We can only obtain analytic solutions to the equations describing nonlinear waves for a very limited set of geometries, constitutive equations, and boundary conditions. For example, the Riemann integrals yield a solution for a one-dimensional simple wave in an infinite volume of nonlinear-elastic material; however, obtaining analytic solutions for the interaction of two simple waves ranges from difficult to impossible depending upon the constitutive equations of the medium. Without the advent of the digital computer and sophisticated methods of numerical analysis, advances in the field of nonlinear wave motion and the development of constitutive theories for dynamic loading of materials would have stagnated over the past 40 years.

The study of waves is important to virtually every branch of science and engineering. Indeed, waves are also important to everyday life. Sound waves allow us to hear, and electromagnetic waves allow us to see. In this book, we restrict our study to one important class of waves often called stress waves. These waves propagate through gases, liquids, and solids. Stress waves in gases and liquids are usually called pressure waves. This nomenclature is derived from the fact that internal forces in solid bodies are represented by a stress tensor, and internal forces in inviscid liquids and gases are represented by a special form of stress called pressure. In Chapter 1, we shall learn the distinction between stress and pressure.

Sound is the most commonly experienced pressure wave. Indeed, anyone standing in a thunderstorm knows that lightening is seen before the report of its thunder is heard. This simple observation reveals the single, most important feature of any wave – its finite velocity. We know there is always a lapse of time between the cause of a sound wave and when we hear it.

Elastic Material

It is a common observation that individual materials react differently to applied loads.

We often calculate nonlinear wave phenomena with complex numerical computer codes. With such powerful capabilities you may be tempted to simply characterize the energy state of a material with large tables of data that yield values of equilibrium stress, temperature, stiffnesses, specifics heats, and thermal expansion coefficients for given values of F and η. Accurate computations of some very common materials rely heavily upon this method. The computation of nonlinear pressure waves in water is an important example. However, even with this capability, questions arise about the constitutive descriptions of these materials. For example, if water is heated by a shock wave, it often spontaneously “flashes” to steam when the pressure drops again in response to a release wave.

Index Notation

We begin by introducing a compact and convenient way to describe quantities in three spatial dimensions with a Cartesian coordinate system. We also show how to transform quantities between different Cartesian coordinate systems.

Cartesian Coordinates and Vectors

A Cartesian coordinate system has three straight coordinates that are mutually perpendicular to each other (see Figure 1.1). The coordinates are named x1, x2, and x3. The subscripts 1, 2, and 3 are called indices. When we let k represent any of these indices, we can refer to the coordinates by the compact notation xk. Each coordinate xk also has a shaded arrow that represents a unit coordinate vectorik. We use a boldface symbol to denote a vector. The length of each ik is equal to unity.

Now consider another vector v that is drawn with arbitrary angles to the three coordinates. We draw a box with sides parallel to ik and with v as a diagonal.

Why must we study thermodynamics? In Chapter 2 we studied a wide range of wave phenomena using the mechanical concepts of force and motion. Using only the constitutive model for an elastic material, we developed detailed analyses of the nonlinear structured wave, the transition of a compression wave into a shock wave, and the decay of a shock wave as it is overtaken by a rarefaction wave. We have analyzed wave motion in solids, liquids, and adiabatic gases.

During July 1994 while I was writing this book, fragments of the comet Shoemaker–Levy 9 crashed into Jupiter. As each fragment entered the Jovian atmosphere, it formed a shock wave, similar to the bow wave in front of a boat. Heat generated within these waves ignited the atmosphere of Jupiter creating fireballs the size of the earth.

Introduction

The flow characteristics of solid particles in a gas–solid suspension vary significantly with the geometric and material properties of the particle. The geometric properties of particles include their size, size distribution, and shape. Particles in a gas–solid flow of practical interest are usually of nonspherical or irregular shapes and polydispersed sizes. The geometric properties of particles affect the particle flow behavior through an interaction with the gas medium as exhibited by the drag force, the distribution of the boundary layer on the particle surface, and the generation and dissipation of wake vortices. The material properties of particles include such characteristics as physical adsorption, elastic and plastic deformation, ductile and brittle fracturing, solid electrification, magnetization, heat conduction and thermal radiation, and optical transmission. The material properties affect the long– and short–range interparticle forces, and particle attrition and erosion behavior in gas–solid flows. The geometric and material properties of particles also represent the basic parameters affecting the flow regimes in gas–solid systems such as fluidized beds.

In this chapter, the basic definitions of the equivalent diameter for an individual particle of irregular shape and its corresponding particle sizing techniques are presented. Typical density functions characterizing the particle size distribution for polydispersed particle systems are introduced. Several formulae expressing the particle size averaging methods are given. Basic characteristics of various material properties are illustrated.

Particle Size and Sizing Methods

The particle size affects the dynamic behavior of a gas–solid flow [Dallavalle, 1948]. An illustration of the relative magnitudes of particle sizes in various multiphase systems is given in Fig. 1.1 [Soo, 1990].

Introduction

Developing a mathematical model for general description of a particulate multiphase flow system has long been a challenging issue for researchers in the field. Although at this stage comprehensive constitutive relationships for modeling have not been fully established, several approaches which have proved useful are available for predicting the gas–solid flow behavior, particularly for engineering application purposes. Among them are the Eulerian continuum approach, Lagrangian trajectory approach, kinetic theory modeling for interparticle collisions, and Darcy's law and the Ergun equation for flows through packed beds. These approaches are the most commonly used and are introduced in this chapter.

