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Transport Processes in Chemically Reacting Flow Systems
By Daniel E. Rosner Dover Publications, Inc.
Copyright © 2000 Daniel E. Rosner
All rights reserved.
ISBN: 978-0-486-15063-5
CHAPTER 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:
a. chemists, physicists, and applied mathematicians with little or no previous experience in the area of transport processes, and/or
b. 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" 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:
a. 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.
b. 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).
c. 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.
d. 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:
1. 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.
2. 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:
a. An increase in the fuel jet velocity does not appreciably lengthen the flame!
b. 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.).
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):
a. The time to completely consume a fuel droplet depends quadratically on the initial droplet diameter;
b. 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.
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 [equivalent to] fuel jet radius,
rf [equivalent to] radial location of the flame zone,
Lf [equivalent to] total flame length,
Lc [equivalent to] 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 [equivalent to] the characteristic diameter of a single molecule,
n-1/3 [equivalent to] the average distance between molecules (where n [equivalent to] number density),
l [equivalent to] the average distance traveled by a single molecule before it encounters another molecule (the "mean-free-path").
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 [congruent to] 1000 K we have, approximately, σ ≈ 4 x 10-1 nm, n1/3 ≈ 5 nm, and ≈ 200 nm (where 1 nm [equivalent to] 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 [equivalent to] time for a representative fluid parcel to traverse the combustor length (≈ Lc/ U, where U is the characteristic axial air velocity);
tdiff [equivalent to] 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 ([equivalent to] 10-3 seconds). In contrast, consider
tinteraction [equivalent to] characteristic time during which a molecule interacts with a collision partner during a single binary encounter,
tcollision [equivalent to] average time elapsed before a molecule encounters another molecule,
tchem [equivalent to] average time between encounters that are successful in bringing about chemical reaction.
(Continues...)
Excerpted from Transport Processes in Chemically Reacting Flow Systems by Daniel E. Rosner. Copyright © 2000 Daniel E. Rosner. Excerpted by permission of Dover Publications, Inc..
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