Introduction

In Chapter 9, the analyses were predicated on the existence of an effective barotropic relation for the homogeneous mixture. Indeed, the construction of the sonic speed in Sections 9.3.1 and 9.3.3 assumes that all the phases are in dynamic equilibrium at all times. For example, in the case of bubbles in liquids, it is assumed that the response of the bubbles to the change in pressure, δp, is an essentially instantaneous change in their volume. In practice this would be the case only if the typical frequencies experienced by the bubbles in the flow are very much smaller than the natural frequencies of the bubbles themselves (see Section 4.4.1). Under these circumstances the bubbles would behave quasistatically and the mixture would be barotropic. However, there are a number of important contexts in which the bubbles are not in equilibrium and in which the nonequilibrium effects have important consequences. One example is the response of a bubbly multiphase mixture to high-frequency excitation. Another is a bubbly cavitating flow where the nonequilibrium bubble dynamics lead to shock waves with substantial noise and damage potential.

In this chapter we therefore examine some flows in which the dynamics of the individual bubbles play an important role. These effects are included by incorporating the Rayleigh–Plesset equation (Rayleigh 1917, Knapp et al. 1970, Brennen 1995) into the global conservation equations for the multiphase flow. Consequently the mixture no longer behaves barotropically.

Introduction

One of the most common requirements of a multiphase flow analysis is the prediction of the energy gains and losses as the flow proceeds through the pipes, valves, pumps, and other components that make up an internal flow system. In this chapter we attempt to provide a few insights into the physical processes that influence these energy conversion processes in a multiphase flow. The literature contains a plethora of engineering correlations for pipe friction and some data for other components such as pumps. This chapter provides an overview and some references to illustrative material but does not pretend to survey these empirical methodologies.

As might be expected, frictional losses in straight uniform pipe flows have been the most widely studied of these energy conversion processes and so we begin with a discussion of that subject, focusing first on disperse or nearly disperse flows and then on separated flows. In the last part of the chapter, we consider multiphase flows in pumps, in part because of the ubiquity of these devices and in part because they provide a second example of the multiphase flow effects in internal flows.

Frictional Loss in Disperse Flow

Horizontal Flow

We begin with a discussion of disperse horizontal flow. There exists a substantial body of data relating to the frictional losses or pressure gradient, (−dp/ds), in a straight pipe of circular cross section (the coordinate s is measured along the axis of the pipe).

Introduction

Sprays are an important constituent of many natural and technological processes and range in scale from the very large dimensions of the global air/sea interaction and the dynamics of spillways and plunge pools to the smaller dimensions of fuel injection and ink-jet systems. In this chapter we first examine the processes by which sprays are formed and some of the resulting features of those sprays. Then, because the combustion of liquid fuels in droplet form constitute such an important component of our industrialized society, we focus on the evaporation and combustion of single droplets and follow that with an examination of the features involved in the combustion of sprays.

Types of Spray Formation

In general, sprays are formed when the interface between a liquid and a gas becomes deformed and droplets of liquid are generated. These then migrate out into the body of the gas. Sometimes the gas plays a negligible role in the kinematics and dynamics of the droplet formation process; this simplifies the analyses of the phenomena. In other circumstances the gas dynamic forces generated can play an important role. This tends to occur when the relative velocity between the gas and the liquid becomes large as is the case, for example, with hurricane-generated ocean spray.

Several prototypical flow geometries are characteristic of the natural and technological circumstances in which spray formation is important. The first prototypical geometry is the flow of a gas over a liquid surface.

Introduction

Scope

In the context of this book, the term multiphase flow is used to refer to any fluid flow consisting of more than one phase or component. For brevity and because they are covered in other texts, we exclude those circumstances in which the components are well mixed above the molecular level. Consequently, the flows considered here have some level of phase or component separation at a scale well above the molecular level. This still leaves an enormous spectrum of different multiphase flows. One could classify them according to the state of the different phases or components and therefore refer to gas/solids flows or liquid/solids flows or gas/particle flows or bubbly flows and so on; many texts exist that limit their attention in this way. Some treatises are defined in terms of a specific type of fluid flow and deal with low-Reynolds-number suspension flows, dusty gas dynamics, and so on. Others focus attention on a specific application such as slurry flows, cavitating flows, aerosols, debris flows, fluidized beds, and so on; again, there are many such texts. In this book we attempt to identify the basic fluid mechanical phenomena and to illustrate those phenomena with examples from a broad range of applications and types of flow.

Parenthetically, it is valuable to reflect on the diverse and ubiquitous challenges of multiphase flow. Virtually every processing technology must deal with multiphase flow, from cavitating pumps and turbines to electrophotographic processes to papermaking to the pellet form of almost all raw plastics.

Introduction

This chapter briefly reviews the issues and problems involved in constructing the equations of motion for individual particles, drops, or bubbles moving through a fluid. For convenience we use the generic name particle to refer to the finite pieces of the disperse phase or component. The analyses are implicitly confined to those circumstances in which the interactions between neighboring particles are negligible. In very dilute multiphase flows in which the particles are very small compared with the global dimensions of the flow and are very far apart compared with the particle size, it is often sufficient to solve for the velocity and pressure, ui (xi, t) and p(xi, t), of the continuous suspending fluid while ignoring the particles or disperse phase. Given this solution one could then solve an equation of motion for the particle to determine its trajectory. This chapter focuses on the construction of such a particle or bubble equation of motion.

The body of fluid mechanical literature on the subject of flows around particles or bodies is very large indeed. Here we present a summary that focuses on a spherical particle of radius R and employs the following common notation. The components of the translational velocity of the center of the particle is denoted by Vi(t). The velocity that the fluid would have had at the location of the particle center in the absence of the particle is denoted by Ui(t).

