Prologue

Particles entrained in a turbulent fluid are dispersed by velocity fluctuations; they assume a motion that is related to the fluid turbulence. If the suspension flows through a conduit, deposition on a wall depends on the particle turbulence. An understanding of these processes is needed to describe the annular flow regime for which liquid flows along the walls and as drops in the gas flow. The fraction of liquid that is entrained by the gas depends on the rate at which the film is atomized and the rate at which drops deposit on the film.

Equations for trajectories of spherical drops and bubbles in a turbulent flow field are developed. These are used to relate the turbulence properties and the dispersion of particles to the turbulence properties of the fluid in which they are entrained. Of particular interest is the development of relations for the influence of drop size on drop turbulence and on drop dispersion.

Prologue

A central issue to be addressed in analyzing the behavior of bubbles in a gas–liquid flow is understanding the free-fall velocity of a spherical solid particle and the rise velocity of a spherical bubble in an infinite stationary fluid. Analytical solutions for these systems are available for very low particle Reynolds numbers (Stokes law and the Hadamard equation). A derivation of Stokes law is presented in the first part of this chapter.

Experiments show that Stokes law is valid for particle Reynolds numbers less than unity. For larger ReP, empirical correlations of the drag coefficient are used. The description of the rise velocity of bubbles is complicated by possible contamination of the interface. Measurements of the rise velocity of single bubbles are usually presented as plots of US versus the bubble size for a given system. The structure of these plots reflects changes in the shape and behavior of the bubbles. Very large bubbles take the shape of a cap. A prediction of the rise velocity of these cap bubbles, developed by Batchelor, is presented in Section 8.7.

Need for a phenomenological understanding

The one-dimensional analysis and the correlations for frictional pressure drop and void fraction (presented in Chapter 1) have been widely used as a starting point for engineering designs. However, these correlations have the handicap that the structure of the phase boundaries is ignored. As a consequence, they often give results which are only a rough approximation and overlook phenomena which could be of first-order importance in understanding the behavior of a system.

It is now recognized that the central issue in developing a scientific approach to gas–liquid flows is the understanding of how the phases are distributed and of how the behavior of a multiphase system is related to this structure (Hanratty et al., 2003). Of particular interest is the finding that macroscopic behavior is dependent on small-scale interactions. An example of this dependence is that the presence of small amounts of high molecular weight polymers can change an annular flow into a stratified flow by damping interfacial waves (Al-Sarkhi & Hanratty, 2001a).

Prologue

Chapter 2 gives considerable attention to slug flow because of its central role in understanding the configuration of the phases in horizontal and inclined pipes. Several criteria have been identified to define the boundaries of this regime: (1) viscous large-wavelength instability of a stratified flow; (2) Kelvin–Helmholtz instability of a stratified flow; (3) stability of a slug; (4) coalescence of large-amplitude waves. Bontozoglou & Hanratty (1990) suggested that a sub-critical non-linear Kelvin–Helmholtz instability could be an effective mechanism in pipes with very large diameters, but this analysis has not been tested. A consideration of the stability of a slug emerges as being particularly important. It explains the initiation of slugs for very viscous liquids, for high-density gases, for gas velocities where wave coalescence is important and for the evolution of pseudo-slugs into slugs. Chapter 2 (Section 2.2.5) outlines an analysis of slug stability which points out the importance of understanding the rate at which slugs shed liquid. Section 9.2 continues this discussion by developing a relation for Qsh and for the critical height of the liquid layer needed to support a stable slug. Section 9.3 develops a tentative model for horizontal slug flow. Section 9.4 considers the frequency of slugging.

Necessary conditions for the existence of slugs

Figure 9.1 presents simplified sketches of the front and the tail of a slug in a pipeline. The front has a velocity cF; the back has a velocity cB. The stratified liquid layer in front of the slug has a velocity and area designated by uL1, AL1. The mean velocity of the liquid in the slug is uL3. The slug is usually aerated; the mean volume fraction of gas in the slug is designated by α. The gas at station 1 is moving from left to right at a velocity uG1. The assumption is made that the velocity fields can be approximated as being uniform.

Preface

Gas–liquid flows are ubiquitous in industrial and environmental processes. Examples are the transportation of petroleum products, the cooling of nuclear reactors, the operation of absorbers, distillation columns, gas lift pumps. Quite often corrosion and process safety depend on the configuration of the phases. Thus, the interest in this area should not be surprising.

The goal of this book is to give an account of scientific tools needed to understand the behavior of gas–liquid systems and to read the scientific literature. Particular emphasis is given to flow in pipelines.

The following brief historical account is taken from a plenary lecture by the author at the Third International Conference on Multiphase Flow, Lyon, France, June 8–12, 1998. (Int. J. Multiphase Flow 26, 169–190, 2000):

Introduction

The “simplest” models for gas–liquid flow systems are ones for which the velocity is uniform over a cross-section and unidirectional. This includes flows in a long straight pipe and steady flows in a nozzle.

