The mass- and momentum-balance techniques described in Chapters 3–5 are general and apply to any control volume (CV). We apply those techniques to a general microscopic control volume in Chapter 6 and use the microscopic balances in Chapters 7 and 8. Microscopic-control-volume calculations yield the equations that govern three-dimensional velocity and stress fields. If the equations can be solved, the information that microscopic balances provide is complete. Solving the microscopic balances is difficult, however, because the continuity equation and the Navier-Stokes equation are a set of four nonlinear, coupled, partial differential equations (PDEs).

For many fluids problems, the information sought is relatively large scale and flow details are not very important. For these problems—such as the calculation of the total force on a wall; overall flow rate in a device; and the total work performed by a pump, a turbine, or a mixer-balancing on a larger CV can be a fast and simple way to arrive at quantities of interest. Macroscopic CV balances are mathematically easier to calculate than microscopic CV balances, although they generally require information that must be determined experimentally.

In this chapter, we derive and learn to use the macroscopic mass, momentum, and energy balances, including the mechanical energy balance (MEB), which is discussed in Chapter 1.

Chapter 2 describes fluid behaviors, and Chapters 1 and 2 introduce basic fluids calculations. We turn now to developing a modeling method that allows us to understand fluids behavior in detail.

Fluids move and deform in predictable ways that are governed by the laws of physics. To apply the laws of physics to fluids, we must develop a mathematical picture or model of fluid motion. With an effective model, we can predict fluid patterns and stresses and apply these predictions to engineering calculations.

To build up the fluid model that we use, we begin with a reminder about how to calculate the motions of individual rigid bodies. To apply these methods to fluids, we then introduce the continuum model, a mathematical picture of fluids in which we consider small packets of fluid to be individual bodies. We discuss how we apply the laws of physics to these small fluid packets or particles to deduce velocities and forces for the fluid particles. Finally, we introduce the control volume, a point of view used for fluid modeling that focuses our calculations on a physical region in space rather than on individual bodies in motion. This difference in strategy—that is, considering a control volume rather than individual bodies—is a key difference between the modeling techniques of fluid mechanics and those of solid-body mechanics.

In Chapter 7, we applied analysis methods to flows inside pipes and other closed conduits. We started with a practical challenge of estimating the extent of a home flood and developed our solution method by thinking about that problem from various angles (Figure 8.1). We first decided on the goal of our analysis; then, star ting with the simplest models, we systematically investigated flows of increasing complexity until we found a solution to the burst-pipe problem through dimensional analysis and data correlations. This protocol is general, and it can be applied to other flows, as demonstrated in this chapter.

We turn now to external flows. External flow is a term used to describe flows over or around obstacles. The wind blowing on a skyscraper is an example of an external flow (see Example 2.5), as is an electric fan cooling a printed circuit board in a computer or a cleaning jet directed past the fender of a freshly painted automobile. Objects moving through fluids also create external flows (see Figure 2.11). Ships on the ocean, mixing blades in viscous liquids, and skydivers (Figure 8.2) are all operating in external flows. External flows are not unidirectional, steady flows; thus, both inertia and viscosity affect flow behavior.

In this text, our goal is to explain flow; this chapter surveys how far we have come. With the completion of nine chapters of study, we find that we can make sense of much of the fluid behavior we observe. A reexamination of those behaviors helps consolidate our knowledge.

Section 10.1 is an integrated summary of the concepts of viscosity, drag, and boundary layers. Section 10.2 provides guidance on how numerical tools are used to pursue advanced flow-field models. In Section 10.3, we turn to turbulent flow, which until now was addressed only through data correlations for friction factor and drag coefficient. Sophisticated applications involving turbulent flow (e.g., airplane flight, mixing, and reactor design) require more detailed understanding of turbulent flow structure than discussed so far. We introduce the statistical study of turbulence in Section 10.3. Lift-briefly introduced in Chapter 8—is studied most effectively with advanced tools such as vorticity and circulation (Section 10.4). Section 10.5 continues with vorticity to show how this tool improves our understanding of curvy flow. Compressible fluid flow is discussed in Section 10.6. The flow behaviors not addressed in this text are accessible through advanced study based on the introductory methods in Chapters 1–9.

