Transport Processes in Chemically Reacting Flow Systems
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Transport Processes in Chemically Reacting Flow Systems - Daniel E. Rosner
English Symbols
About the Author
Dr. Daniel E. Rosner is Professor and former Chairman (1984–1987, 1993–1996) of the Department of Chemical Engineering at Yale University, where he also holds a joint appointment in Mechanical Engineering. He is founder and Director of the Yale High Temperature Chemical Reaction Engineering Laboratory, and is an engineering consultant to many corporations and consortia, which have included Alcoa, Babcock & Wilcox, Dresser-Rand, Du Pont, EPRI, Exxon, General Electric, IFPRI, Pfaudler, Praxair, SCM-Chemicals, and Union Carbide. His research activities include convective energy and mass transport, interfacial chemical reactions, phase transformations, gas dynamics, fine particle technology, and combustion—subjects on which he has published over 235 papers and the present ASEE-1988 award-winning textbook-treatise, here brought up-to-date via supplemental material for each of the original eight chapters. He joined the Yale University engineering faculty after eleven years of industrial research experience, having completed his undergraduate and Ph.D. engineering degrees at City College of New York and Princeton University, respectively. Dr. Rosner was just named winner of the 1999 D. Sinclair senior scientist research award of the American Association of Aerosol Research.
Dedication
To my wife, Susan
Copyright
Copyright © 1986 by Butterworth-Heinemann Copyright © 2000 by Daniel E. Rosner
All rights reserved under Pan American and International Copyright Conventions.
Published in Canada by General Publishing Company, Ltd., 30 Lesmill Road, Don Mills, Toronto, Ontario.
Bibliographical Note
This Dover edition of Transport Processes in Chemically Reacting Flow Systems, first published in 2000, is an unabridged republication of the work originally published in 1986 by Butterworth-Heinemann, Boston. A new Preface and Supplement, which effectively bring the work up to date, have been specially prepared for this edition.
Library of Congress Cataloging-in-Publication Data
Rosner, Daniel E.
Transport processes in chemically reacting flow systems / Daniel E. Rosner.
p. cm.
Originally published: Boston : Butterworths, c1986. With new pref. and suppl.
Includes bibliographical references.
9780486150635
1. Mass transfer. 2. Transport theory. 3. Fluid dynamics. I. Title.
TP156.M3 R67 2000
660’.28423—dc21
00-022701
Manufactured in the United States of America
Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y 11501
Preface to the Dover Edition
It is a pleasure to have the opportunity to update my 1986 edition with the appended Supplement to the Dover Edition
(pages 541–587), which provides new material for each of the original eight chapters. I have retained my original purpose of providing fruitful viewpoints for tackling engineering problems involving transport phenomena in chemically reacting flow systems. Indeed, the diversity of new applications involving chemically reacting, multi-phase flow systems, including combustion, ¹ requires a fundamental and quite interdisciplinary approach. In the author’s view too many recent books, especially those by a single author, explicitly or implicitly present a rather limited perspective, which can lead the unwary student or researcher into unforeseen pitfalls (see, e.g., the student exercises in Chapters 2S, 3S, 4S, 5S; S
after the chapter number refers to the new supplementary text). Moreover, those authors dealing with some of the more advanced material contained herein (e.g., volume-averaging,
or the balance equations governing an interfacial phase,
etc.) often do so in a much more formal, albeit rigorous, manner which can obscure the principal ideas, and, consequently, leave the student’s intuition relatively undeveloped. Hopefully, the present (necessarily brief) accounts and comments—along with the representative recent references cited—will provide an inviting entrée into these developing topics, all of which transcend the specific field of combustion.¹
The student exercises appearing throughout this book and its new Supplement are representative of those developed at Yale for the graduate courses, Energy, Mass and Momentum Transport Processes,
and its sequels, Topics in Multi-Phase Chemical Reaction Engineering,
and Combustion for Synthesis and Materials Processing
(Rosner, 1997; see Supplement, 1S). Their principal purpose is to demonstrate the important practical implications of the seemingly abstract concepts/laws discussed in the text and Supplement, 1S-8S. The short true/false
(T/F) questions focus mainly on important concepts mentioned (or implied) in the main text or Supplement, and, occasionally, either answer can be defended! The quantitative exercises that follow them drive home some of the important points (cf. Supplement, 8S), often demonstrating that rather simple but quite rational calculations can be made to, e.g., estimate approximately how large a combustion reactor must be (in order to produce, say, a metric ton of high-value product every hour). Such preliminary design
calculations are prudent first steps before considering much more detailed follow-on calculations (often using proprietary codes developed within each industry). All exercises have been designed to develop a young engineer’s intuition and curiosity, and provide interesting examples of the important roles of the core subjects: chemical thermodynamics, homogeneous/heterogeneous chemical kinetics, fluid dynamics, heat and mass transport processes, separation processes, and chemical reaction engineering, which we draw upon and illustrate herein.