A treatment of pipe flow with a constant cross-section is initiated by reviewing analyses of incompressible and compressible single-phase flows. A simple way to use these results is to describe gas–liquid flows with a homogeneous model that assumes the phases are uniformly distributed, that there is no slip between the phases and that the phases are in thermodynamic equilibrium. The volume fraction of the gas, α, is then directly related to the relative mass flows of the phases. However, the assumption of no slip, S = 1, can introduce considerable error. This has prompted a consideration of a separated flow model, where uniform flows of gas and liquid are pictured as moving parallel to one another with different velocities and to be in thermodynamic equilibrium.

This book is intended as a combination of a reference book for those who work with cavitation or bubble dynamics and as a monograph for advanced students interested in some of the basic problems associated with this category of multiphase flows. A book like this has many roots. It began many years ago when, as a young postdoctoral Fellow at the California Institute of Technology, I was asked to prepare a series of lectures on cavitation for a graduate course cum seminar series. It was truly a baptism by fire, for the audience included three of the great names in cavitation research, Milton Plesset, Allan Acosta, and Theodore Wu, none of whom readily accepted superficial explanations. For that, I am immensely grateful. The course and I survived, and it evolved into one part of a graduate program in multiphase flows.

There are many people to whom I owe a debt of gratitude for the roles they played in making this book possible. It was my great good fortune to have known and studied with six outstanding scholars, Les Woods, George Gadd, Milton Plesset, Allan Acosta, Ted Wu, and Rolf Sabersky. I benefited immensely from their scholarship and their friendship. I also owe much to my many colleagues in the American Society of Mechanical Engineers whose insights fill many of the pages of this monograph. The support of my research program by the Office of Naval Research is also greatly appreciated.

Introduction

When the concentration of bubbles in a flow exceeds some small value the bubbles will begin to have a substantial effect on the fluid dynamics of the suspending liquid. Analyses of the dynamics of this multiphase mixture then become significantly more complicated and important new phenomena may be manifest. In this chapter we discuss some of the analyses and phenomena that may occur in bubbly multiphase flow.

In the larger context of practical multiphase (or multicomponent) flows one finds a wide range of homogeneities, from those consisting of one phase (or component) that is very finely dispersed within the other phase (or component) to those that consist of two separate streams of the two phases (or components). In between are topologies that are less readily defined. The two asymptotic states are conveniently referred to as homogeneous and separated flow. One of the consequences of the topology is the extent to which relative motion between the phases can occur. It is clear that two different streams can readily travel at different velocities, and indeed such relative motion is an implicit part of the study of separated flows. On the other hand, it is clear from the results of Section 5.11 that any two phases could, in theory, be sufficiently well mixed and the disperse particle size sufficiently small so as to eliminate any significant relative motion.

Introduction

This first chapter will focus on the mechanisms of formation of two-phase mixtures of vapor and liquid. Particular attention will be given to the process of the creation of vapor bubbles in a liquid. In doing so we will attempt to meld together several overlapping areas of research activity. First, there are the studies of the fundamental physics of nucleation as epitomized by the books of Frenkel (1955) and Skripov (1974). These deal largely with very pure liquids and clean environments in order to isolate the behavior of pure liquids. On the other hand, most engineering systems are impure or contaminated in ways that have important effects on the process of nucleation. The later part of the chapter will deal with the physics of nucleation in such engineering environments. This engineering knowledge tends to be divided into two somewhat separate fields of interest, cavitation and boiling. A rough but useful way of distinguishing these two processes is to define cavitation as the process of nucleation in a liquid when the pressure falls below the vapor pressure, while boiling is the process of nucleation that ocurs when the temperature is raised above the saturated vapor/liquid temperature. Of course, from a basic physical point of view, there is little difference between the two processes, and we shall attempt to review the two processes of nucleation simultaneously.

Introduction

In the preceding chapter some of the equations of bubble dynamics were developed and applied to problems of bubble growth. In this chapter we continue the discussion of bubble dynamics but switch attention to the dynamics of collapse and, in particular, consider the consequences of the violent collapse of vapor-filled cavitation bubbles.

Bubble Collapse

Bubble collapse is a particularly important subject because of the noise and material damage that can be caused by the high velocities, pressures, and temperatures that may result from that collapse. The analysis of Section 2.4 allowed approximate evaluation of the magnitudes of those velocities, pressures, and temperatures (Equations (2.36), (2.38), (2.39)) under a number of assumptions including that the bubble remains spherical. It will be shown in Section 3.5 that collapsing bubbles do not remain spherical. Moreover, as we shall see in Chapter 7, bubbles that occur in a cavitating flow are often far from spherical. However, it is often argued that the spherical analysis represents the maximum possible consequences of bubble collapse in terms of the pressure, temperature, noise, or damage potential. Departure from sphericity can diffuse the focus of the collapse and reduce the maximum pressures and temperatures that might result.