This book forms the basis of a one-semester introductory course in fluid mechanics for engineers and scientists. Students working with this text are expected to have a background in multivariable calculus, linear algebra, and differential equations; review of these topics as applied to fluid mechanics is provided in Chapter 1. Problem solving is taught by example throughout the text. We include numerous solved examples and end-of-chapter problems, and a complete solution manual is available for instructors.

Fluid mechanics can be a difficult subject. Nonlinear physics governs flow, and thus we often resort to a variety of simplifications to obtain solutions. Different simplifications are used under different conditions, making fluid mechanics intimidating, at least to a beginner. An Introduction to Fluid Mechanics presents the topic through a discovery process, as described in this preface, that mimics engineering practice. The process used seeks solutions by answering the following questions:

1. What is the problem?

2. What do we need to know, and do, to address the problem?

3. What is the solution to the problem?

4. What other problems/opportunities may be addressed now that we have solved this problem?

This organizational choice builds critical thinking skills by emphasizing the thought processes that lead to model development. The book is divided into four parts that answer these four questions for the study of fluid mechanics.

Getting motivated

Flows are beautiful and complex. A swollen creek tumbles over rocks and through crevasses, swirling and foaming. A child plays with sticky taffy, stretching and reshaping the candy as she pulls and twists it in various ways. Both the water and the taffy are fluids, and their motions are governed by the laws of nature. Our goal is to introduce readers to the analysis of flows using the laws of physics and the language of mathematics. On mastering this material, readers can harness flow to practical ends or create beauty through fluid design.

In this text we delve into the mathematical analysis of flows; however, before beginning, it is reasonable to ask if it is necessary to make this significant mathematical effort. After all, we can appreciate a flowing stream without understanding why it behaves as it does. We also can operate machines that rely on fluid behavior—drive a car, for example—without understanding the fluid dynamics of the engine. We can even repair and maintain engines, piping networks, and other complex systems without having studied the mathematics of flow. What is the purpose, then, of learning to mathematically describe fluid behavior?

The answer is quite practical: Knowing the patterns that fluids form and why they are for med, and knowing the stresses that fluids generate and why they are generated, is essential to designing and optimizing modern systems and devices.

This chapter illustrates and discusses some of the most widely used mesoscale models for describing particulate processes. The reader should keep in mind that the chapter is not a comprehensive discussion of all the possible mesoscale models, but is simply a collection of some example models, used in this context to highlight the major issues typically encountered in the simulation of multiphase systems. Although some of the models illustrated in the next sections have general validity, they typically assume slightly different forms when applied to the simulation of solid–liquid systems (e.g. crystallization and precipitation processes), solid–gas systems (e.g. fluidized suspensions, aerosol reactors), liquid–liquid systems (e.g. emulsions), gas–liquid systems (e.g. bubble columns and gas–liquid stirred tanks), and liquid–gas systems (e.g. evaporating and non-evaporating sprays). In what follows we will often refer to the elements of the disperse phase as “particles” to indicate both solid particles (such as crystals, solid amorphous particles, and solid aggregates) and fluid particles (such as droplets and bubbles). The remainder of the chapter is organized as follows. After providing an overview of the philosophy behind the development of mesoscale models in Section 5.1, specific examples of phase-and real-space advection processes (and the consequent diffusion processes) are discussed in Sections 5.2–5.5. Subsequently, phase-space point (discontinuous) processes are presented in Sections 5.6–5.8. For each of these processes, the corresponding functional form of the mesoscale model appearing in the final GPBE is derived and discussed.