Some authors inadvertently give the impression that direct numerical simulation
will provide all of the answers that will be needed by the engineer of the future! Fortunately, this is not the case, and while computer simulation will, undoubtedly, play an increasingly important role in research and development, this book has been written with the conviction that fundamentally-based physical and chemical insight will remain in great demand, and that our rate of progress will be determined by the creative interplay and exploitation of experiment, theory, and computation based on physically realistic sub-models. Thus, my very brief discussions of numerical methods (Chapters 8 and 8S) are merely intended to convey to the reader what kinds of tools are currently available to the engineer, their underlying basis, and what is the natural niche of each of the major available methods. Remarkably, this perspective is often missing from more specialized treatises, which tend to pitch one particular method.
The prospect of a wide audience for this economically priced and updated Dover paperback edition is especially gratifying. It has already benefitted enormously from the feedback and questions of a very diverse group of perceptive students and researchers. I hope that a broad spectrum of new readers will find my exposition interesting, enlightening, and even useful! In any case, please convey to me your comments and suggestions on any aspect of the present manuscript.
Daniel E. Rosner
New Haven
daniel.rosner@yale.edu
December. 1999
Table of Contents
About the Author
Title Page
Dedication
Copyright Page
Preface to the Dover Edition
List of Primary Figures
List of Primary Tables
Preface
1 - Introduction to Transport Processes in Chemically Reactive Systems
2 - Governing Conservation Principles
3 - Constitutive Laws: The Diffusion Flux Laws and Their Coefficients
4 - Momentum Transport Mechanisms, Rates, and Coefficients
5 - Energy Transport Mechanisms, Rates, and Coefficients
6 - Mass Transport Mechanisms, Rates, and Coefficients
7 - Similitude Analysis with Application to Chemically Reactive Systems Overview of the Role of Experiment and Theory
8 - Problem-Solving Techniques, Aids, Philosophy: Forced Convective Heat and Mass Transfer to a Tube in Cross-Flow
Appendix 8.1 - Recommendations on Problem Solving
Appendix 8.2 - Outline of the Method of Finite Differences (MFD) for the Numerical Solution of Partial Differential (Field) Equations (PDEs) and Ancillary Boundary Conditions (BCs)
Appendix 8.3 - Outline of the Method of Finite Elements (MFE) for the Numerical Solution of PDEs on Domains of Complicated Shape
Appendix 8.4 - Outline of the Method of Weighted Residuals (MWR) for the Approximate Solution of Partial Differential (Field) Equations and Ancillary Conditions (BCs, ICs)
Appendix 8.5 - Physical Constants
Appendix 8.6 - Metric System Notes/Conversion Factors
Solutions to Selected Exercises
Index
Supplement to the Dover Edition
List of Primary Figures
Figure 2.1-2 Coordinate surfaces in the cylindrical polar coordinate system. Here the position of each point in space is defined by the three numbers: r, θ, z. The volume element about this point is (Δr) • ( r Δθ) •(Δz) ≡ ΔV. (See, also, Figure 2.5-4(a).) Each local vector (say, the velocity v) can, appropriately, be resolved into its r, θ, z components (υr, υθ, υz, respectively).
Figure 2.5-1 Absolute
molar enthalpies for several important ideal gases containing the chemical elements C, H, O, and N.
Figure 2.5-3 Summary of friction-loss factors (Re > 10⁵) for common fluid-filled system components (after Beek and Muttzall (1975)).
Figure 2.5-4 Orthogonal polar coordinate systems: (a) cylindrical (r, θ, z); (b) spherical (r, θ, φ).
Figure 3.2-2 Temperature dependence of the dynamic viscosity, µ, of selected liquids (near atmospheric pressure).
Figure 3.2-3 Corresponding states correlation for the viscosity of simple fluids; based primarily on data for group VIIIA elements (after Bird, et al. (1960)).
Figure 3.3-1 Thermal conductivities of various substances (adapted from Rohsenow and Choi (1961)).
Figure 3.3-2 Corresponding-states correlation for the thermal conductivity of simple fluids; based primarily on data for group VIIIA elements (after Bird, et al. (1960)).
Figure 3.4-1 Predicted small-particle Schmidt number (v/Dp) for SiO2 particles in 1500 K air; transition between free-molecule (Eq. (3.4-8)) and continuum (Eq. (3.4-14)) behavior (Rosner and Fernandez de la Mora (1982)).
Figure 4.3-2 Steady one-dimensional isentropic flow of a perfect gas with γ = 1.3 (adapted from Shapiro (1953)).
Figure 4.3-3 Steady one-dimensional flow of a perfect gas (with γ = 1.3) in a constant-area duct; frictionless flow with heat addition (adapted from Shapiro (1953)).
Figure 4.3-4 Steady one-dimensional flow of a perfect gas (with γ = 1.3) in a constant-area duct; adiabatic flow with friction (adapted from Shapiro (1953)).
