubbles and immersed tubes in the bed Gas in the emulsion and immersed tubes in the bed Gas and tubes in the freeboard Solids and tubes in the bed Solids and tubes in the freeboard Phases and jackets around the bed and freeboard in similar relationships Bed and distributor Bed and internal reactor wall Gas in the freeboard and internal reactor walls External reactor wall and the environment Internal wall of the tubes in the bed or in the freeboard and the luid lowing inside them • Reactor and jackets around it Relationships that can be applied for each case are presented below.

15.3.1

BUBBLES AND GAS IN THE EMULSION

A speciic method to calculate the heat transfer by convection between gas in the bubbles and gas in the emulsion has not been found. Therefore, as suggested in Kunii

347

Auxiliary Parameters Related to Fluidized-Bed Processes

and Levenspiel [5], an analogy with the equivalent mass transfer can be assumed to give 1/ 2

α CGBGE =

U mf ρG,avC G,av λ ε U ρ c  + 2  G,av mf B G,av G,av  . dB 3  

(15.13)

Here, the average values of properties should be computed for the local cup-mix conditions between emulsion and bubble gas phases. The above correlation can be used to compute the heat transfer between those phases as R CGBGE = α CGBGE (TGB − TGE )

dA B . dVB

(15.14)

The bubble area-volume ratio is approximated by Equation 14.23 (Chapter 14).

15.3.2

SOLIDS AND GAS IN THE EMULSION

Equation 13.5 shows two terms related to heat transfer between gas and particles of solid type m. The irst (RCSEGE,m) is due to convection, and the second (R hSEGE,m) is related to energy transfer due to mass transfer between the phases. The method for computation of the second term (R hSEGE,m) is similar to that shown in case of moving beds and given by Equation 11.53 (Chapter 11). The heat transfer by convection between the solid particle m and the gas in the emulsion can be given by R CSEGE,m = α CSEGE,m (TSE ,m − TGE )

dA PE,m dVGE

(15.15)

where α CSEGE,m =

N Nu,m c GE . dm

(15.16)

The Nusselt number for this situation is deined [13] by N Nu,m = 0.3N1Re.3,m

(15.17)

which is valid for Nre,m < 100. Here, N Re,m =

FG d m . Sµ G

(15.18)

For Nre,m ≥ 100, the Nusselt number is given [14] by N  N Nu,m = 0.4  Re,m   εE 

2/3

N1Pr/ 3,G .

(15.19)

348

15.3.3

Solid Fuels Combustion and Gasification

SOLIDS AND SOLIDS

15.3.3.1 Heat Transfer by Radiation The thermal radiative transfer between the various solids in the bed is a complex problem. There are several procedures to compute such heat exchanges; among them is the lux method [15–25]. This method allows the inclusion of terms related to radiative transfer in the differential energy balances. On the other hand, the complexity of the mathematical treatment increases signiicantly and is beyond the scope of this introductory text to modeling. In addition, having in mind the degree of approximations used at the present level, a simpler approach can be applied to compute that heat transfer. Besides, after comparisons with applications of the lux method, at least for cases of luidized bubbling beds, the line of attack used here has proved to be quite reasonable [26–30]. It assumes the following: 1. The gas layers between the particles are transparent. 2. A particle kind m is surrounded by all sorts of particles present in the bed. Therefore, the other particles of type n take a certain fraction of its total surrounding viewed area. That area fraction is assumed to be the same as fn′′, which is given by A fn′′ = 3 n ∑ Am m =1

where Am is the total external area of particle of solid type m in the surrounding position at which the calculation is being performed. 3. All particles behave as gray bodies. These assumptions lead to the following equation: R RSESE,m , n =

(

4 4 σ TPE , m − TPE , n

)

dA PE,m 1 − ε′m 1 − ε′n fm′′ 1 dVSE,m + + ε′m ε′n fn′′ fn′′

(15.20)

where m and n indicate the possible particle types present in the bed, and ε ′ indicates their emissivities. A simple deduction for the radiation heat transfer between gray bodies can be found in any text in the area [31–34]. A similar equation is applicable for the radiative heat transfers in the freeboard. 15.3.3.2 Heat Transfer by Convection The processes of heat transfer by convection between the various solid particles can be treated by applying an analogy to the gas–solid convective transfer. This is possible because, from the macroscopic point of view, the luidized bed of particles can be treated as a luid. In this way, the referred heat transfer is computed by R CSESE,m , n = α SESE,m , n fn′′ ( TPE,m − TPE, n )

dA PE,m dVSE,m

where m and n indicate the possible kinds of particles present in the bed.

(15.21)

349

Auxiliary Parameters Related to Fluidized-Bed Processes

The heat transfer coeficient is given by −2

α CSESE ,m ,n = 4.51 × 10 d

−1.22 m ,n

 U   U  mf

−0.56 6

.

(15.22)

This has been adapted [35, 36] from the work by Delvosalle and Vanderschuren [37] in order to allow the calculations for the coeficient between particles with different sizes. Therefore, the deinition of an average diameter dm,n is necessary. A possible formula is d m,n =

15.3.4

1 . fm fn + dm dn

(15.23)

TUBES AND THE BED

The heat transfer between the surface of the tubes and the bed is accounted for by the three terms: 1. Convection with the gas in the emulsion (RCGETD) 2. Convection with the gas in the bubbles (RCGBTD) 3. Radiation with the particles in the emulsion (RRSETD) The convection between particles and tubes is indirectly computed by the convection between emulsion and tubes. This is so because the empirical correlation does not separate these phenomena. The models present in Chapters 7 and 13 neglect transfers by radiation from or to gases. This is justiiable because of the small thickness of the interstitial gas layers between particles. This assumption has been shown to be reasonable [26–29, 38], at least in the bed region of combustors and gasiiers. 15.3.4.1 Convection between Tubes and the Bed The heat transfer by convection between emulsion interstitial gas and tubes is described as R CGETD = α EOTD ( TGE − TWOTD )

dA OTD . dVGE

(15.24)

dA OTD . dVGB

(15.25)

The equivalent for the gas in the bubbles is given by R CGBTD = α BOTD ( TGB − TWOTD )

Heat transfer coeficients can be found in the literature [39], which shows the rule for splitting the total heat exchanged between bed and immersed tubes into the parts derived from each phase in the bed. The expressions presented here have been

350

Solid Fuels Combustion and Gasification

adapted to match the assumption that heat transfer by convection between tubes and bed is accomplished by the gas in the emulsion and in the bubbles. For the emulsion phase, it are given by  (πB1 )1/ 2 2B1/ 2  erfc B2 exp B22 − 1 + 1  α EOTD = α′  1 + B1   B2 (1 + B1 )

(

)

(15.26)

UB U G − U mf + U B

(15.27)

where  λ ρ c ( U − U mf )  α′ = 2  mf mf mf G  πd OTD   B1 =

1/ 2

U G − U mf 0.35(gd D )1/ 2

(15.28)

α′′ α′(πb1 )1/ 2

(15.29)

λ G,av . δfilm

(15.30)

B2 =

α′′ =

Here, the ilm thickness for inclined tube has been taken as an average between that relative to the horizontal and vertical values, as follows:  cos(i TD ) 1 − cos(i TD )  δfilm = d p,av  + . 10   4

(15.31)

The deinitions of physical properties at the minimum luidization condition (index mf) and average gas temperature and composition—used during computations of physical properties—are described as follows: λ mf = λ G,av

1 − ε mf + 0.1ρG,avc G,avd p,av U mf λ 0.04 + G,av λS

(15.32)

ρmf = ρp (1 − εmf )

(15.33)

c mf = c S

(15.34)

5

∑c

Tav = T* + m =1

F (Tm − T*)

m m 5

∑ cm Fm

.

(15.35)

k =1

In the above, the index m designates each phase (1 for carbonaceous solid, 2 for limestone [if present], 3 for inert solid [if present], 4 for emulsion gas, and 5 for gas

351

Auxiliary Parameters Related to Fluidized-Bed Processes

in the bubbles). The enthalpy reference temperature T* is 298.15 K. The mass lows Fk for solid phases should be interpreted as the circulation lows FH,k. For the bubble phase, the following correlation can be used:  4λ ρ c U  α BOTD =  mf G,av G,av mf  πd OTD  

1/ 2

(15.36)

In those equations, the gas properties (cG,av, ρG,av, λG,av) should be computed at each vertical position z and using average temperature and composition found in the emulsion and bubble gases. The speciic heat for solid phase (cS) is the mass-average (or using fm as weighting factor) value taken among all solids in the bed. The overall coeficient for the heat transfer between tube and bed can be computed by [39] α OTD = α EOTD (1 − 0.7125ε B ) + α BOTD 0.7125ε B.

(15.37)

At each vertical position z, the temperature at the external surface of the tubes immersed in the bed can be calculated with the aid of tube internal and external heat transfer coeficients as TWOTD =

α OTD TGA + α JTD THJTD . α OTD + α JTD

(15.38)

15.3.4.2 Convection between Internal Tube Wall and Fluid A method to estimate the heat transfer rates between the luid lowing in the tubes and their internal surfaces is presented here. Any method should take into account the various possibilities of the thermodynamic state of the lowing water inside the tubes: compressed liquid, vapor-liquid mixtures, and superheated steam. Usually, a large portion of the tube length is used for nucleate boiling. However, the other stages—pure liquid and pure steam convections—are important as well. Convergence procedures will be involved in computations of heat transfer between tubes and bed. The diagram shown in Figure 15.1 illustrates a simplistic example. The equations referred to there are listed below [40]:

(

N Nu, JT = 3.663 + 1.613 N Pe

)

1/ 3

 µ fl     µ JWT 

0.14

(15.39)

 d  4/5 N Nu, JT = 0.0214 N Re − 100 N 2Pr/ 5  1 + JT  LT  

(

(

N Nu, JT = 0.012 N

)

0.87 Re

)

− 280 N

2/5 Pr

 d JT  1 + L  T  

2/3

(15.40)

2/3

.

(15.41)

Equation 15.40 should be used when NPr is less than 1.5; otherwise, Equation 15.41 must be used. In addition,

352

Solid Fuels Combustion and Gasification

Read data

No heat transfer

= Liquid convection


>

Yes

TW : Tsat
0.7 m), since the heat transfer between the furnace and the respective wall is much lower for the freeboard than for the bed region.