The genesis of solid friction

The laws of friction

Friction is the resistance encountered when one body moves tangentially over another with which it is in contact. The work expended against friction is often redundant, that is, it makes no useful contribution to the overall operation of the device of which the bodies are part, and ultimately must be dissipated as waste heat. Consequently, in most tribological designs our aim is to keep these frictional forces as small as possible. Of course there are exceptions to this general rule, occasions when sufficient friction is essential to continued progress and there are many practical devices which rely on the frictional transmission of power: automobile tyres on a roadway, vehicle brakes and clutches, as well as several of the variable-speed transmission systems now finding wider application. When two objects are to be held together, the only alternative to methods which rely on friction is the formation of some sort of chemical or metallurgical bond between them. The development of this sort of technique–adhesives and ‘superglues’, and even welding and brazing–are relatively recent; ‘traditional’ forms of fixing rely almost exclusively on friction. A nail hammered into a piece of wood is held in place by frictional effects along its length; if the frictional interaction were substantially reduced, the nail would be squeezed out. Similarly, the grip between a nut and a bolt depends on adequate friction between them.

Introduction

With a few important exceptions, engineering devices which involve the contact of loaded, sliding surfaces will only operate satisfactorily, that is, without giving rise to unacceptable amounts of surface damage or wear, when they are provided with adequate lubrication. The lubricant can act in two distinct, but not necessarily mutually exclusive, ways. The first of its functions may be to physically separate the surfaces by interposing between them a coherent, viscous film which is relatively thick (i.e. significantly larger than the size of likely surface asperities). In hydrostatic bearings this film is provided by an external pump and so its presence depends on the continuous operation of an external source of energy. In hydrodynamic bearings its generation relies only on the geometry and motion of the surfaces (hence the term dynamic) together with the viscous nature of the fluid. The second role of the lubricant may be to generate an additional thin, protective coating on one or both of the solid surfaces, preventing, or at least limiting, the formation of strong, adhesive and so potentially damaging friction junctions between the underlying solids at locations of particularly acute loading. If this protective coating has a comparatively low shear strength then the ultimate tangential force of friction can be much reduced: this mechanism of friction limitation is generally known as boundary lubrication. Such boundary films are generally very thin, perhaps only a few (albeit very large) molecules thick, and their formation and survival depends very much on the physical and chemical interactions between components of the lubricant and the solid surfaces.

Introduction

All real materials possess some microstructure which becomes apparent when a representative specimen is viewed with a sufficient degree of magnification. However, engineers usually work at a comparatively large scale and so use macroscopic models of material behaviour: these are likely to involve a very large number of the individual microstructural units which contribute to the overall material response. Engineering properties of the bulk represent average or integrated values, and when dealing with either solids or liquids the material is often treated as a homogeneous continuum. In the case of the linear elastic theory of solids what is required is Hooke's law and the observation that most metals undergo only a very small strain before yielding. Similarly, a Newtonian compressible fluid is described by the linear relationships between the applied shear stress and the resultant shear strain rate, and between density and the applied hydrostatic pressure. Many liquid lubricants, including hydrocarbon mineral oils, are effectively incompressible at moderate pressures so that their density remains constant, and this simplifies the analysis still further.

There is no fundamental reason why the lubricant in a hydrodynamic bearing should not be a vapour or a gas (rather than a liquid such as oil or water); indeed considered as a potential lubricant, air has a number of advantages–such as cleanliness and ease of supply–and it has been used in aerostatic and aerodynamic bearings for several decades.

The nature of engineering surfaces

No real engineering surface, no matter how carefully, or indeed expensively, prepared can possess perfect geometry. As well as errors in the form or shape of the component there will always be a roughness on the surface which is apparent when this is examined at a sufficiently high magnification. When two such surfaces are loaded together it is the tips of the surface roughnesses or asperities that must first carry the applied load: the geometry of individual contact spots and the way in which these islands of real contact are distributed throughout the nominal or apparent contact area is clearly of interest to tribologists in attempting to predict the overall performance, or likely life history, of the contact.

The geometric texture of an engineering surface reflects both its production route and the nature of the underlying material. It is possible to produce a truly smooth surface (for example, cleaving specimens of mica can produce a surface with roughness only on the atomic scale) and if two such surfaces are loaded together real and apparent areas are very nearly equal. The asperities on the surface of a very compliant surface, such as a soft rubber, may, if sufficiently small, be squeezed flat by quite modest contact loads, and in this way there can again be equality between real and apparent areas of contact. However, these are special cases; in general, useful metal surfaces exhibit a range of surface fluctuations which, although large compared to molecular dimensions, are small compared to the dimensions of most engineering components.

The term tribology is scarcely twenty-five years old and yet there can be few university or college courses in mechanical engineering which do not now include material under this heading. Of course, the problem of producing bearings, slides, seals, and other tribological systems to give smooth machine running and long component lives is one which has faced practitioners for generations, and consideration of their design has always played a part in the education of mechanical engineers. What has become increasingly obvious in recent years is the inherently interdisciplinary nature of the tribologist's task; as well as involving practising and academic engineers, advances in the subject have drawn upon the ingenuity and expertise of physicists, chemists, metallurgists, and material scientists. Consequently, although envisaged principally for use by final year undergraduates and post-graduate students in mechanical engineering, I hope that this volume may be of interest to students and specialists in these other related areas. Tribology is still very much an area of active research and the published literature in the fields of lubrication, friction and wear–already dauntingly voluminous–continues to grow at an alarming rate. I have made no attempt to produce a research monograph but rather to provide a framework of fundamental analytical tools which can be used in a wide variety of different physical situations. Each chapter is concluded with a short list of suggestions for further reading which provide access to the more specialised literature.