Figure 4.3-8 Normal shock property ratios as a function of upstream (normal) Mach number Ma (for γ = 1.3).
Figure 4.4-3 Experimental values for the overall drag coefficient (dimensionless total drag) for a cylinder (in cross-flow), over the Reynolds’ number range 10–1 ≤ Re < 10⁶ (adapted from Schlichting (1979)).
Figure 4.4-4 Experimental values for the overall drag coefficient (dimensionless total drag) for a sphere over the Reynolds’ number range 10–1 ≤ Re ≤10⁶ (adapted from Schlichting (1979)).
Figure 4.4-5 Road map" of common methods of solution to problems in transport (convection/diffusion) theory.
Figure 4.5-3 Experimental and theoretical friction coefficients for incompressible Newtonian fluid flow in a straight smooth-walled circular duct of constant cross section (after Denn (1980)).
Figure 4.5-6 Tangential velocity profile within the laminar BL on a flat plate at zero incidence (after Blasius [1908] and Schlichting (1979)).
Figure 4.7-1 Experimentally determined dependence of fixed bed friction factor ƒbed on the bed Reynolds’ number (adapted from Ergun (1952)).
Figure 5.3-3 Experimentally determined local Nusselt numbers for cross-flow of a Newtonian fluid (Pr = 0.7) about a circular cylinder at various Reynolds’ numbers (adapted from Giedt, W. H. Trans. ASME 71, 378 (1949) and Van Meel (1962)).
Figure 5.3-4 Polar plot of experimentally determined local SthPr²/³, Cf and Cp-distributions for cross-flow of a Newtonian fluid about a circular cylinder at Re = 1.7 × 10⁵ (adapted from Fage and Falkner (1931) and Giedt (1959)).
Figure 5.5-1 Nusselt number distribution in a straight circular duct with fully developed viscous (Newtonian) fluid flow.
Figure 5.5-2 Experimentally determined dependence of packed-bed Nusselt number, Nuh.bed, on the bed Reynolds’ number, Rebed, and fluid Prandtl number, Pr (adapted from Whittaker (1972)).
Figure 5.6-1 Area-averaged natural convective heat-transfer data for vertical flat surfaces in an otherwise quiescent Newtonian fluid. Note transition to turbulence (within the thermal BL) at Rah-values (based on plate length) above ca. 10⁹ (adapted from Eckert and Jackson (1950)).
Figure 5.7-1 Constant velocity and temperature contours for a turbulent round jet in a co-flowing stream of velocity, Us, and temperature Ts(adapted from Forstall and Shapiro (1950)).
Figure 5.7-2 Universal velocity profile near the wall for fully developed turbulent pipe flow of a Newtonian fluid.
Figure 5.8E Chart for predicting the centerline temperature of an infinitely long cylinder (adapted from Heisler (1947)).
Figure 5.9-1 Dependence of total hemispheric emittance on surface temperature for several refractory materials (log-log scale) (adapted from Rosner (1964)).
Figure 5.9-2 Predicted view-factors between two parallel coaxial disks (adapted from Sparrow and Cess (1978)).
Figure 5.9-3 Predicted view-factors for the concentric cylinder geometry: (a) outer cylinder to inner cylinder; (b) outer cylinder to itself.
• L (adapted from Eckert (1937)).
Figure 6.4-3 Transfer-coefficient reduction factor due to Stefan flow blowing
(Bm > 0) and enhancement factor due to Stefan suction
(–1 ≤Bm < 0) (after Bird, Stewart, and Lightfoot (1960)).
Figure 6.4-4 Catalyst effectiveness factor for first-order chemical reaction in a porous solid sphere (adapted from Weisz and Hicks (1962)).
Figure 6.4-5 Catalyst effectiveness factor vs. experimentally observable (modified) Thiele modulus (adapted from Weisz and Hicks (1962)).
Figure 6.4E Fraction, f, of initial solute content which has escaped vs. the dimensionless time τ ≡ Dtfor an isothermal sphere.
Figure 6.5-2 Possible diffusion flame shapes for a coaxial fuel jet discharging into a duct surrounded by an equal-velocity, uniform oxidizer stream (adapted from Burke and Schumann (1928)).
Figure 6.6-1 Particle capture fraction correlation for ideal (Re → ∞) flow past a transverse circular cylinder (Israel and Rosner (1983)). Here tflow ≡ (d/2)/U.
Figure 7.1-2 Corresponding states
correlation for the compressibility pV/(RT) of ten vapors (after G.-J. Su (1946)).
Figure 7.2-1 Correlation of perimeter-averaged natural convection
heat transfer from/to a horizontal circular cylinder in a Newtonian fluid (adapted from McAdams (1954)).
Figure 7.2-2 Pressure dependence of methane/air laminar flame speed (adapted from Diedrichsen and Wolfhard (1956)).
Figure 7.2-3 Dependence of laminar flame speed on burned gas temperature for several (Φ = 0.8) fuel/air mixtures (adapted from Kaskan (1951)).