16.2.2 INSTITUTE OF GAS TECHNOLOGY RENUGAS® UNIT During the 1980s and 1990s, the Institute of Gas Technology (IGT) (Chicago, Illinois) developed the RENUGAS process for gasiication of biomass. A sizable pilot

389

Bubbling Fluidized-Bed Simulation Program and Results 1100 Wall

Inside 1000

Temparature (K)

900 800 700 600 500 400 300 200 0

0.5

1

1.5

2

2.5

3

3.5

Height (m)

FIGURE 16.31 Temperature proiles of water inside the jacket and the wall separating jacket and furnace (Test 26, Babcock and Wilcox unit).

operating under a pressurized bubbling luidized bed was used to develop the gasiication process for various biomasses, including woods and sugar cane bagasse. Several tests were performed using mixtures of oxygen and steam, and some conditions and results of those experimental tests have been published [12]. One of those cases, for which more details are given, is used here for comparison against simulation results. 16.2.2.1 Plant Description and Simulation Inputs The main geometry and operational conditions for tests T12-1 and T12-3a are presented in Table 16.9.* Table 16.10 shows the particle size distributions of the solid fed into the bed. The pressurized atmosphere inside that pilot provides a good opportunity to verify the simulation performance in such cases. During operations, nitrogen was injected through a jacket involving the central reactor to prevent fuel gas leakage to the surrounding atmosphere. However, that nitrogen stream found its way into the bed, thereby diluting the produced gas and decreasing the gasiication eficiency. After those tests, the problem was corrected, and nitrogen injections were no longer necessary in the industrial version of the RENUGAS process. Sand was used as the material for startups, and the bed was heated until the bed reached ignition temperatures. The usual procedures are to inject natural gas or any other lammable gas through the distributor or to direct lames from torches burning fuel gas to the bubbling bed surface. This is a common startup method for bubbling *

Unfortunately, a few data or details were not available in the IGT report [12] and have to be assumed from the most probable values.

390

Solid Fuels Combustion and Gasification

TABLE 16.9 Proximate Analysis of Wood (Tests T12-1 and T12-3a, IGT) Test Detail

T12-1

T12-3a

Fuel Proximate analysis (% w.b.)a Moisture Volatiles Fixed carbon Ash Ultimate analysis (% d.b.) C H N O S High heating value (HHV) (d.b.) (MJ/kg) Particle apparent density (kg/m3) Particle real or skeletal density (kg/m3) Basic Geometry Bed equivalent diameter (m) Bed heightb (m) Freeboard equivalent diameter (m) Freeboard height (m) Fuel feeding position (m) Injected fuel and gases Feeding low of fuel (kg/s) Oxygen low through distributor (kg/s) Steam low through distributor (kg/s) Temperature of injected oxygen (K) Temperature of injected steam (K) Nitrogen intermediate injection (kg/s) Position of N2 injection (m)b Average pressure in the bed (kPa)

Wood

Wood

4.94 79.39 14.90 0.77

9.14 75.88 14.24 0.74

48.4 6.31 0.21 44.23 0.03 19.14 720 1750

48.4 6.31 0.21 44.23 0.03 19.14 720 1750

a b

0.292 1.585 0.451 6.147 0.381 8.113 × 10–2 2.058 × 10–2 4.922 × 10–2 644 672 4.3772 × 10–2 0.381 2170

0.292 1.585 0.451 6.147 0.381 8.922 × 10–2 2.320 × 10–2 6.156 × 10–2 644 672 3.658 × 10–2 0.381 2292

w.b., wet basis; d.b., dry basis. measured from the surface of the distributor.

luidized beds, and it was used in the IPT unit (Institute of Technological Research, São Paulo, Brazil) as well. The sand was heated to around 800 K before the feeding of carbonaceous fuel started. After that, the inert solid was no longer fed, and therefore its concentration decreased because material from the bed was continuously (or at intervals) withdrawn from the system. However, one should keep in mind that until all sand was withdrawn from the bed, the operations were not truly at a steady-state regime. During the transitional period, comparisons between experimental data with simulations—assuming a steady-state regime—can be misleading since an apparent

391

Bubbling Fluidized-Bed Simulation Program and Results

TABLE 16.10 Particle Size Distributions of Solids Fed into the Bed (Tests T12-1 and T12-3a, IGT) Wood Diameter (µm)a 1680 841 354 250 177

a

Inert

Mass Percentage

Diameter (µm)a

5.0 28.4 30.4 21.7 14.5

420 350 297 250 210 177

Mass Percentage 0.3 54.4 37.1 7.8 0.3 0.1

Average of the slice with the respective mass fraction.

steady state may be achieved where no signiicant variations occur. The relatively large and denser sand particles remain in the bed, and their withdrawal is too slow. Another source of deviations between real operations and simulations may rest on the fact that the functional group (FG) model (see Chapter 10) was applied here with the same boundary conditions as for bituminous coal. No speciic data for wood or any biomass applicable to the FG model was found in the literature. This is a point left for future improvements. 16.2.2.2 Real Operation and Simulation Results The tables below summarize the real operational data [12] and selected simulation results. The most important aspects of simulations are presented in Figures 16.32 through 16.54. 16.2.2.3 Discussion Many important aspects of the luidization process have been already discussed for the case of the Babcock and Wilcox test. In the present section, only the most interesting remarks related to gasiication are made, as follows: a) According to Tables 16.11 and 16.12, CSFMB was capable of reproducing fairly well the operational conditions of both tests. The relatively good approximations for the rate of tar production show that the combination of FG with the DISKIN model can provide reasonable predictions in that respect, even applying the equations and conditions set for bituminous coal. b) The deviations in elutriation or rate of entrainment at the top of freeboard (Table 16.10) are acceptable in view of uncertainties in densities of fuel solid and the possible presence of remaining sand in the bed. c) Figures 16.32 through 16.34 show the temperature proiles in the bed for the two cases. Unfortunately, only the average temperature in the bed is reported by IGT [12]. CSFMB was able to estimate values close to the measured ones

392

Solid Fuels Combustion and Gasification

TABLE 16.11 Main Process Parameters (Tests T12-1 and T12-3a, IGT) T12-1 Process Parameter Flue gas low (kg/s) Flow of tar in exiting gas (kg/s) Flow of entrained solids (kg/s) Supericial velocitya (m/s) Carbon conversion (%) Average temperature at distributor (K) Average temperature at bed middle (K) Average temperature at bed top (K) Temperature of emulsion gas (K)a Temperature of bubble gas (K)a Average temperature at freeboard top (K) Temperature of carbonaceous particles (K)a Mass held in the bed (kg) Static bed height (m) Average residence time of particles (based on feeding rate) (minutes) TDHc (m) (measured from the bed surface) Mixing index Pressure loss across the bed (kPa) a b c

T12-3a

Real

Simulation

Real

Simulation

0.184 2.28 × 10–3 1.94 × 10–3 n.d.b 90.2 n.d.b 1105 n.d.b n.d.b n.d.b n.d.b n.d.b n.d.b n.d.b n.d.b

0.185 2.53 × 10–3 1.22 × 10–4 0.233 91.13 879.2 1197.3 1196.3 1198.0 1151.6 1206.6 1197.8 86.1 1.231 17.68

0.205 9.07 × 10–4 8.32 × 10–4 n.d.b 96.2 n.d.b 1184 n.d.b n.d.b n.d.b n.d.b n.d.b n.d.b n.d.b n.d.b

0.198 2.76 × 10–3 1.36 × 10–4 0.612 88.94 900.0 1193.5 1205.7 1194.9 1109.9 1205.7 1194.7 84.6 1.220 15.80

n.d.b n.d.b n.d.b

4.485 1.000 12.60

n.d.b n.d.b n.d.b

4.529 1.000 12.39

At the middle of the bed. n.d., not determined or not reported. TDH, transport disengaging height

(Table 16.11). As in the case of combustors (Figure 16.3), those igures also present the large difference of temperature between carbonaceous solid and gas phases (bubble and emulsion) at the distributor. The reasons for that have already been explained, and Figures 16.36 and 16.37 (the latter in logarithmic scale) show how fast oxygen is consumed because of combustion of the solid fuel particles near the distributor (height = 0). The swift oxidation of solid fuel also led to the quick increase in the temperatures of the gas phases. In the present case, oxygen was rapidly consumed in the bubbles as well, as illustrated by Figures 16.38 and 16.39 (the latter in logarithmic scale). In those cases, the bubbles are relatively small (Figure 16.49) because the luidization conditions near the distributor remained close to the minimum luidization ones (Figure 16.50). Hence, intense heat and mass transfers between emulsion and bubbles took place. This was possible because during the operations, the injected gas streams were preheated, which prevented fast expansion near the distributor. Since all oxygen was consumed very near the distributor, the remainder of the bed could be kept under reducing conditions, which allowed for the accumulation of fuel gases. This, in turn,

393

Bubbling Fluidized-Bed Simulation Program and Results

TABLE 16.12 Composition of Produced Gas (Tests T12.1 and T12-3a, IGT) T12-1

T12-3a

Component

Real

Simulation

Real

Simulation

H2 CO CO2 CH4 H2O H2S NH3 NO NO2 N2 N2O SO2 HCN C2H4 C2H6 C3H6 C3H8 C6H6

12.05 8.00 17.06 7.37 35.82 n.d.a n.d. n.d. n.d. 19.18 n.d. n.d. n.d. 0.03 0.22 n.d. 0.00 0.27

12.8760 9.0891 16.6307 7.8565 33.9538 0.0066 0.0801 0.0000 0.0000 19.4361 0.0000 0.0018 0.0004 0.0346 0.0138 0.0000 0.0000 0.0204

12.69 7.22 17.42 7.60 40.30 n.d. n.d. n.d. n.d. 14.34 n.d. n.d. n.d. 0.00 0.02 n.d. 0.00 0.41

12.3306 7.2059 16.8287 7.2054 41.4342 0.0053 0.0843 0.0000 0.0000 14.8382 0.0000 0.0019 0.0003 0.0327 0.0130 0.0000 0.0000 0.0196

n.d., not determined or not reported.

a

1300 1200

Temperature (K)

1100 1000 900 800 700 Emuls.gas

Bubble

Carbonac.

Inert

Average

600 0

0.2

0.4

0.6

0.8

1

1.2

Height (m)

FIGURE 16.32

Temperature proiles in the bed (Test T12-3a, IGT).

