The subject of multiphase flows encompasses a vast field, a host of different technological contexts, a wide spectrum of different scales, a broad range of engineering disciplines, and a multitude of different analytical approaches. Not surprisingly, the number of books dealing with the subject is voluminous. For the student or researcher in the field of multiphase flow this broad spectrum presents a problem for the experimental or analytical methodologies that might be appropriate for his/her interests can be widely scattered and difficult to find. The aim of the present text is to try to bring much of this fundamental understanding together into one book and to present a unifying approach to the fundamental ideas of multiphase flows. Consequently the book summarizes those fundamental concepts with relevance to a broad spectrum of multiphase flows. It does not pretend to present a comprehensive review of the details of any one multiphase flow or technological context, although reference to books providing such reviews is included where appropriate. This book is targeted at graduate students and researchers at the cutting edge of investigations into the fundamental nature of multiphase flows; it is intended as a reference book for the basic methods used in the treatment of multiphase flows.

I am deeply grateful to all my many friends and fellow researchers in the field of multiphase flows whose ideas fill these pages. I am particularly indebted to my close colleagues Allan Acosta, Ted Wu, Rolf Sabersky, Melany Hunt, Tim Colonius, and the late Milton Plesset, all of whom made my professional life a real pleasure.

Introduction

From a practical engineering point of view one of the major design difficulties in dealing with multiphase flow is that the mass, momentum, and energy transfer rates and processes can be quite sensitive to the geometric distribution or topology of the components within the flow. For example, the geometry may strongly effect the interfacial area available for mass, momentum, or energy exchange between the phases. Moreover, the flow within each phase or component will clearly depend on that geometric distribution. Thus we recognize that there is a complicated two-way coupling between the flow in each of the phases or components and the geometry of the flow (as well as the rates of change of that geometry). The complexity of this two-way coupling presents a major challenge in the study of multiphase flows and there is much that remains to be done before even a superficial understanding is achieved.

An appropriate starting point is a phenomenological description of the geometric distributions or flow patterns that are observed in common multiphase flows. This chapter describes the flow patterns observed in horizontal and vertical pipes and identifies a number of the instabilities that lead to transition from one flow pattern to another.

Topologies of Multiphase Flow

Multiphase Flow Patterns

A particular type of geometric distribution of the components is called a flow pattern or flow regime and many of the names given to these flow patterns (such as annular flow or bubbly flow) are now quite standard.

Introduction

One of the characteristics of multiphase flows with which the engineer has to contend is that they often manifest instabilities that have no equivalent in single-phase flow (see, for example, Boure et al. 1973, Ishii 1982, Gouesbet and Berlemont 1993). Often the result is the occurence of large pressure, flow-rate, or volume-fraction oscillations that, at best, disrupt the expected behavior of the multiphase flow system (and thus decrease the reliability and life of the components, Makay and Szamody 1978) and, at worst, can lead to serious flow stoppage or structural failure (see, for example, NASA 1970, Wade 1974). Moreover, in many systems (such as pump and turbine installations) the trend toward higher rotational speeds and higher power densities increases the severity of the problem because higher flow velocities increase the potential for fluid/structure interaction problems. This chapter focuses on internal flow systems and the multiphase flow instabilities that occur in them.

System Structure

In the discussion and analysis of system stability, we consider that the system has been divided into its components, each identified by its index, k, as shown in Figure 15.1 where each component is represented by a box. The connecting lines do not depict lengths of pipe that are themselves components. Rather, the lines simply show how the components are connected. More specifically they represent specific locations at which the system has been divided up; these points are called the nodes of the system and are denoted by the index, i.

Introduction

The one-dimensional theory of sedimentation was introduced in a classic article by Kynch (1952), and the methods he used have since been expanded to cover a wide range of other multiphase flows. In Chapter 14 we introduced the concept of drift flux models and showed how these can be used to analyze and understand a class of steady flows in which the relative motion between the phases is determined by external forces and the component properties. The present chapter introduces the use of the drift flux method to analyze the formation, propagation, and stability of concentration (or kinematic) waves. For a survey of this material, the reader may wish to consult Wallis (1969).

The general concept of a kinematic wave was first introduced by Lighthill and Whitham (1955) and the reader is referred to Whitham (1974) for a rigorous treatment of the subject. Generically, kinematic waves occur when a functional relation connects the fluid density with the flux of some physically conserved quantity such as mass. In the present context a kinematic (or concentration) wave is a gradient or discontinuity in the volume fraction, α. We refer to such gradients or discontinuities as local structure in the flow; only multiphase flows with a constant and uniform volume fraction are devoid of such structure. Of course, in the absence of any relative motion between the phases or components, the structure is simply convected at the common velocity in the mixture.