Figure 7.2-6 Correlation for the GT combustor efficiency vs. parameter proportional to (inverse) Damköhler number (adapted from S. Way (1956)).
Figure 7.2-7 Correlation of GT combustor stability limits vs. parameter proportional to (inverse) Damköhler number (after D. Stewart (1956)).
Figure 7.2-8 Correlation of flashback limits for premixed combustible gases in tubes (after Putnam and Jensen (1949)).
Figure 7.2-10 Test of proposed correlation of the dimensionless blow-off
velocity, UboL/αu vs. SuL/αu for a flame stabilized by a bluff body of transverse dimension L in a uniform, premixed gas stream (adapted from Spalding (1955)).
Figure 7.2-11 Correlation of laminar jet diffusion flame lengths (adapted from Altenkirch et al. (1977)).
Figure 7.2-12 Approximate correlation of fuel-droplet burning rate constants at elevated pressures (based on data of Kadota and Hiroyasu (1981)).
Figure 8.1-1 Correlation of heat loss/gain by circular cylinder in a steady cross-flow of air (after McAdams (1954)).
Figure 8.2-1 Correlation of inertial capture of particles by a circular cylinder in cross-flow (Israel and Rosner (1983)).
List of Primary Tables
Table 2.5-1 Scale factors for the three most commonly used orthogonal coordinate systems.
Table 3.2-1 Lennard-Jones potential parameters (after Svehla (1962)).
Table 3.2-2 Correlation between Lennard-Jones parameters and accessible macroscopic parameters.
Table 3.4-1 Power-law curve-fit to available Dij data for some low-density binary gas mixtures.
Table 3.13E Ion diffusion coefficients in 25°C water.
Table 5.4-1 Steady-state, source-free energy diffusion in one dimension.
Table 5.5-1 Heat transfer and friction for fully developed laminar Newtonian flow through straight ducts of specified cross-section (after Shah and London (1978)).
Table 5.9-1 Black body radiant emission from surfaces at various temperatures.
Table 5.9-2 Approximate temperature dependence of total radiant-energy flux from heated solid surfaces (cf. Rosner (1964)).
Table 6.4-1 Representative parameter values for some heterogeneous catalytic reactions (after Hlavacek et al. (1969)).
Table 6.5-1 Physical and combustion properties of selected fuels in air (after NACA 1300 and Fristrom and Westenberg (1965)).
Table 6.6-1 Critical Stokes’ numbers for pure
inertial impaction.
Table 6.7-1 Some estimates of overall combustion kinetics parameters (after Kanury (1975)).
Table 8.1-1 Thermodynamic and transport properties of air at 20 atm (after Poferl and Svehla (1973)).
Preface
as the convective mass , etc. (see Nomenclature).
J. W. Gibbs remarked that the role of theory in any science is to find the perspective from which the subject appears in its simplest form. My purpose is to present in a simple language but rather general form, principles and approaches that have proven to be very fruitful, and that will doubtless remain so in solving the challenging problems still ahead of us. Thus, while our perspective and scope is broader than that found in many previous transport textbooks (especially those intended for undergraduates), the presentation here is deliberately concise and very selective, leaving many details
for student exercises. I hope the result provides the dedicated reader with the fundamentally oriented yet up-to-date background needed to tackle more advanced, specialized topics. In any event, I am confident it will put the reader in a position to properly formulate and solve many important problems involving rates of energy, mass, or momentum transport in fluids that may be reacting chemically.
The pedagogical choice of combustion for many of the examples is not merely the result of my own research background. For the reasons outlined below I am convinced that combustion is an excellent prototype
for presenting the important concepts of transport in chemically reacting fluid flows. First, it is perhaps the only area of chemically reacting flows not only common to chemical engineering, mechanical engineering, and aeronautical engineering, but also familiar in the daily experience of all applied scientists. Second, while avoiding the dazzling variety of phases, states, and chemical species encountered in present-day ChE reactor applications, combustors exhibit all of the important qualitative features of nonideal, transport-limited, nonisothermal reactors used to synthesize valuable chemicals—indeed, many chemicals (C2H2, HCl, P2O5, TiO2, etc.) are routinely produced in flame
reactors. Finally, it should not be necessary to remind the reader of the economic importance of the efficient use of our remaining fossil fuels, and the prevention of combustion-related accidents. Since one of my primary objectives is to lay a proper foundation for subsequent study and R & D, in this introductory treatment I have deliberately avoided many topics, more heavily dependent on empiricism, associated with interacting multiphase transport (e.g., boiling, bubbling fluidized bed dynamics, etc.). However, as indicated in Section 2.6.4, the macroscopic conservation conditions (see the Introduction to Chapter 2) on which we systematically build our understanding of single-phase flow systems also provide the starting point for rational pseudo-continuum theories of dispersed multiphase situations. Therefore, it is appropriate that these underlying principles first be mastered in the context of either single-phase flows, or simple limiting cases of two-phase flows (e.g., steady flow through isothermal porous media) or packed beds (Sections 4.7, 5.5.5, and 6.5.1) and diffusion with chemical reaction in porous solid media (Section 6.4.4). [Study of Section 2.5 can be postponed without a loss in continuity; however, several of the derived forms of the conservation equations given here will prove useful in Chapters 4, 5, and 6.]