1.4

1.6

394

Solid Fuels Combustion and Gasification 1300 1200

Temperature (K)

1100 1000 900 800 700 Emuls.gas

Bubble

Carbonac.

Inert

Average

600 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Height (m)

FIGURE 16.33 Temperature proiles in the bed (Test T12-1, IGT).

1300 Emuls.gas

Bubble

Carbonac.

Inert

Average

1200

Temperature (K)

1100 1000 900 800 700 600 1.E-10 1.E-9 1.E-8 1.E-7 1.E-6 1.E-5 1.E-4 1.E-3 1.E-2 1.E-1 1.E-0

1.E1

Height (m)

FIGURE 16.34

Temperature proiles in the bed (logarithmic scale) (Test T12-1, IGT).

395

Bubbling Fluidized-Bed Simulation Program and Results 1220

Temperature (K)

1215

1210

1205

1200

1195 Gas 1190

1

2

3

4

Carbonac.

Average

5

6

7

Height (m)

FIGURE 16.35

Temperature proiles in the freeboard (Test T12-3a, IGT).

0.2 CO2

CO

O2

Molar fraction

0.15

0.1

0.05

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Height (m)

FIGURE 16.36 IGT).

Concentration proiles of CO, CO2, and O2 in the emulsion (Test T12-1,

396

Solid Fuels Combustion and Gasification 0.2

Molar fraction

0.15

0.1

0.05 CO2

CO

O2

0 1.E-10 1.E-9 1.E-8 1.E-7 1.E-6 1.E-5 1.E-4 1.E-3 1.E-2 1.E-1 Height (m)

1.E0

1.E1

FIGURE 16.37 Concentration proiles of CO, CO2, and O2 in the emulsion (logarithmic scale) (Test T12-1, IGT).

0.2 CO2

CO

O2

Molar fraction

0.15

0.1

0.05

0

0

FIGURE 16.38 IGT).

0.2

0.4

0.6

0.8 Height (m)

1

1.2

1.4

1.6

Concentration proiles of CO, CO2, and O2 in the bubbles (Test T12-1,

397

Bubbling Fluidized-Bed Simulation Program and Results 0.2

Molar fraction

0.15

0.1

0.05 CO2

CO

O2

0 1.E-10 1.E-9 1.E-8 1.E-7 1.E-6 1.E-5 1.E-4 1.E-3 1.E-2 1.E-1

1.E0

1.E1

Height (m)

FIGURE 16.39 Concentration proiles of CO, CO2, and O2 in the bubbles (logarithmic scale) (Test T12-1, IGT).

0.2

Molar fraction

0.15

0.1

0.05 CO2 0 0.E1

1.E0

2.E0

3.E0 4.E0 Height (m)

5.E0

CO

6.E0

O2 7.E0

FIGURE 16.40 Concentration proiles of CO, CO2, and O2 in the system (Test T12-1, IGT).

398

Solid Fuels Combustion and Gasification 0.9

H2O

H2

CH4

0.8 0.7

Molar fraction

0.6 0.5 0.4 0.3 0.2 0.1 0 0

1

2

3

4

5

6

7

Height (m)

FIGURE 16.41 Concentration proiles of H2O, H2, and CH4 in the system (Test T12-1, IGT).

9.E-2 H2S

NH3

Tar/Oil

8.E-2 7.E-2

Molar fraction

6.E-2 5.E-2 4.E-2 3.E-2 2.E-2 1.E-2 0.E1 0

1

2

3

4

5

6

7

Height (m)

FIGURE 16.42 IGT).

Concentration proile of tar throughout the bed and freeboard (Test T12-1,

399

Bubbling Fluidized-Bed Simulation Program and Results C + O2

C + H2O

C + CO2

C + H2

C + NO

1.E0 1.E-1 1.E-2

Rates (kmol/m2/s)

1.E-3 1.E-4 1.E-5 1.E-6 1.E-7 1.E-8 1.E-9 1.E-10 1.E-11 1.E-12 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Height (m)

FIGURE 16.43 Rates of main heterogeneous reactions in the bed (Test T12-1, IGT).

C + O2

C + H2O

C + CO2

C + H2

C + NO

6.E-9

Rates (kmol/m2/s)

5.E-9

4.E-9

3.E-9

2.E-9

1.E-9

0.E1 1

2

3

4

5

6

7

Height (m)

FIGURE 16.44 Rates of main heterogeneous reactions in the freeboard (Test T12-1, IGT).

400

Solid Fuels Combustion and Gasification CO + H2O

CO + O2

H2 + O2

CH4 + O2

C2H6 + O2

1.E1 1.E0 1.E-1 1.E-2 Rates (kmol/m3/s)

1.E-3 1.E-4 1.E-5 1.E-6 1.E-7 1.E-8 1.E-9 1.E-10 1.E-11 1.E-12 1.E-13 1.E-10 1.E-9

1.E-8

1.E-7 1.E-6 1.E-5

1.E-4

1.E-3

1.E-2

1.E-1

1.E0

1.E1

Height (m)

FIGURE 16.45 Rates of main homogeneous reactions in the emulsion (Test T12-1, IGT).

CO + H2O

CO + O2

H2 + O2

CH4 + O2

C2H6 + O2

1.E0 1.E-1 1.E-2

Rates (kmol/m3/S)

1.E-3 1.E-4 1.E-5 1.E-6 1.E-7 1.E-8 1.E-9 1.E-10 1.E-11 1.E-12 1.E-10 1.E-9

1.E-8

1.E-7

1.E-6

1.E-5 1.E-4 1.E-3 Height (m)

1.E-2

1.E-1

1.E0

1.E1

FIGURE 16.46 Rates of main homogeneous reactions in the bubbles (Test T12-1, IGT).

401

Bubbling Fluidized-Bed Simulation Program and Results 1.E-2

Rates (kmol/m3/S)

CO + H2O

CO + O2

H2 + O2

CH4 + O2

C2H6 + O2

1.E-3

1.E-4

1.E-5

1

2

3

4 Height (m)

5

6

7

FIGURE 16.47 Rates of main homogeneous reactions in the freeboard (Test T12-1, IGT).

1.E0 C + N2O

1.E-1

V = Tar + Gas

Tar = Char

C-drying

V + H2

Rates (kmol/m2/S)

1.E-2 1.E-3 1.E-4 1.E-5 1.E-6 1.E-7 1.E-8 1.E-9 0

0.2

0.4

0.6

0.8 Height (m)

1

1.2

1.4

FIGURE 16.48 Rates of other important reactions in the emulsion (Test T12-1, IGT).

1.6

402

Solid Fuels Combustion and Gasification

1.4 Diameter (m)

Velocity (m/s)

1.2 1 0.8 0.6 0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Height (m)

FIGURE 16.49

Bubble sizes and velocity throughout the bed (Test T12-1, IGT).

0.3 Min. Fluidiz.

Actual

0.25

0.2

0.15

0.1

0.05 1.E-10

1.E-9

1.E-8

1.E-7

1.E-6

1.E-5

1.E-4

1.E-3

1.E-2

1.E-1

1.E0

1.E1

Height (m)

FIGURE 16.50 Actual and minimum luidization supericial velocities throughout the bed (Test T12-1, IGT).

403

Bubbling Fluidized-Bed Simulation Program and Results 1.E0 C + N2O

V = Tar + Gas

Tar = Char

C-drying

V + H2

1.E-1

Rates (kmol/m2/S)

1.E-2 1.E-3 1.E-4 1.E-5 1.E-6 1.E-7 1.E-8 1

2

3

4

5

6

7

Height (m)

FIGURE 16.51 Rates of other heterogeneous reactions in the freeboard (Test T12-1, IGT).

Tar + H2

C2H4 + O2

C3H6 + O2

C3H8 + O2

C6H6 + O2

HCN + O2

2.5E-2

Rates (kmol/m3/S)

2.E-2

1.5E-2

1.E-2

5.E-3

0.E1 1

FIGURE 16.52

2

3

4 Height (m)

5

6

7

Rates of other homogeneous reactions in the freeboard (Test T12-1, IGT).

404

Solid Fuels Combustion and Gasification

Tar + H2

C2H4 + O2

C3H6 + O2

C3H8 + O2

C6H6 + O2

HCN + O2

1.E0 1.E-1 1.E-2

Rates (kmol/m3/S)

1.E-3 1.E-4 1.E-5 1.E-6 1.E-7 1.E-8 1.E-9 1.E-10 1.E-11 1.E-12

0

0.2

0.4

0.6

0.8 Height (m)

1

1.2

1.4

1.6

FIGURE 16.53 Rates of other homogeneous reactions in the emulsion (Test T12-1, IGT).

120 Carbonac.

Absorbent

Inert

Circulation flux (kg/m2/S)

100

80

60

40

20

0

0

0.2

0.4

0.6

0.8 Height (m)

1

1.2

1.4

FIGURE 16.54 Circulation rates of solid particles in the bed (Test T12-1, IGT).