Introduction

This chapter addresses the class of compressible flows in which a gaseous continuous phase is seeded with droplets or particles and in which it is necessary to evaluate the relative motion between the disperse and continuous phases for a variety of possible reasons. In many such flows, the motivation is the erosion of the flow boundaries by particles or drops and this is directly related to the relative motion. In other cases, the purpose is to evaluate the change in the performance of the system or device. Still another motivation is the desire to evaluate changes in the instability boundaries caused by the presence of the disperse phase.

Examples include the potential for serious damage to steam turbine blades by impacting water droplets (e.g., Gardner 1963, Smith et al. 1967). In the context of aircraft engines, desert sand storms or clouds of volcanic dust can not only cause serious erosion to the gas turbine compressor (Tabakoff and Hussein 1971, Smialek et al. 1994, Dunn et al. 1996, Tabakoff and Hamed 1986) but can also deleteriously effect the stall margin and cause engine shutdown (Batcho et al. 1987). Other examples include the consequences of seeding the fuel of a solid-propelled rocket with metal particles to enhance its performance. This is a particularly complicated example because the particles may also melt and oxidize in the flow (Shorr and Zaehringer 1967).

In recent years considerable advancements have been made in the numerical models and methods available for the solution of dilute particle-laden flows.

Introduction

Unlike solid particles or liquid droplets, gas/vapor bubbles can grow or collapse in a flow and in doing so manifest a host of phenomena with technological importance. We devote this chapter to the fundamental dynamics of a growing or collapsing bubble in an infinite domain of liquid that is at rest far from the bubble. Although the assumption of spherical symmetry is violated in several important processes, it is necessary to first develop this baseline. The dynamics of clouds of bubbles or of bubbly flows are treated in later chapters.

Bubble Growth and Collapse

Rayleigh–Plesset Equation

Consider a spherical bubble of radius, R(t) (where t is time), in an infinite domain of liquid whose temperature and pressure far from the bubble are T∞ and p∞(t) respectively. The temperature, T∞, is assumed to be a simple constant because temperature gradients are not considered. Conversely, the pressure, p∞(t), is assumed to be a known (and perhaps controlled) input that regulates the growth or collapse of the bubble.

Though compressibility of the liquid can be important in the context of bubble collapse, it will, for the present, be assumed that the liquid density, ρL, is a constant. Furthermore, the dynamic viscosity, µL, is assumed constant and uniform. It will also be assumed that the contents of the bubble are homogeneous and that the temperature, TB(t), and pressure, pB(t), within the bubble are always uniform. These assumptions may not be justified in circumstances that will be identified as the analysis proceeds.

Introduction

Dense fluid-particle flows in which the direct particle/particle interactions are a dominant feature encompass a diverse range of industrial and geophysical contexts (Jaeger et al. 1996), including, for example, slurry pipelines (Shook and Roco 1991), fluidized beds (Davidson and Harrison 1971), mining and milling operations, ploughing (Weighardt 1975), abrasive water jet machining, food processing, debris flows (Iverson 1997), avalanches (Hutter 1993), landslides, sediment transport, and earthquake-induced soil liquefaction. In many of these applications, stress is transmitted both by shear stresses in the fluid and by momentum exchange during direct particle/particle interactions. Many of the other chapters in this book analyze flow in which the particle concentration is sufficiently low that the particle-particle momentum exchange is negligible.

In this chapter we address those circumstances, usually at high particle concentrations, in which the direct particle/particle interactions play an important role in determining the flow properties. When those interactions dominate the mechanics, the motions are called granular flows and the flow patterns can be quite different from those of conventional fluids. An example is included as Figure 13.1, which shows the downward flow of sand around a circular cylinder. Note the upstream wake of stagnant material in front of the cylinder and the empty cavity behind it.

Within the domain of granular flows, there are, as we shall see, several very different types of flow distinguished by the fraction of time for which particles are in contact. For most slow flows, the particles are in contact most of the time.

Introduction

In Chapter 2 it was assumed that the particles were rigid and therefore not deformed, fissioned, or otherwise modified by the flow. However, there are many instances in which the particles that comprise the disperse phase are radically modified by the forces imposed by the continuous phase. Sometimes those modifications are radical enough to, in turn, affect the flow of the continuous phase. For example, the shear rates in the continuous phase may be sufficient to cause fission of the particles and this, in turn, may reduce the relative motion and therefore alter the global extent of phase separation in the flow.

The purpose of this chapter is to identify additional phenomena and issues that arise when the translating disperse phase consists of deformable particles, namely bubbles, droplets, or fissionable solid grains.

Deformation due to Translation

Dimensional Analysis

Because the fluid stresses due to translation may deform the bubbles, drops, or deformable solid particles that make up the disperse phase, we should consider not only the parameters governing the deformation but also the consequences in terms of the translation velocity and the shape. We concentrate here on bubbles and drops in which surface tension, S, acts as the force restraining deformation. However, the reader will realize that there would exist a similar analysis for deformable elastic particles. Furthermore, the discussion is limited to the case of steady translation, caused by gravity, g. Clearly the results could be extended to cover translation due to fluid acceleration by using an effective value of g as indicated in Section 2.4.2.