Also deliberately excluded is explicit material on what might be called the systems
aspects of heat/mass exchangers, chemical reactors, and networks thereof. Thus, while we formulate and exploit the principles on which individual exchangers and chemical reactors are selected and designed (e.g., sized), explicit consideration of the economic optimization of specific devices, or the integration of many separate devices (as in multistage arrangements, or chemical plants
) would take us too far from our central themes.
While Chapters 4, 5, and 6 deal successively with momentum, energy, and mass transport, we clearly develop, state, and exploit useful quantitative analogies
between these transport phenomena, including interrelationships that remain valid even in the presence of homogeneous or heterogeneous chemical reactions (Sections 6.5.3 and 6.5.5). Moreover, we include a separate chapter (7) on the use of transport theory in the systematization and generalization of experimental data on chemically reacting systems, emphasizing similitude
methods that go far beyond ordinary dimensional analysis.
Because of our present emphasis on the transport mechanisms of convection and diffusion, which operate for momentum, energy, and (species) mass, the somewhat singular
subject of radiative energy transport (Section 5.9) is only briefly included. While some chemical reactors are intended to produce photons (e.g., combustion-driven furnaces or chemical lasers), radiation is often an incidental by-product.
These factors, together with the one-way
nature of the fluid dynamics–radiative energy coupling in most engineering devices (i.e., the fluid-momentum, energy, and species density
fields are needed to predict the radiative behavior, but not vice versa), account for the brevity of this section. Nevertheless, what little is included is intended to indicate the nature of the radiative transport problem, and to suggest fruitful alternative approaches to deal with it.
Following a concise overview
(Chapter 7, Summary) of the main points of each chapter, many of these principles and methods are then brought together in a comprehensive numerical example (Chapter 8) intended to also serve as a prototype (see Appendix 8.1, Recommendations on Problem-Solving) for student solutions to the novel problems posed at the end of each chapter. These exercises,
which are an extremely important part of this textbook from the viewpoint of a student’s education, have been designed to bring out important qualitative and quantitative engineering implications of the topics treated in each chapter. Unless otherwise specified they were developed by the author in connection with his previous teaching, research, and consulting; however, in some cases (clearly cited), they are elaborations or revisions of similar problems included in earlier textbooks or treatises. Several complete solutions are provided to demonstrate the specific use of seemingly abstract
concepts, mathematical formulae, and/or graphical or tabular data provided in each chapter. While our preference is for metric units (m-kg-s, or cm-g-s), some examples are deliberately included in other commonly used engineering unit systems (for conversion factors, see Appendix 8.6). Most equations derived or quoted herein are either dimensionless or, if dimensional, stated in a form in which they are valid in any self-consistent unit set.
In summary, the principles developed and often illustrated here for combustion systems are important not only for the rational design and development of engineering equipment (e.g., chemical reactors, heat exchangers, mass exchangers) but also for scientific research involving coupled transport processes and chemical reaction in flow systems. Moreover, the groundwork is laid for the systematic further study of more specialized topics (chemical reactor analysis/design, separation processes, multiphase transport, radiative energy transport, computational fluid mechanics, combustion science and technology, etc.). Indeed, while developed primarily for use as a graduate (and undergraduate) textbook in transport processes (energy, mass, and momentum), our emphasis on fluids containing molecules capable of undergoing chemical reaction (e.g., combustion) should make this book useful in more specialized engineering courses, especially chemical reaction engineering and combustion fundamentals. Specific sequences of topics in each of these possible courses are identified in Tables P1 and P2. In each case it is assumed that the relevant background in the underlying sciences of chemical thermodynamics and chemical kinetics can be provided via readily available texts in these classical areas.
By this time the reader will have noted that this text is concerned with the principles underlying the development of comprehensive rational computer models of chemically reacting flow systems, rather than the description of recently developed computer aids to engineering design. Thus, our emphasis is on the use of fundamental laws in the clever exploitation of a judicious blend of experiment, analysis, and numerical methods to first develop the requisite understanding, and, ultimately, to develop mathematical models for the essential portions of engineering problems involving energy, mass, and/or momentum exchange. In this respect, the particular problems and solutions I have chosen to explicitly include here should be regarded merely as instructive prototypes
for dealing with the challenging new engineering problems that face us.
Much of my own learning occurs in the process of doing research in the general area of transport processes in chemically reacting systems. For this reason I wish to acknowledge the Office of Scientific Research of the U.S. Air Force and NASA-Lewis Research Laboratories for their financial support of research that has strongly influenced the orientation and content of this book. I am also indebted to many colleagues at Yale University and EXXON Corporation for their helpful comments, and to the members of Technion–Israel Institute of Technology for their hospitality during the Fall of 1982, when this manuscript was essentially put into its present form. However, the author takes full responsibility for any errors of commission or omission associated with this first edition, and will welcome the written feedback of students, faculty, and practicing engineers and applied scientists who use this book.