1.6

Bubbling Fluidized-Bed Simulation Program and Results

d)

e)

f)

g)

h)

405

contributed to high gasiication eficiencies. In cases where such preheating is not provided, fast expansion of gas occurs and bubbles are large, leading to relatively slow increases of temperature of that phase, and the bubble retains oxygen. The slow mass transfer between large bubbles and emulsion provides oxidant conditions even at points far from the distributor, whereas within that region, fuel gases (such as hydrogen and carbon monoxide) produced by reducing gasiication reactions are oxidized. As we have seen, preheating improves the gasiication conditions. Usually, tar is an undesirable component, and efforts should be made to eliminate or minimize its concentration in the produced gas. Among other alternatives, this can be achieved by operating the bed at a relatively high temperature and by feeding the carbonaceous fuel at lower positions in the bed, which provides longer residence time for tar coking and cracking before reaching the freeboard. In case of gasiiers, the solid fuel feeding should be located not only in lower positions but also above the oxidation region. This is so because pyrolysis releases valuable gases such as hydrogen and carbon monoxide that would be consumed if released in the region where oxygen would be still available. During the IGT operations, the biomass was fed above the oxidation region. As shown by comparing Figures 16.40 and 16.42, the peak of tar occurs after the total consumption of oxygen. That allowed for accumulation of hydrogen and other gases, as illustrated by Figures 16.40 and 16.41. However, the conditions were not enough for the complete destruction of tar inside the equipment (Figure 16.42 and Table 16.11). As seen in Chapter 3, care should be taken to avoid segregation among solid species in the bed. That may be done by using high mixing rates. If segregation is allowed in beds with biomass and sand, part of the feeding fuel would likely loat on the bed surface. Therefore, during its devolatilization, the released tar would be carried to the freeboard by the incoming gas stream from below. Usual residence times in the freeboard are not as long as in the bed. Therefore, not enough time would be available for complete or near-complete destruction of tar in that region. Proper simulation programs are powerful tools to predict segregations and to test various designs and operational conditions to avoid them. Figures 16.32 and 16.33 also show the brisk changes of temperatures near the solid fuel feeding position. This is mainly due to sudden destruction of tar by Reactions R.9, R.50, and R.54 (Chapter 8, Tables 8.4 and 8.5), as illustrated by Figures 16.42, 16.48, and 16.53. Figure 16.35 shows the temperature proile in the freeboard during Test T12-1. The raising of temperature in the region above the bed is understood to be due to the still fast tar cooking, cracking, and hydrogenation (Figures 16.48, 16.51, and 16.52). This happens despite the endothermic reactions occurring at that region (Figures 16.44 and 16.47). Only when the tar decomposition reached lower rates the endothermic were reactions able to lower the temperature, which happened for approximately z > 4.5. According to the previously discussion, in many situations the temperature inside bubbles tends to stay below the average in the bed before ignition of

406

Solid Fuels Combustion and Gasification

accumulated gases in that phase takes place. However, this picture may not be followed in regions where the local supericial velocity is close to the minimum luidization velocity and bubbles are relatively small. The resulting high energy and mass transfers with the emulsion forces the temperature of bubbles to remain close to that of the emulsion. This is the present situation, as illustrated by Figures 16.32 and 16.33. For the regions near the feeding point, two factors are the most important regarding the bubble temperatures: 1. Heat and mass transfers with the emulsion 2. Injection of nitrogen, which was exceptionally made in the pilot operations at IGT The irst factor above tends to maintain bubble and emulsion temperatures close to each other, whereas the second leads to bubble cooling because nitrogen was injected at a lower temperature than the average of the bed. Depending on the local conditions, factors 1 and 2 compete, which explains why during Test T12-3a (Figure 16.32) the bubble phase temperature decreased near the nitrogen position, whereas this was not observed during Test T12-1 (Figure 16.33). i) Figure 16.34 illustrates the dramatic difference between temperatures near the gas distributor surface (z = 0). Similar differences are found in almost all cases of combustion (see Figure 16.3) or gasiication reactors operating with oxygen or air (or mixtures) as the luidizing gas. Of course, this does not happen in cases when the gasifying agent (for instance, CO2) does not contain oxygen. As noted previously, a relatively high temperature of carbonaceous fuel near the distributor might indicate the possibility of particle agglomeration and bed collapsing. Nevertheless, even in those cases, if the residence time of particles at a high temperature is very short, the probability of agglomeration is low. This also occurs here since that region occupies less than 1 mm of bed height. j) It is important to stress that pyrolysis is responsible for a signiicant part of fuel gas production gases rich in CO, CH4, and other hydrocarbons. Therefore, higher gasiication eficiencies can be achieved by avoiding oxidation of those gases. On the other hand, tar should be avoided in the exiting gas. To summarize, those goals can be reached by the following: 1. Keeping the fuel feeding just above the oxidizing region. 2. Maintaining the actual and minimum luidization supericial velocity as closely as possible, at least for regions near the distributor. This will lead to small bubbles and fast heat and mass transfer between bubbles and emulsion, which in turn minimizes the height of the oxidizing region and therefore maximizes the volumes where fuel gases are not oxidized. Closeness between actual and minimum luidization can be achieved by the following: • Preheating the gas streams injected into the gasiier, which would minimize the gas expansion • Using larger bed diameters • Increasing the fuel and other solid particle sizes fed into the bed

Bubbling Fluidized-Bed Simulation Program and Results

407

3. Working with deeper beds with fuel feeding far from their tops in order to ensure enough residence time for tar coking and cracking before it reaches the freeboard. The relatively high eficiency of the RENUGAS process is due to the above conditions. In any case, comprehensive simulation is the fastest way to achieve optimization and avoid exhaustive and costly empirical work. k) Figures 16.44, 16.47, 16.51, and 16.52 demonstrate that many reactions do not reach equilibrium, i.e., do not attain a rate of 0. For instance, the shift reaction (Figure 16.47) maintains high rates throughout the whole process, and the carbon-steam reaction (Figure 16.44) continues to contribute even at points near the gas exit. This fact shows how inaccurate zero-dimensional models are, since they necessarily assume chemical equilibrium. l) The circulation rates of each solid particulate phase in the bed are presented in Figure 16.54. Comments regarding the importance of those rates in the process have already been made in Section 16.2.1.3.

16.3

EXERCISES

16.3.1

PROBLEM 16.1**

Implement correlations into model presented in Chapter 13 to account for heat transfer between the bed and a surrounding jacket [13–15]. Assume that liquid water is running in the jacket and that no phase change is achieved. Develop a method to compute the average temperature inside the jacket at each position (z).

16.3.2

PROBLEM 16.2***

Repeat Problem 16.1 for a case in which water in the jacket reaches boiling conditions and nucleate boiling takes place.

16.3.3

PROBLEM 16.3**

Improve the system of boundary conditions, shown in Chapter 13 (Section 13.3.1), in order to allow intermediate injections of gases into the bed or freeboard.

16.3.4

PROBLEM 16.4****

Write a simulation program for a luidized-bed gasiier that is supposed to operate under the following conditions: • There is a steady-state regime. • The gasiier is continuously fed with porous spherical particles of pure graphite. • Air is continuously fed through the distributor or bottom of the bed. • Assume, for instance, the following data: • Mass low of feeding particles is 1.0 × 10 –2 kg s–1. • Calculate the mass low of air in order to be 25% of that necessary for complete or stoichiometric combustion of feeding fuel. • Gasiier internal diameter is 0.5 m. • Bed height is 2.0 m.

408

Solid Fuels Combustion and Gasification • • • • •

Apparent density of feeding particles is 1000 kg m–3. Real density of feeding particles is 2000 kg m–3. Global bed density is 700 kg m–3. Operating pressure is 100 kPa. Average diameter of feeding particles that would require minimum luidization velocity (at 298 K) is around 90% of the supericial velocity of the gas stream injected through the distributor. • Feeding particles are dry and volatile free. Therefore, there is no need to introduce routines or calculations regarding those processes. • In order to allow the combustion of CO, assume that the concentration of water in the entering gas is 1/100 of the concentration of oxygen. • Assume an adiabatic rector; i.e., neglect the heat transfer to environment. • Use the routines developed in previous chapters or use the available literature and Appendix B to set the calculations for the physical-chemical properties. Guiding steps are as follows: 1. 2. 3. 4.

Write the basic equations or the mass and energy differential equations. Write the boundary conditions. Develop a simpliied block diagram for the computation. Write the auxiliary equations for the chemical kinetics, equilibrium, physical and chemical properties, enthalpy and heat transfer calculations, etc. 5. Develop more detailed block diagrams for each subroutine or part of your model. 6. Apply commercially available packages (such as IMSL•) for the convergence routines and for methods of solving the systems of differential equations. 7. After completion, obtain the numerical results for the concentration and temperature proiles throughout the bed.

16.3.5

PROBLEM 16.5****

Improve the previous program by adding a routine for computation of rates and composition of pyrolysis yields. For a irst approximation, just use the Loison and Chauvin correlations as shown in Chapter 10. Integrate that into the program and run it for the case of charcoal. Use the input described for case T12-1 from IGT.

16.3.6

PROBLEM 16.6****

Repeat the previous problem using a combination of the FG and DISKIN models (see Chapter 10).

16.3.7

PROBLEM 16.7****

Improve your program even further to allow simulations of boilers. For a test, use data as given above for the case of a Babcock and Wilcox unit.

Bubbling Fluidized-Bed Simulation Program and Results

409

REFERENCES 1. de Souza-Santos, M.L., Modelling and Simulation of Fluidized-Bed Boilers and Gasiiers for Carbonaceous Solids, PhD thesis, University of Shefield, Shefield, United Kingdom, 1987. 2. de Souza-Santos, M.L., Comprehensive modelling and simulation of luidized-bed boilers and gasiiers, Fuel, 68, 1507–1521, 1989. 3. de Souza-Santos, M.L., Solid Fuels Combustion and Gasiication: Modeling, Simulation, and Equipment Operation, Marcel Dekker (CRC Press), New York, 2004. 4. de Souza-Santos, M.L., Comprehensive simulator (CSFMB) applied to circulating luidized bed boilers and gasiiers, The Open Chemical Engineering Journal, 2, 106–118, 2008. 5. de Souza-Santos, M. L., CSFB applied to luidized-bed gasiication of special fuels, Fuel, 88(5), 826–833, 2009. 6. Rabi, J.A., and de Souza-Santos, M.L., Comparison of two model approaches implemented in a comprehensive luidized-bed simulator to predict radiative heat transfer: results for a coal-fed boiler, International Journal on Computer and Experimental Simulations in Engineering and Science, 3, 87–105, 2008. 7. Babcock and Wilcox Co., SO2 Absorption in Fluidized Bed Combustor of Coal—Effect of Limestone Particle Size, Report EPRI FP-667, Project 719-1, 1978. 8. Babcock and Wilcox Co., Summary Evaluation of Atmospheric Pressure Fluidized Bed Combustion Applied to Electric Utility Large Steam Generators, Report EPRI FP-308, Vol. II: Appendices, 1976. 9. Overturf, B.W., and Reklaitis, G.V., Fluidized-bed reactor model with generalized particle balances. Part I: Formulation and solution, AIChE J., 29(5), 813–820, 1983. 10. Overturf, B.W., and Reklaitis, G.V., Fluidized-bed reactor model with generalized particle balances. Part II: Coal combustion application, AIChE J., 29(5), 820–829, 1983. 11. Ergun S., Coal classiication and characterization, in Coal Conversion Technology, Wen, C.Y., and Lee. E.S., Eds., Addison-Wesley, Reading, MA, 1979, pp. 1–53. 12. Evans, R.J., Knight, R.A., Onischak, M., and Babu, S.P., Process and environmental assessment of the RENUAS process, presented at Symposium on Energy from Biomass and Wastes, sponsored by the Institute of Gas Technology, Washington, DC, April 6–10, 1986. 13. Perry, J.H., Green, D.W., and Maloney, J.O., in Perry’s Chemical Engineers Handbook, 7th Ed., Perry, J.H., Ed. McGraw-Hill, New York, 1997, pp. 12-1–12-90. 14. Incropera, F.P., and DeWitt, D.P., Fundamentals of Heat and Mass Transfer, 4th Ed., John Wiley, New York, 1996. 15. Rohsenow, W.M., and Hartnett, J.P., Handbook of Heat Transfer, McGraw-Hill, New York, 1973.