The Eulerian continuum approach, based on a continuum assumption of phases, provides a “field” description of the dynamics of each phase. The Lagrangian trajectory approach, from the study of motions of individual particles, is able to yield historical trajectories of the particles. The kinetic theory modeling for interparticle collisions, extended from the kinetic theory of gases, can be applied to dense suspension systems where the transport in the particle phase is dominated by interparticle collisions. The Ergun equation provides important flow relationships, which are useful not only for packed bed systems, but also for some situations in fluidized bed systems.

Introduction

As discussed in Chapter 9, dense-phase fluidization other than particulate fluidization is characterized by the presence of an emulsion phase and a discrete gas bubble/void phase. At relatively low gas velocities in dense-phase fluidization, the upper surface of the bed is distinguishable. As the gas velocity increases, the bubble/void phase gradually becomes indistinguishable from the emulsion phase. The bubble/void phase eventually disappears and the gas evolves into the continuous phase with further increasing gas velocities. In a dense-phase fluidized bed, the particle entrainment rate is low and increases with increasing gas velocity. As the gas flow rate increases beyond the point corresponding to the disappearance of the bubble/void phase, a drastic increase in the entrainment rate of the particles occurs such that a continuous feeding of particles into the fluidized bed is required to maintain a steady solids flow. Fluidization at this state, in contrast to dense-phase fluidization, is denoted lean-phase fluidization.

Lean-phase fluidization encompasses two flow regimes, i.e., the fast fluidization regime and the dilute transport regime. The fast fluidization regime is characterized by a heterogeneous flow structure, whereas the dilute transport regime is characterized by a homogeneous flow structure. The circulating fluidized bed (CFB) is a fluidized bed system in which solid particles circulate between the riser and the downcomer. The fast fluidization regime is the principal regime under which the circulating fluidized bed is operated. The operating variables for a circulating fluidized bed system include both the gas flow rate and the solids circulation rate, in contrast to the gas flow rate only in a dense–phase fluidized bed system. The solids circulation is established by a high velocity of gas flow. Note that the dilute transport regime is discussed in Chapter 11, in the context of pneumatic conveying.

Gas–solid flows are involved in numerous industrial processes and occur in various natural phenomena. For example, in solid fuel combustion, gas–solid flows are involved in pulverized coal combustion, solid waste incineration, and rocket propellant combustion. Gas–solid flows are encountered in pneumatic conveying of particulates commonly used in pharmaceutical, food, coal, and mineral powder processing. Fluidization is a common gas–solid flow operation with numerous important applications such as catalytic cracking of intermediate hydrocarbons, and Fischer–Tropsch synthesis for chemicals and liquid fuel production. Gas–solid flows occur in gas–particle separations, as exemplified by cyclones, electrostatic precipitators, gravity settling, and filtration operations. Fine powder–gas flows are closely associated with material processing, as in chemical vapor deposition for ceramics and silicon production, plasma coating, and xerography. In heat transfer applications, gas–solid flows are involved in nuclear reactor cooling and solar energy transport using graphite suspension flows. Solid dispersion flows are common in pigment sprays, dust explosions and settlement, and nozzle flows. The natural phenomena accompanied by gas–solid flows are typified by sand storms, moving sand dunes, aerodynamic ablation, and cosmic dusts. The optimum design of the industrial processes and accurate account of the natural phenomena that involve gas–solid flows as exemplified previously require a thorough knowledge of the principles governing these flows.

This book is intended to address basic principles and fundamental phenomena associated with gas–solid flows, as well as characteristics of selected gas–solid flow systems. It covers the typical range of particle sizes of interest to gas–solid flows, i.e., 1 μm–10 cm, recognizing that flow characteristics for submicrometer particles are also of great industrial importance.

Introduction

There are inherent phenomena associated with gas–solid flows. Such phenomena are of considerable interest to a wide spectrum of process applications involving gas–solid flows and can be exemplified by erosion, attrition, pressure-wave propagation, flow instability, and gas–solid turbulence modulation.

Mechanical erosion may cause severe damage to a pipe wall or any moving part in a gas–solid system. Particle attrition yields fine particles, which may alter the flow conditions of the system and may become a source of particulate emission. Understanding the basic modes of mechanical wear and the mechanism of attrition is essential to the control of their behavior. Pressure waves through a gas–solid suspension are directly related to nozzle flows (such as in jet combustion) and measurement techniques associated with acoustic waves and shock waves. Understanding the propagation of the speed of sound and normal shock waves in a dilute gas–solid suspension is, therefore, important. Flow instability such as wavelike motion represents an intrinsic feature of a gas–solid flow. An instability analysis can be made using the pseudocontinuum approach. For most gas–solid flows, particles move in a strong turbulent gas stream. The gas turbulence can be significantly modulated by the solid particles. Thus, understanding the particle-turbulence interactive phenomenon is necessary.

This chapter addresses the various phenomena indicated. In addition, the thermodynamic laws governing physical properties of the gas–solid mixture such as density, pressure, internal energy, and specific heat are introduced.