Table P1 Chemical Reaction Engineering
Table P2 Combustion Fundamentals
Daniel E. Rosner
New Haven
1
Introduction to Transport Processes in Chemically Reactive Systems
INTRODUCTION
The information needed to design and control engineering devices for carrying out chemical reactions will be seen below to extend well beyond the obviously relevant underlying sciences of:
i. thermochemistry/stoichiometry, and
ii. chemical kinetics,
already encountered in each student’s preparatory courses. Even the basic data of these underlying sciences are generated by using idealized laboratory configurations (e.g., closed calorimeters (bombs
), well-stirred reactors, etc.) bearing little outward resemblance to practical chemical reactors.
This book deals with the role, in chemically reacting flow systems, of transport processes—particularly the transport of momentum, energy, and (chemical species) mass in fluids (gases and liquids). The laws governing such transport will be seen to influence:
i. the local rates at which reactants encounter one another;
ii. the ability of the reactants to be raised to a temperature at which the rates of the essential chemical reactions are appreciable;
iii. the volume (or area) required to carry out the ensuing chemical reaction(s) at the desired rate; and
iv. the amount and the fate of unwanted (by-) products (e.g., pollutants) produced.
For systems in which only physical changes occur (e.g., energy and/or mass exchange, perhaps accompanied by phase change), the same general principles can be used to design (e.g., size) or analyze equipment, usually with considerable simplifications. Indeed, we will show that the transport laws governing nonreactive systems can often be used to make rational predictions of the behavior of analogous
chemically reacting systems. For this reason, and for obvious pedagogical reasons, the simplest illustrations of momentum, energy, and mass transport (in Chapters 4, 5, and 6, respectively) will first deal with nonreacting systems, but using an approach and a viewpoint amenable to our later applications or extensions to chemically reacting systems. This strategy is virtually an educational necessity, since chemical reactions are now routinely encountered not only by chemical engineers, but also by many mechanical engineers, aeronautical engineers, civil engineers, and researchers in the applied sciences (materials, geology, etc.).
Engineers frequently study momentum, energy, and mass transport in three separate, sequential, one-semester courses, as listed in the accompanying table.
Here the essential viewpoints and features of these three subjects will be concisely presented from a unified perspective (Chapters 2 through 7), with emphasis on their relevance to the quantitative understanding of chemically reacting flow systems (Chapters 6 and 7). Our goal is to complete the foundation necessary for dealing with modern engineering problems and more specialized topics useful to:
chemists, physicists, and applied mathematicians with little or no previous experience in the area of transport processes, and/or
engineers who have studied certain aspects of transport processes (e.g., including fluid mechanics, and heat transfer) but in isolation and/or divorced from their immediate application to chemically reacting (e.g., combustion) flow systems.
For several reasons (see the Preface) we will illustrate the principles of chemically reacting flows and reactor design and analysis using examples drawn primarily from the field of combustion—i.e., that branch of the engineering of chemical reactions in which the net exoergic chemical change accompanying the mixing and reaction of fuel
and oxidizer
³ is exploited for such specific purposes as power generation, propulsion, heat exchange, photon production, chemical synthesis, etc. Combustion examples have the merits that:
i. The field is encountered by virtually all engineers and applied scientists;
ii. These examples exhibit most of the important features of nonideal, transport-limited, nonisothermal chemical reactors; and
iii. They deal with an interdisciplinary subject of enormous industrial and strategic importance.
Convenient chemical fuels (easy to store, clean burning, energetic per unit volume, etc.) are today an especially vital commodity, and the need to efficiently synthesize and utilize such fuels continues to provide a powerful incentive to the study and advancement of combustion science and technology.
In the remainder of this introductory chapter we will first briefly illustrate the role of physical transport processes in several reasonably familiar combustion applications to which we will return in Chapters 6 and 7. We then introduce the basic strategy we will adopt to formulate and solve technologically important transport problems with (or without) chemical reactions.
1.1 PHYSICAL FACTORS GOVERNING REACTION RATES AND POLLUTANT EMISSION: EXAMPLES OF PARTIAL OR TOTAL MIXING
RATE LIMITATIONS
The following specific but representative examples illustrate the important role of physical rate processes, and will serve to motivate our subsequent treatment of the quantitative laws of momentum, energy, and mass transport in flow systems with chemical reaction. For definiteness (see Preface) they deal, respectively, with a premixed (fuel + oxidizer) system, an initially unmixed gaseous fuel + oxidizer system, and an initially unmixed condensed fuel (heterogeneous
combustion) system.