Fluidized17 Circulating Bed Combustion and Gasification Model CONTENTS 17.1 Introduction ................................................................................................. 411 17.2 Mathematical Model ................................................................................... 411 17.3 Overall Model ............................................................................................. 414 References .............................................................................................................. 416

17.1

INTRODUCTION

As with everything, there are positive and negative aspects of applications of bubbling and circulating luidization techniques. Given a process or an objective, the decision of selecting the most appropriate process is not simple. Each case should be carefully studied, which in itself is a strong justiication for the development and use of comprehensive simulations. To complement the material already presented for bubbling luidization, the present chapter shows the basic aspects of a mathematical model for circulating luidized-bed equipment. Similarly to the previous cases, the model has generated codes included in the Comprehensive Simulator for Fluidized and Moving Beds equipment (CSFMB)* software.

17.2

MATHEMATICAL MODEL

As described in Chapter 3 and detailed below, there are several differences between bubbling and circulating regimes. However, it has also been shown that a onedimensional approach can provide reasonable representation for equipment operating under the circulating version, as shown in Chapter 18. Assume the case of the model chart presented in Figure 17.1. It is important to note that the terms dense region and lean region in circulating beds are the equivalent of bed and freeboard in the case of the bubbling technique, respectively. Therefore, if the results refer to bed and freeboard, they should be respectively understood as dense and lean regions.

*

Also called CeSFaMB (http://www.csfmb.com).

411

412

Solid Fuels Combustion and Gasification Cyclone

Lean region

PF

Tubes

entrained particles

Solids

Carbonaceous PF

Gas

Absorbent

Gas-gas reactions

Inert

Dense region

Tubes

PF

Gas-solid reactions

PF

Returning particles

Emulsion phase

no or very small bubble phase

Particles and gases

PF Gas-gas reactions

PF Gas-gas reactions

Carbonaceous UC Solids Absorbent

Inert

UC

UC

Gas-solid reactions Reinjection Aeration

Gas inlet

Heat transfer

Heat & mass transfers

PF = Plug-flow

UC = Uniform composition

FIGURE 17.1 Simpliied diagram showing the basic aspects of and assumptions for the model of circulating luidized bed equipment.

Details of bubbling luidized beds have been described in Chapters 13, 14, and 15. Accordingly, the program is also applicable to bubbling luidized units operating with partial or complete reinjection into the bed of particles collected at the cyclone system. Such operations are similar in several ways to the circulating bed ones; however, differences are noticeable, and the most important are briely commented on below: 1. For the same average particle size and density of particles in a bed, circulating equipment operates under higher supericial gas velocities than bubbling ones. On the other hand, there is a large intersection between the dynamics of bubbling and circulating bed luidization [1, 2]. 2. Circulating beds usually operate within the fast luidization regime, where no bubbles are discernible. Some operations may occur at turbulent regime region, i.e., with supericial velocity between the irst and second turbulent limits (Chapter 4). In any case, the simulator computes the supericial velocity and veriies the regime point-by-point of the dense phase. If the velocity surpasses the irst turbulent limits, the model assumes bubbles with decreasing sizes, until size is negligible when the second turbulent limit is reached. Above the second turbulent regime, the bubble phase virtually disappears. Therefore, the emulsion takes the whole dense region.

Circulating Fluidized-Bed Combustion and Gasification Model

413

3. The void fraction of the emulsion greatly departs from the minimum luidization one. At each point of the dense region, the void fractions of emulsion and bubbles are computed assuming the following relations: εE = and

ε − εB 1 − εB

ε B = ε B, tur1ftur .

(17.1)

(17.2)

Here, εB,tur1 is the fraction of the bed occupied by the bubble phase just after the irst turbulent limit (or still at slow luidization). The turbulence factor is given by the following: ftur = ftur = 1

U tur2 − U . U tur2 − U tur1 for U < Utur1

ftur = 10 –6 for U > Utur2 .

(17.3) (17.4) (17.5)

The value for ftur is never set as 0 to avoid mathematical incoherencies. In any case, above the second turbulent limit, the model assumes a fast luidization regime. As we have seen, this approach allows continuous transition from slow to fast luidization. It also permits CSFMB to simulate intermediate turbulent regimes, even when one is not aware where a given set of data could lead in terms of luidization dynamics. In addition, depending on the imposed or informed operational conditions, slow luidization can be found in relatively cold regions of the bed—such as are found near the distributor—and turbulent or fast luidization above that. According to the approach proposed here, the simulation would progress throughout the bed or dense regions without unrealistic imposed discontinuities or jumps from one luidization regime to another. 4. The lean phase of circulating beds and the freeboard of bubbling beds present similarities as well as differences. The presence of clusters of solid particulate becomes more pronounced for higher differences between gas and solid ascending velocities. Therefore, clusters are more commonly observed in lean regions of circulating beds. That point is discussed below. After the differences between bubbling and circulating luidization were analyzed, it became clear that the model developed for bubbling process [1, 3–26] could be adapted to simulate circulating ones as long some aspects were revised. The irst point was that already explained in item 3 above, concerning the modiications to take into account the breakage and eventual virtual disappearance of bubbles. The second task was to verify the best model to predict the void fraction proile in the lean region. Of course, a large number of papers on this subject can be found in the literature, and excellent reviews are presented by Pécora [27] and Pécora and Goldstein [28, 29].

414

Solid Fuels Combustion and Gasification

It has been veriied [30–43] that one-dimensional attacks also provide reasonable descriptions of void fraction throughout the lean region; in particular, the work of Rhodes and Geldart [36] is notable. Their main conclusion was that in general, the correlations describing the void fraction proiles at the freeboard of bubbling beds are applicable to describe the void fraction proiles in the lean region of circulating beds. Based on that, Wen and Chen’s relation [44]—used since the irst version of CSFMB [4, 5] and described in Chapters 13 through 15—describing the lux of solid against the height in the freeboard is also applied here for the lean region. Another point to insert into the model and simulation program is a method by which to estimate the height of dense region. This can be accomplished by setting equations that ensure the equalization of pressure between the raising column and the returning column at the reinjection position. In other words, the pressure at the entrance into the raiser should be equal to the pressure losses imposed by the dense region, lean region, piping, cyclone, and valve. For that, the work of Pécora [27] and Pécora and Goldstein [28, 29] was applied. Of course, the relations concerning the energy and mass balances for the dense and lean regions continue to be valid and similar to those previously set [1, 3] and detailed in Chapter 13. In addition, the relationships used to predict the operation of the cyclone system were used similarly (see Chapter 14). The computation of heat transfers between the returning leg and surroundings are easily set. As described below, after each iterative computation of the whole process, the temperature of particles at the top of the lean region becomes known. Simple correlations for heat transfers in this situation can be found in the classical literature [45–49] and are left as an exercise to the reader. That strategy allows the computation of average temperature at each point inside the returning leg.

17.3

OVERALL MODEL

The adaptation of the model developed for bubbling luidized beds and presented in Chapter 13 for cases of circulating follows a strategy similar to that illustrated in Figure 17.1. Among the main differences between bubbling and circulating beds is that in the latter, the transport velocity of average particle is surpassed, and most solid material is lowing through the lean region. Therefore, the residence time of gases and particles in the dense region is much shorter than the time spent in the lean region. Consequently, most important chemical transformations occur in the lean region rather than in dense region. In view of that and differently from the strategy adopted in the case of bubbling beds (Chapter 16), the convergence regarding conversion of solid species into gas components includes not only the dense region (or bed) but also the lean one (or freeboard). Although bubbles cannot be recognized in fast luidization regimes, many circulating beds operate below or within turbulent conditions near the distributor. This is so because the injected gases are not fully heated or expanded, and the supericial velocity may stay below the transport velocity or even below the irst turbulent limit. Therefore, bubbles are formed to be decreased in volume as the transition to turbulent takes place in regions above the distributor. Bubbles would vanish when the second turbulent limit is achieved.

Circulating Fluidized-Bed Combustion and Gasification Model

415

The basic assumptions are similar to the case of bubbling modeling. However, a few differences should be noted. In order to emphasize speciic aspects of circulating beds, the main assumptions and computational strategy can be summarized as follows: 1. The unit operates in a steady-state regime. 2. The equipment is separated in two main regions: the dense region (or bed, in the case of the bubbling condition) and the lean region (or freeboard, in bubbling processes). 3. The dense region is divided into two main phases: bubble and emulsion. 4. Until the irst turbulent limit of supericial velocity is reached, the void fraction of the bed occupied by bubbles and their dimensions are similar to those computed for bubbling beds. 5. For supericial velocities between the irst and second turbulent limits, the void fraction of the dense region occupied by bubbles and their dimensions decrease according to the rules shown above in this chapter. 6. For supericial velocities above the second turbulent limit, the bubble phase vanishes, and the dense region is occupied only by the emulsion. Of course, the void fraction in the emulsion is well above that found in the minimum luidization condition. 7. The emulsion is composed of solid particles and percolating gas. 8. There are three possible solid phases: fuel, inert, and sulfur-absorbent (such as limestone, dolomite, or mixture of those). Ash, eventually detached from the spent fuel, would constitute part of the inert solid phase. 9. Emulsion gas is considered inviscid; therefore, it rises through the bed in a plug-low regime. 10. The same as the above is assumed for the bubble gas in regions where the supericial velocity is below the second turbulent limit. During that period or region of dense phase, the dimensions, raising velocity, fraction of bed volume occupied by bubbles, and other characteristics of bubbles are computed as shown in Chapter 14. 11. While bubbles are present, the exchange of mass and heat with the emulsion is computed by the equations and correlations shown in Chapter 15. 12. Mass and heat transfers also take place between particles and emulsion gas. 13. Heat transfers also occur between particles. 14. Gases are assumed to be transparent regarding radiative heat transfers. 15. Emulsion gas exchanges heat with the vessel or reactor walls. Therefore, all heat transfers between the walls and other phases (bubbles and particles) take place indirectly through the emulsion gas. 16. All phases exchange heat with eventually immersed surfaces (such as tube banks) in the dense and lean regions.* *

Because of the high velocities of particles and consequent fast erosion of immersed surfaces, most circulating boilers do not keep tube banks in the dense or lean region. However, CSFMB is prepared to contemplate even this remote possibility. Thus, all equations related to heat transfers from the bed to the tube surfaces causing phase changes inside the tubes are added to the simulator.

416

Solid Fuels Combustion and Gasification

17. An average uniform composition for each solid particle is assumed in the dense region. 18. The average composition of each solid phase in the dense region is computed by reiterative method as shown for the case of bubbling beds. 19. Compositions and temperatures of all gas and solid phases vary in the lean region and are computed using complete differential and energy balances (Chapter 13). 20. Particle size distributions are modiied because of chemical reactions, attritions between particles themselves, and the entrainment and recirculation processes. All those are taken into account to compute the size distributions of each solid phase in the dense and lean regions. 21. Heat and mass transfers in the axial or vertical direction within each phase are considered negligible compared with the respective transfers in the radial or horizontal direction between a phase and neighboring ones. 22. Aeration gas (with any set composition) is added to the dense or lean region at the particle reinjection position. This is set by CSFMB as the intermediary injection. 23. At each axial position (z), mass transfers between phases are computed. As soon as chemical species are consumed or formed by reactions, they are subtracted from or added to the respective phase. Hence, these effects appear as sink or source terms in the mass continuity equations for each phase (Chapter 13). 24. At each position (z), heat transfers between phases result from differences of temperature at each phase. These terms would appear as sinks or sources in the energy conservation equations. Those equations are described in Chapter 13. 25. At the basis of the dense region (z = 0), the two-phase model [50] is applied to determine the splitting of the injected gas stream between emulsion and bubble phases. 26. For points above the distributor surface (z > 0), the mass low in each phase is determined by fundamental equations of transport phenomena. Those include mass transfers between the various phases, as well as homogeneous and heterogeneous reactions. 27. Boundary conditions for the gas phases concerning temperature, pressure, and composition at (z = 0) are given by the values of the injected gas stream. 28. For every iteration, boundary conditions for the three possible solid phases (carbonaceous, sulfur absorbent, and inert) are obtained after differential energy balances involving conduction, convection, and radiative heat transfers between the distributor surface and the various phases. As we have seen, the present model is a signiicant improvement over a former version [51].

REFERENCES 1. de Souza-Santos, M.L., Solid Fuels Combustion and Gasiication: Modeling, Simulation, and Equipment Operation, Marcel Dekker, New York, 2004.

Circulating Fluidized-Bed Combustion and Gasification Model

417

2. Geldart, D., Gas Fluidization Technology, John Wiley, Chichester, U.K., 1986. 3. de Souza-Santos, M.L., A new version of CSFB, Comprehensive Simulator for Fluidized Bed Equipment, Fuel, 86, 1684–1709, 2007. 4. de Souza-Santos, M.L., Modelling and Simulation of Fluidized-Bed Boilers and Gasiiers for Carbonaceous Solids, PhD thesis, University of Shefield, Shefield, United Kingdom, 1987. 5. de Souza-Santos, M.L., Comprehensive modeling and simulation of luidized-bed boilers and gasiiers, Fuel, 68, 1507–1521, 1989. 6. de Souza-Santos, M.L., Cincoto M.A., Pikman B., Ushima, A.H., and de Souza, M.E., Fluidized-bed combustion of rice husks; experimental tests, in Proc. IV Brazilian Congress on Energy, Rio de Janeiro, Brazil, November 5–9, 1990. 7. Ushima, A.H., Guardani, R., and de Souza-Santos, M.L., Determination of operational conditions for the reduction of CaSO4 to CaS in a luidized bed, presented at the 9th Brazilian Congress on Chemical Engineering, Salvador, Bahia, Brazil, 1992. 8. de Souza-Santos, M.L., Comprehensive Modeling and Simulation of Fluidized-Bed Reactors Used in Energy Generation Processes, presented at the Meeting on Energy Modeling: Optimizing Information and Resources, Institute of Gas Technology, Chicago, 1993. 9. de Souza-Santos, M.L., Application of comprehensive simulation to pressurized luidized bed hydroretorting of shale, Fuel, 73, 1459–1465, 1994. 10. de Souza-Santos, M.L., Application of comprehensive simulation of luidized-bed reactors to the pressurized gasiication of biomass, Journal of the Brazilian Society of Mechanical Sciences, 16(4), 376–383, 1994. 11. de Souza-Santos, M.L., A study on pressurized luidized-bed gasiication of biomass through the use of comprehensive simulation, in Proc. Fourth International Conference on Technologies and Combustion for a Clean Environment, Lisbon, Portugal, paper 25.2, vol. II, 1997. 12. de Souza-Santos, M.L., Search for favorable conditions of atmospheric luidized-bed gasiication of sugar-cane bagasse through comprehensive simulation, presented at the 14th Brazilian Congress of Mechanical Engineering, Bauru, São Paulo, Brazil, 1997. 13. de Souza-Santos, M.L., Search for favorable conditions of pressurized luidized-bed gasiication of sugar-cane bagasse through comprehensive simulation, presented at the 14th Brazilian Congress of Mechanical Engineering, Bauru, São Paulo, Brazil, 1997. 14. de Souza-Santos, M.L., A study on pressurized luidized-bed gasiication of biomass through the use of comprehensive simulation, in Proc. Combustion Technologies for a Clean Environment, Vol. 4, Carvalho, M.G., Fiveland, W.A., Lockwood, F.C. and Papadopoulos, C., Eds. Gordon and Breach, Amsterdam, Holland, 1998, Chapter 4. 15. de Souza-Santos, M.L., A feasibility study on an alternative power generation system based on biomass gasiication/gas turbine concept, Fuel, 78, 529–538, 1999. 16. Costa, A.M.S., and de Souza-Santos, M.L., Studies on the mathematical modeling of circulation rate of particles in bubbling luidized beds, Powder Technology, 103, 110– 116, 1999. 17. Leveson, P.D., and de Souza-Santos, M.L., Improvements on the modeling of sulphur dioxide emissions from luidized bed combustors and gasiiers, presented at ENCIT 2000—8th Brazilian Congress of Thermal Engineering and Sciences, Porto Alegre, Brazil, 2000. 18. Rabi, J.A., and de Souza-Santos, M.L., A two-parameter preliminary optimization study for a luidized-bed boiler through a comprehensive mathematical simulator, presented at ENCIT 2000—8th Brazilian Congress of Thermal Engineering and Sciences, Porto Alegre, Brazil, 2000. 19. de Souza-Santos, M.L., Two trial models for the rate of combustion and gasiication of liquid fuels sprayed on inert luidized particles, presented at the Sixth International Conference on Technologies and Combustion for a Clean Environment, Porto, Portugal, 2001.

418

Solid Fuels Combustion and Gasification

20. de Souza-Santos, M.L., Two trial models for the rate of combustion and gasiication of slurries or liquid biomass sprayed on inert luidized particles, presented at the Fifth Biomass Conference of the Americas, Orlando, FL, 2001. 21. Rabi, J.A., and de Souza-Santos, M.L., Preliminary improvements in the radiative heat transfer modeling for luidized bed biomass gasiication: bed section, presented at the Fifth Biomass Conference of the Americas, Orlando, FL, 2001. 22. Rabi, J.A., and de Souza-Santos, M.L., Incorporation of a 2-lux model for radiative heat transfer in a comprehensive luidized bed simulator. Part I: Preliminary model equations, presented at the 9th Brazilian Congress on Engineering and Thermal Sciences, Caxambu, Brazil, 2002. 23. Rabi, J.A., and de Souza-Santos, M.L., Incorporation of a 2-lux model for radiative heat transfer in a comprehensive luidized bed simulator. Part II: Numerical results and assessment, presented at the 9th Brazilian Congress on Engineering and Thermal Sciences, Caxambu, Brazil, 2002. 24. Rabi, J.A., and de Souza-Santos, M.L., Incorporation of a two-lux model for radiative heat transfer in a comprehensive luidized bed simulator. Part I: Preliminary theoretical investigations, Thermal Engineering, 3, 64–70, 2003. 25. Rabi, J.A., and de Souza-Santos, M.L., Incorporation of a two-lux model for radiative heat transfer in a comprehensive luidized bed simulator. Part II: Numerical results and assessment, Thermal Engineering, 4, 49–54, 2004. 26. van den Enden, P.J., and Lora, E.S., Design approach for a biomass fed luidized bed gasiier using the simulation software CSFB, Biomass and Bioenergy, 26, 281–287, 2004. 27. Pécora, A.A.B., Study of the Fluid Dynamics of Circulating Fluidized Beds with Secondary Air Injection, PhD thesis, University of Campinas, Campinas, Brazil, 1995. 28. Pécora, A.A.B., and Goldstein, L., Jr., A luid dynamics study of circulating luidized beds with secondary air injection, ASME Heat Transfer Division, 334, 287–292, 1996. 29. Pécora, A.A.B., and Goldstein, L., Jr., A luid dynamics model of circulating luidized bed with secondary air injection, Proc. VII Latin American Congress on Heat and Mass Transfer (LATCYM), 1, 245–249, 1998. 30. Kato, K., Shibasaki, H., Tamura, K., Arita, S., Wang, C., and Takarada, T., Particle holdup in a fast luidized bed, Journal of Chemical Engineering of Japan, 22, 130–136, 1989. 31. Choi, J.H., Yi, C.K., and Son, J.E., Axial voidage proile in a cold mode circulating luidized bed, Proc. 3rd Int. Conf. On Circulating Fludized Beds, Nagoya, Japan, 1991, 4.9.1–4.9.6. 32. Li, Y., Chen, B., Wang, F., Wang, Y., and Kwauk, M., Rapid luidization, International Chemical Engineering, 21, 67–678, 1981. 33. Kwauk, M., Wang, N., Li, Y., Chen, B., and Shen, Z., Fast luidization at ICM, in Circulating Fluidized Bed Technology, Basu, P., Ed., Pergamon Press, New York, 33–62, 1986. 34. Li, J., Tung, Y., and Kwang, M., Axial voidage proiles of fast luidized beds in different operating regions, in Circulating Fluidized Bed Technology II, Basu, P., and Large, J.F., Eds., Pergamon Press, New York, 193–203, 1988. 35. Subbarao, D., Clusters and lean-phase behavior, Powder Technology, 46, 101–107, 1986. 36. Rhodes, M.J., and Geldart, D., A model for the circulating luidized bed, Powder Technology, 53, 155–162, 1987. 37. Kunii, D., and Levenspiel, O., Flow modeling of fast luidized beds, Proc. 3rd International Conference of Circulating Fluidized Beds, Nagoya, Japan, 1990, 4.7.1–4.7.8. 38. Zhang, H., Xie, Y., Chen, Y., and Hasatani, M., Mathematical modeling for longitudinal voidage distribution of fast luidized beds, Proc. 3rd International Conference of Circulating Fluidized Beds, Nagoya, Japan, 1990, 4.13.1–4.13.6. 39. Rhodes, M.J., Modeling the low structure of upward-lowing gas–solid suspensions, Powder Technology, 60, 27–38, 1990.