1.1.1 Flame Spread across IC Engine Cylinder
Consider events in a well-carbureted internal combustion (IC or piston
) engine cylinder following the firing of the spark plug (Figure 1.1-1). Of interest is the adequacy of the spark for ignition, and the time required for the combustion reaction to consume the fuel vapor + air mixture in the cylinder space (defined, in part, by the instantaneous piston location). One finds that if the spark energy is adequate the localized combustion reaction is able to spread outward, consuming the fresh reactants in a propagating combustion wave,
which usually appears to be a wrinkled
discontinuity in a high-speed photograph. What factors govern the necessary spark energy deposition rate per unit/volume, and flame
propagation rate across the chamber? Certainly the rates and exo-ergicities of the participating chemical reactions play a role, however; so do the transport processes which participate in determining the local compositions and temperatures. Thus:
If the spark energy deposition rate is too small, or the gas velocities past the gap too large, no local region of the gas mixture will be heated to a temperature high enough to allow the combustion region to continuously spread into the unburned gases—i.e., the fledgling combustion zone will extinguish
in response to an unfavorable ratio of the rate of heat loss to that of heat generation.
Figure 1.1-1 Flame spread across a carbureted internal combustion (piston) engine.
Even where the unburned premixed gas is locally motionless, the rate at which it is heated to temperatures at which combustion reactions become appreciable is determined in part by the rate of energy diffusion (conduction) from the already burned gas (on the hot
side of the wave).
Chaotic (turbulent) gas motion may augment the local rates of energy transfer to the unburned gas by suddenly projecting pockets of burned gas into the unburned region. This effect, combined with an augmentation in the (now wrinkled) flame area, act to increase the rate of propagation of the combustion wave across the unburned gas space.
In the immediate vicinity of the water-cooled cylinder walls the premixed gas loses energy to the wall, becoming considerably more difficult to ignite. This can cause local extinction without the combustion wave being able to consume all of the unburned fuel vapors originally in the chamber.
Clearly, each of these important facets of internal-combustion engine performance involves transport phenomena in a central way.
1.1.2 Gaseous Fuel Jet
Figure 1.1-2 shows an ignited, horizontal, turbulent gaseous fuel jet introduced into a surrounding air stream at a pressure level near 1 atm. Here, in contrast to Figure 1.1-1, the fuel and oxidizer vapors do not coexist initially, but must first find each other
in a narrow reaction zone that separates the unreacted fuel region from the surrounding air. Hot combustion products, generated in this narrow reaction zone, mix in both directions, locally diluting
but heating both fuel and oxidizer streams.
Initially unmixed fuel/oxidizer systems of this type have two principal advantages over their pre-mixed
counterparts:
There is little explosion hazard involved in recycling
energy (that might have been wasted in the effluent stream) into one or both of the reactant feed streams.
In furnaces
(where the goal is to transfer as large a fraction as possible of the reaction energy to heat sinks
placed within the reactor), such (diffusion
) flames are found to be better heat radiators (owing to the transient presence of hot soot particles) when carbonaceous fuels are burned (see Section 5.9.2).
The shape and the length of flame required to completely burn the fuel vapor are here dominated by turbulent transport, rather than chemical kinetic factors. This is suggested by the following interesting behavior:
An increase in the fuel jet velocity does not appreciably lengthen the flame!
Large changes in the chemical nature of the fuel have only a small influence on flame length and shape characteristics.
Thus, while (as is often the case) chemical kinetic factors may play an important role in pollutant emission (e.g., NO(g), soot), physical transport processes control the overall volumetric energy release rate (flame length, etc.).
Figure 1.1-2 Horizontal gaseous fuel jet issuing into a turbulent, co-flowing, oxidizer-containing gas stream.
1.1.3 Single Fuel Droplet and Fuel Droplet Spray Combustion
Many useful fuels are not only conveniently stored as liquids, but they can be effectively burned (without complete pre-vaporization
) by spraying them directly into the combustion space (e.g., oil-fired furnaces, diesel-engine cylinders). If the ambient oxidizer concentration is adequate and the relative velocity between the droplet and gas is sufficiently small, such a droplet may be surrounded by an envelope
diffusion flame (Figure 1.1-3) in which fuel vapors generated at the droplet surface meet inflowing oxygen. The situation is reminiscent of Section 1.1.2 in that the fuel and oxidizer vapor are separated from one another, meeting only at a thin reaction front.
However, here the energy generated at the flame zone must also be fed back to sustain the endothermic fuel vaporization process itself. Again, these physical processes usually control the overall combustion rate, as evidenced by the following behavior (cf. Rosner (1972) and Section 6.5.5.7):
The time to completely consume a fuel droplet depends quadratically on the initial droplet diameter;
The droplet lifetime is only weakly dependent on ambient gas temperature and pressure level, and even chemical characteristics of the fuel.