Circulating Fluidized-Bed Combustion and Gasification Model

419

40. Yang, W.C., A model for the dynamics of circulating luidized bed loop, in Circulating Fluidized Bed Technology II, Large, J.F., and Basu P., Eds., Pergamon Press, New York, 1988, pp. 181–191. 41. Kefa, C., Jianren, F., Zhongyang, L., Jianhua, Y., and Mingjiang, N., The prediction and measurement of particle behavior in circulating luidized beds, in Circulating Fluidized Bed Technology II, Large, J.F., and Basu, P., Eds. Pergamon Press, New York, 1988, pp. 27–131. 42. Berruti, F., and Kalogerakis, N., Modeling the internal low structure of circulating luidized beds, The Canadian Journal of Chemical Engineering, 67, 1010–1014, 1989. 43. Guardani, R., Characterization of Fluid Dynamics and Mathematical Modeling of Circulating Fluidized Bed Reactor, PhD thesis, Department of Chemical Engineering, Polytechnic School, University of Sao Paulo, Sao Paulo, Brazil, 1989. 44. Wen, C.Y., and Chen, L.H., Fluidized bed freeboard phenomena: entrainment and elutriation, AIChE J., 28, 117, 1982. 45. Rohsenow, W.M., and Hartnett, J.P., Handbook of Heat Transfer, McGraw-Hill, New York, 1973. 46. Luikov, A.V., Heat and Mass Transfer, Mir, Moscow, 1980. 47. Isachenko, V.P., Osipova, V.A., and Sukomel, A.S., Heat Transfer, Mir, Moscow, 1977. 48. Incropera, F.P., and DeWitt, D.P., Fundamentals of Heat and Mass Transfer, 4th Ed. John Wiley, New York, 1996. 49. Schmidt, F.W., Henderson, R.E., and Wolgemuth, C.H., Introduction to Thermal Sciences, 2nd Ed., John Wiley, New York, 1984. 50. Toomey, R.D., Johnstone, H.F., Gaseous luidization of solid particles, Chem. Eng. Progress, 48(5), 220–226, 1952. 51. de Souza-Santos, M.L., Comprehensive simulator (CSFMB) applied to circulating ludized-bed boilers and gasiiers, The Open Chemical Engineering Journal, 2, 106– 118, 2008.

Fluidized18 Circulating Bed Simulation Program and Results CONTENTS 18.1 18.2 18.3

Introduction ................................................................................................ 421 Simulation Procedure ................................................................................. 421 Simulation Results ...................................................................................... 421 18.3.1 Plant Description and Simulation Inputs...................................... 422 18.3.2 Real Operation and Simulation Results ....................................... 422 18.4 Discussion ................................................................................................... 430 References .............................................................................................................. 432

18.1

INTRODUCTION

Similarly to the previous cases of moving and bubbling luidized-bed reactors, the present chapter shows the simulation strategy devised for building the program based on the model presented in the previous chapter. In addition, simulation results are compared for a few operational conditions of a circulating luidized gasiier. Comments regarding key aspects of this process are also included.

18.2

SIMULATION PROCEDURE

As before, the task of building a program requires the program chart. This is described in Figure 18.1 for cases of circulating luidized-bed equipment. Again, the reader is encouraged to imagine other possibilities, as well to improve on them. It is worth noting the differences between program charts of bubbling (Figure 16.1) and circulating beds: • In the present case, the bed height is not read as input but is computed in order to equilibrate the pressures at the particle reinjection point, as explained in Chapter 17. • As the most important conversions take place in the lean region, the carbon convergence procedure always includes both the dense region and lean region.

18.3

SIMULATION RESULTS

Two main operational conditions of a circulating luidized-bed gasiier are shown and compared with the simulation results. 421

422

Solid Fuels Combustion and Gasification

A Data reading

Assume carbon conversion

Preliminary calculations

First-trial solid-phase temperatures at z = 0

Call convergence

Set boundary conditions for gas phases

Preliminary overall energy balance

Estimate dense region height

Compute particle size distribution

N Convergence for solid-phase temperatures

End

Print results

Has it converged?

Y

Solve system mass & energy differential eq. of dense region

Y

Cyclone & heat transfer eqs. evaluate conditions at returning leg

Assumed CC Computed CC?

N

Set boundary conditions for lean region

Solve system mass & energy differential eq. of lean region

A

FIGURE 18.1 Program chart for cases of circulating luidized-bed equipment.

18.3.1

PLANT DESCRIPTION AND SIMULATION INPUTS

A pilot unit has been developed by the Department of Chemical and Biological Engineering at the University of British Columbia (UBC) in collaboration with Chinese and Korean research institutions. It consists of a 6.5-m-high and 0.1-m internal diameter circulating-bed pilot unit operating near atmospheric pressure. During the tests, the reinjection of particles from the returning leg to the main column or raiser was controlled by a loop-seal valve. Air, injected through a porous plate, was used as the gasiication agent of various biomasses. Tests were conducted within a temperature interval of 970–1120 K and a relatively wide range of feeding rates. Despite the many reported [1, 2] details and operational conditions, the author is very grateful to the research team at UBC for providing many other information required by the Comprehensive Simulator for Fluidized and Moving Beds equipment (CSFMB) simulator. The main operational conditions are also summarized in Tables 18.1 and 18.2.

18.3.2

REAL OPERATION AND SIMULATION RESULTS

The most signiicant operational conditions and their respective simulation results are summarized in Tables 18.3 and 18.4. In addition, various proiles of several important variables are illustrated in Figures 18.2 through 18.10.

423

Circulating Fluidized-Bed Simulation Program and Results

TABLE 18.1 Main Design Data and Test Operational Conditions of UBC Circulating Gasifier Unit Test Detail Fuel

01

07

Cypress

Hemlock

Proximate analysis (w.b. %) Moisture Volatiles Fixed carbon Ash

9.70 75.69 13.98 0.63

8.80 76.67 14.17 0.36

Ultimate analysis (% d.b.) C H N O S High heating value (HHV) (d.b) (MJ/kg) Particle apparent density (kg/m3)a Particle real or skeletal density (kg/m3)a

51.60 6.20 0.65 40.40 0.45 20.3 720 1750

51.82 6.20 0.60 40.6 0.38 20.3 720 1750

Basic geometry Dense region equivalent diameter (m) Lean region equivalent diameter (m) Equipment height (m) Fuel feeding position (m)

0.100 0.100 6.500 0.770

0.100 0.100 6.500 0.770

Injected fuel and gases Feeding low of fuel (kg/s) Air low through distributor (kg/s) Temperature of injected air (K) Steam low injected at 2 m above distributor (kg/s) Temperature of injected steam (K) Average pressure in the equipment (kPa) a

6.40 × 10–3 1.63 × 10–2 433 0.0000 105

8.00 × 10–3 1.75 × 10–2 433 1.938 × 10–4 425 105

Approximate value.

CSFMB performance was also veriied against data obtained from a pilot circulating luidized bed installed at the University of Campinas (Unicamp) [3, 4]. Table 18.5 shows the main dimensions and operational conditions of a test. Several luidization parameters are computed by the simulator, among them the dynamic height of dense region with value of 0.4 m and the mass low of particles. The simulation result for the latter was 4.41 × 10 –2 kg/s, whereas the value measured during experiments was 3.86 × 10 –2 kg/s; therefore, deviation was around 10%.

424

Solid Fuels Combustion and Gasification

TABLE 18.2 Particle Size Distributions of Solid Fed into the UBC Circulating Gasifier Units Test 01 Biomass

Test 07 Inert

Biomass

Inert

Mass Diameter Mass Diameter Mass Diameter Mass Diameter percentage (µm)a percentage (µm)a percentage (µm)a percentage (µm)a 4770 2590 2190 1850 1550 1060 564 334 170 45

a

22.97 34.47 11.42 6.14 7.0 12.24 2.84 1.56 0.95 0.41

1400 710 417 250 90

0.70 73.13 22.61 3.46 0.10

6730 4770 2590 2190 1850 1550 1060 564 334 170 45

0.92 7.21 10.81 13.47 6.51 10.57 34.84 8.98 4.6 1.46 0.63

1400 710 417 250 90

0.70 73.13 22.61 3.46 0.10

Average of the slice with the respective mass fraction.

TABLE 18.3 Composition of Gas Produced during Real Operations of UBC Circulating Gasifier Units and the Respective Simulation Results Test 01 Composition

Real

Simulation

Test 07 Real

Simulation

H2

5.6

4.9995

5.5

5.8822

H2S

n.d.a n.d. n.d. n.d. 68.0 n.d. n.d. n.d. 6.9 18.1 n.d. 1.4 n.d. n.d. n.d. n.d. n.d.

0.0416

n.d.

0.0465

0.1250 0.1914 0.0000 65.9791 0.0000 0.0000 0.0551 9.2428 16.9791 0.0018 2.3634 0.0107 0.0041 0.0000 0.0000 0.0063

n.d. n.d. n.d. 59.5 n.d. n.d. n.d. 16.6 15.0 n.d. 3.4 n.d. n.d. n.d. n.d. n.d.

0.1586 0.1645 0.0000 62.0648 0.0000 0.0000 0.0440 11.4872 16.5799 0.0032 3.5563 0.0065 0.0025 0.0000 0.0000 0.0038

NH3 NO NO2 N2 N2O O2 SO2 CO CO2 HCN CH4 C2H4 C2H6 C3H6 C3H8 C6H6 a

n.d., not determined or not reported.