Actually, a fuel droplet usually finds itself in a local environment that cannot support an individual envelope diffusion flame. Rather, the conditions of oxidizer transport into the droplet cloud are such that most droplets collectively vaporize in fuel-rich environments which then supply a single vapor-phase jet diffusion flame, much like that shown in Figure 1.1-2. Again, this overall behavior is quite insensitive to the intrinsic chemical kinetic properties of these fuel/air systems. These considerations not only pertain to conventional (hydrocarbon) liquid fuels but also to fuels
such as liquid sulfur or liquid phosphorus burned in spray devices like that described here to produce, respectively, SO2(g) (one step in the production of sulfuric acid), or P2O5 for fertilizer production.
Figure 1.1-3 Envelope flame model of isolated fuel droplet combustion in an ambient gas containing oxidizer (adapted from Rosner (1972)).
1.2 CONTINUUM (VS. MOLECULAR) VIEWPOINT: LENGTH AND TIME SCALES OF FLUID-DYNAMIC INTEREST
Recall that from an experimental (phenomenological) point of view the laws of thermochemistry and chemical kinetics can be developed without postulating a molecular model of matter. This macroscopic
point of view can be extended to deal with continuously deformable media (fluids) encountered in engineering applications. The resulting subject is then called continuum
mechanics, or fluid mechanics, except that we must deal with fluids whose composition (and other state properties) change from point to point and with time.
Historically, the laws of mechanics and thermodynamics were first developed for discrete amounts of matter—e.g., a mass point,
an artillery projectile, the moon, etc. These same laws can, however, be extended to apply to fluids that appear continuous on a macroscopic level (e.g., the entire gas phase in Figures 1.1-1, 1.1-2), as shown in Chapter 2. This program, initiated by Cauchy, Euler, Lagrange, and Fourier, among others, provides the basis for quantitatively understanding and even predicting complex fluid motions, without or with simultaneous chemical reaction.
More generally, the combustion space is filled with more than one phase (e.g., Figure 1.1-3) and hence is discontinuous on a scale that is coarser than microscopic. Even such flows can be treated as continuous on the length scales of combustor interest (e.g., many centimeters or meters). However, to complete such a pseudo-continuum formulation, information is required on the local interactions between the co-existing phases.
Often this is provided from an intermediate scale analysis in which the region is considered piecewise continuous, and conservation laws are imposed within each continuum.
For definiteness, consider the gaseous-fuel jet situation sketched in Figure 1.2-1. A number of macroscopic dimensions are indicated, e.g.:
rj ≡ fuel jet radius,
rf ≡ radial location of the flame zone,
Lf ≡ total flame length,
Lc ≡ characteristic dimension of the combustion space.
Typically rj < rf < Lf < Lc; however, what is more important here is that each such macroscopic dimension is very large compared to the following "microscopic" lengths:
σ ≡ the characteristic diameter of a single molecule,
n–1/3≡ the average distance between molecules (where n ≡ number density),
≡ the average distance traveled by a single molecule before it encounters another molecule (the mean-free-path
).
Figure 1.2-1 Gaseous fuel jet in confined furnace space: characteristic macroscopic lengths.
This disparity makes it possible to use a continuum formulation to treat macroscopic problems, explicitly ignoring the molecularity
or granularity
of the (gaseous) medium (e.g., at a point where p = 1 atm and T ≅ 1000 K we have, approximately, σ ≈ 4 x 10-¹ nm, n≈ 200 nm (where 1 nm ≡ 10–9 m)).
An equivalent statement can be made in terms of characteristic times, about which more will be said in Chapter 7. Thus, we have the macroscopic characteristic times:
tflow ≡ time for a representative fluid parcel to traverse the combustor length (≈ Lc/U, where U is the characteristic axial air velocity);
tdiff ≡ time for a representative fluid parcel (or tracer constituent) to diffuse from the jet centerline to the flame zone,
etc. These macroscopic times are usually much longer than 1 ms (≡ 10–3 seconds). In contrast, consider
tinteraction ≡ characteristic time during which a molecule interacts with a collision partner during a single binary encounter,
tcollision ≡ average time elapsed before a molecule encounters another molecule,
tchem ≡ average time between encounters that are successful in bringing about chemical reaction.
Ordinarily ttcollision « tchem; however, we will be concerned here with flows for which the first two of these times are very small compared to tchem and the macroscopic tdiff and tflow. (For a gaseous system with p ≈ 1 atm, T ≈ 1000 K, ordinarily tinteraction ≈ 0.5 ps and tcollision ≈ 0.2 ns (1 ps = 10–12 s).)
Once we suppress the granularity
of real matter by treating it as continuous on the scale of our interest, we simplify enormously the local description of the fluid. For example, rather than attempting to describe the instantaneous translational, rotational, and vibrational motion of each of the 7 x 10¹⁸ molecules in a typical cubic centimeter (!) of gas mixture (at 1 atm, 1000 K, say), we merely have to consider certain average properties per unit volume of the prevailing fluid mixture, such as:
ρ ≡ prevailing gas-mixture mass density
(mass per unit volume)
ρv ≡ prevailing gas-mixture linear momentum density