425

Circulating Fluidized-Bed Simulation Program and Results

TABLE 18.4 Several Additional Operational Conditions during Real Operations of UBC Circulating Gasifier Units and the Respective Simulation Results Test 01 Process parameter

Real

Flue gas low (kg/s) Flow of tar in exiting gas (kg/s) Flow of solids at top of freeboard (kg/s) Minimum luidization velocity (m/s)b Fluidization supericial velocity (m/s)b Average temperature at distributor (K) Average temperature in dense region (K)b Average temperature at top of dense region (K) Average temperature at top of lean region (K) Temperature of carbonaceous particles (K)b Temperature at the recycling injection (K) Dynamic bed height (m) a b c

2.20 × 10 n.d.a n.d. n.d. n.d. n.d. n.d. 1013c n.d. n.d. n.d. n.d.

–2

Test 07

Simulation

Real

2.19 × 10 9.92 × 10–8 5.31 × 10–2 0.050 6.382 1166.2 1166.0 1153.0 1362.0 1166.4 1335.5 0.870

2.18 × 10 n.d. n.d. n.d. n.d. n.d. n.d. 991 n.d. n.d. n.d. n.d.

–2

Simulation –2

2.46 × 10–2 2.03 × 10–7 4.88 × 10–2 0.031 5.178 1057.1 1160.6 1154.5 1383.6 1164.7 1360.2 0.780

n.d., not determined or not reported. At the middle of the dense region. Position not clearly speciied in the report.

2800 Gas

Carbonac.

Inert

Average

2600 2400

Temperature (K)

2200 2000 1800 1600 1400 1200 1000 800 0

1

2

3

4

5

6

7

Height (m)

FIGURE 18.2 Temperature proiles in the lean section, obtained by simulation of Test 07, UBC rig.

426

Solid Fuels Combustion and Gasification 0.25 CO2

CO

O2

Molar fraction

0.2

0.15

0.1

0.05

0 0.E1

1.E0

2.E0

3.E0 4.E0 Height (m)

5.E0

6.E0

7.E0

FIGURE 18.3 CO2, CO, and O2 concentration proiles through the entire equipment, obtained by simulation of Test 07, UBC rig.

1.E0 H2O

1.E-1

H2

CH4

1.E-2 1.E-3 Molar fraction

1.E-4 1.E-5 1.E-6 1.E-7 1.E-8 1.E-9 1.E-10 1.E-11 1.E-12 0

0.1

0.2

0.3

0.4 Height (m)

0.5

0.6

0.7

0.8

FIGURE 18.4 H2O, H2, and CH4 concentration proiles through the entire equipment, obtained by simulation of Test 07, UBC rig.

427

Circulating Fluidized-Bed Simulation Program and Results 5.E-2 H2 S

NH3

Tar/oil

Molar fraction

4.E-2

3.E-2

2.E-2

1.E-2

1.E1 0

1

2

3 4 Height (m)

5

6

7

FIGURE 18.5 Tar (or oil) and other species concentration proiles through the entire equipment, obtained by simulation of Test 07, UBC rig.

1.E-2 Carbonaceous

Absorbent

Inert

Average

Diameter (m)

1.E-3

1.E-4

1.E-5 0.E1

1.E0

2.E0

3.E0 4.E0 Height (m)

5.E0

6.E0

7.E0

FIGURE 18.6 Average diameter of particle species against vertical position during the simulation of Test 07, UBC rig.

428

Solid Fuels Combustion and Gasification 6.E-2 Carbonac.

Absorbent

Inert

5.E-2

Flow (kg/s)

4.E-2

3.E-2

2.E-2

1.E-2

0.E1 0

1

2

3 4 Height (m)

5

6

7

FIGURE 18.7 Upward mass low of solid particles against vertical position in the lean region, obtained by simulation of Test 07, UBC unit.

0.025 0.024 0.023

Flow (kg/s)

0.022 0.021 0.02 0.019 0.018 0.017 0

1

2

3 4 Height (m)

5

6

7

FIGURE 18.8 Upward low of gases against the vertical position in the lean region, obtained by simulation of Test 07, UBC unit.

429

Circulating Fluidized-Bed Simulation Program and Results 0.018 0.016

Mass flow (kg/s)

0.014 0.012 0.01 0.008 0.006 0.004 0.002 Emulsion

Bubble

Total

0 0

0.1

0.2

0.3

0.4 Height (m)

0.5

0.6

0.7

0.8

FIGURE 18.9 Flow of gases in the dense region, obtained by simulation of Test 07, UBC unit.

0.018 0.016

Mass flow (kg/s)

0.014 0.012 0.01 0.008 0.006 0.004 0.002 Emulsion

Bubble

Total

0 0

0.1

0.2

0.3

0.4 0.5 Height (m)

0.6

0.7

0.8

0.9

FIGURE 18.10 Flow of gases in the dense region, obtained by simulation of Test 01, UBC unit.

430

Solid Fuels Combustion and Gasification

TABLE 18.5 Conditions of Circulating-Bed Fluidization Carried at the University of Campinas (Unicamp) Detail

Test 43

Inert (sand) particle characteristics Particle apparent density (kg/m3)a Particle real or skeletal density (kg/m3) Mass percentage of fed particles with average diameter around 0.920 mm Mass percentage of fed particles with average diameter around 0.200 mm

700 1394 80.00 20.00

Equipment basic geometry Dense region equivalent diameter (m) Lean region equivalent diameter (m) Equipment height (m)

0.100 0.100 4.000

Reinjection of particles Cyclone diameter (m) Height of cyclone cylindrical section (m) Height of cyclone conical section (m) Internal diameter of recycling leg (m) Length of recycling tube (m) Position (above the distributor) of particle reinjection in the bed (m)

0.164 0.328 0.328 0.0603 3.10 0.415

Injected gas Air low through distributor (kg/s) Temperature of injected air (K) Average pressure in the equipment (kPa)

0.043 436 120

a

18.4

Approximate value.

DISCUSSION

The results conirm that, unlike most bubbling cases, the dense region—which is the equivalent of the bed in bubbling luidization—is not the most important region in the case of circulating units. The reasons for that are described in Chapter 17. After relatively low and homogeneous average conditions in the dense region, fast combustion of suspended fuel leads to steep variations in temperature (Figure 18.2) and concentrations of several gases, and abrupt consumption of oxygen (Figures 18.3 and 18.4) takes place in the lean region. The sudden variations in temperature and concentrations in the gas phase are due not only to combustion but also to fast pyrolysis, which occurs around the same position. Figure 18.5 illustrates the tar production and its fast destruction by cracking and cooking near the devolatilization peak. Thus, part of the fuel gases released by pyrolysis found an oxidizing atmosphere, which contributes to the sharp increase in temperatures (Figure 18.2). However, the average temperature shows little variation, as is usually conirmed by thermocouples placed throughout the equipment.

431

Circulating Fluidized-Bed Simulation Program and Results

The test in the cold condition also showed that the present modeling leads to results not too far from reality. Further improvements would probably allow better representations of the very complex dynamics taking place in the lean region of circulating luidized equipment. It is also important to note a few aspects related to the dynamics in the lean region. Of course, the size of particles and therefore the average diameter tends to decrease with the height. The variation is relatively gentle under cold conditions (Figure 18.11), and the usual correlations for the upward mass low of particles in the lean region lead to the precise exponential decline of that low against the height (Figure 18.12). However, variations in temperature (Figure 18.2) combined with fast changes in concentration due to chemical reactions (Figures 18.3 through 18.5) may distort that picture, as illustrated in Figure 18.6. The fast gas–solid reactions either decrease the size of carbonaceous particles (in the exposed-core model) or their density (in the unreacted-core model) (see Chapter 9). Since the unreacted-core model is adopted in the lean region, the particles retain their respective diameter, but their density decreases. Therefore, particles that otherwise would return to the dense region might be carried upward by the fast-increasing mass low of produced gases (Figure 18.8). This is characteristic of circulating luidization equipment, in which the transport velocity is usually surpassed. After the peak of temperature, the process leads to small variations in the average particle diameter (Figure 18.6) and their total upward mass low (Figure 18.7) for higher positions in the lean region. Another important aspect is the decline of mass low through the bubble phase in the dense region, as illustrated in Figures 18.9 and 18.10. The decreases start at the point where the gas supericial velocities surpass the irst turbulent limit. Bubbles 1.E-2

Diameter (m)

Carbonaceous

Absorbent

Inert

Average

1.E-3

1.E-4 0.E1

5.E-1

1.E0

1.5E0

2.E0 Height (m)

2.5E0

3.E0

3.5E0

4.E0

FIGURE 18.11 Particle size averages throughout the dense and lean regions of a cold operation (test carried out at Unicamp).

432

Solid Fuels Combustion and Gasification 6.E-2 Carbonac.

Absorbent

Inert

5.E-2

Flow (kg/s)

4.E-2

3.E-2

2.E-2

1.E-2

0.E1 0

0.5

1

1.5

2 Height (m)

2.5

3

3.5

4

FIGURE 18.12 Upward low of solid particles throughout the lean region during cold operation (test carried out at Unicamp).

may completely vanish if the second limit is surpassed, as happened during Test 01, shown in Figure 18.10. This exempliies the applicability and usefulness of the present comprehensive approaches, because they consider all factors combined in order to produce more realistic representations of real equipment operations.

REFERENCES 1. Li, X.T., Grace, J.R., Lim, C.J., Watkinson, A.P., Chen, H.P., and Kim, J.R., Biomass gasiication in a circulating bed, Biomass and Bioenergy, 26, 171–193, 2004. 2. Grace, J.R., Brereton, C.M.H., Lim, C.J., Legros, R., Shao, J., Senior, R.C., Wu, R.L., Muir, J.R., and Engman, R., Circulating Fluidized Bed Combustion of Western Canadian Fuels, Final Report prepared for the Energy Mines and Resources of Canada under contract 52ss.23440-7-9136, 1989. 3. Pécora, A.A.B., Study of the luid dynamics of circulating luidized beds with secondary air injection, PhD thesis, University of Campinas, Campinas, SP, Brazil, 1995. 4. Pécora, A.A.B., Goldstein Jr., L., A luid dynamics study of circulating luidized beds with secondary air injection, ASME Heat Transfer Division, 334, 287–292, 1996.