Disperse multiphase flows

The majority of the equipment used in the chemical process industry employs multiphase flow. Bubble columns, fluidized beds, flame reactors, and equipment for liquid–liquid extraction, for solid drying, and size enlargement or reduction are common examples. In order to efficiently design, optimize, and scale up industrial systems, computational tools for simulating multiphase flows are very important. Polydisperse multiphase flows are also common in other areas, such as fuel sprays in auto and aircraft engines, brown-out conditions in aerospace vehicles and particulate flows occurring in the environment. Although at first glance the multifarious industrial and environmental multiphase flows appear to be very different from each other, they have a very important common element: it is possible to identify a continuous phase and a disperse phase (usually with a distribution of characteristic “particle sizes”).

Historically the development of the theoretical framework and of computational models for disperse multiphase flows has focused on two different aspects: (i) the evolution of the disperse phase (e.g. breakage and coalescence of bubbles or droplets, particle–particle collisions, etc.) and (ii) multiphase fluid dynamics. The first class of models is mainly concerned with the description of the disperse phase, and is based on the solution of the spatially homogeneous population balance equation (PBE). A PBE is a continuity statement written in terms of a number density function (NDF), which will be described in detail in Chapter 2.

In this chapter, the governing equations needed to describe polydisperse multiphase flows are presented without a rigorous derivation from the microscale model. (See Chapter 4 for a complete derivation.) For clarity, the discussion of the governing equations in this chapter will be limited to particulate systems (e.g. crystallizers, fluidized beds, and aerosol processes). However, the reader familiar with disperse multiphase flow modeling will recognize that our comments hold in a much more general context. Indeed, the extension of the modeling concepts developed in this chapter to many other multiphase systems is straightforward, and will be discussed in later chapters.

The primary purpose of this chapter is to introduce the key concepts and notation needed to develop models for polydisperse multiphase flows. We thus begin with a general discussion of the number-density function (NDF) in its various forms, followed by example transport equations for the NDF with known (PBE) and computed (GPBE) particle velocity. These transport equations are written in terms of “averaged” quantities whose precise definitions will be presented in Chapter 4. We then consider the moment-transport equations that are derived from the NDF transport equation by integration over phase space. Finally, we briefly describe how turbulence modeling can be undertaken starting from the moment-transport equations.

Number-density functions (NDF)

The disperse phase is constituted by discrete elements. One of the main assumptions of our analysis is that the characteristic length scales of the elements are smaller than the characteristic length scale of the variation of properties of interest (i.e. chemical species concentration, temperature, continuous phase velocities).

In this chapter we discuss issues specific to applying moment methods with spatially inhomogeneous systems. In particular, we focus on the spatial transport of moment sets by advection, diffusion, and free transport. In Chapter 7, issues related to transport in phase space are thoroughly treated and, here, we will discuss such terms only inasmuch as they affect spatial transport. In Section 8.1, the principal modeling issues that arise with spatially inhomogeneous systems are briefly reviewed. In the sections that follow, we discuss separately moment methods for (i) the inhomogeneous population-balance equation (PBE) (i.e. where the internal coordinates do not include or affect the velocity) in Section 8.3, (ii) the inhomogeneous kinetic equation (KE) (i.e. where the only internal coordinate is velocity) in Section 8.4, and finally (iii) the full inhomogeneous generalized population-balance equation (GPBE) in Section 8.5. Concrete examples, and the corresponding discretized formulas, are provided for each type of system in order for the reader to understand fully the issues that arise when simulating inhomogeneous systems. An important theme running through the entire chapter is the issue of realizable moment sets, and how realizability is affected by spatial transport. Thus, in order to have explicit examples of the numerical issues, we introduce kinetics-based finite-volume methods (KBFVM) for moment sets in Section 8.2. Nevertheless, the reader should keep in mind that these numerical issues are generic to moment transport and will arise with all spatial-discretization methods.

In this chapter we discuss the basic theory of Gaussian quadrature, which is at the heart of quadrature-based moment methods (QBMM). Proofs for most of the results are not included and, for readers requiring more extensive analytical treatments, references to the literature are made. In addition to a summary of the relevant theory, different algorithms to calculate the abscissas (or nodes) and the weights of the quadrature approximation from a known set of moments are presented, and their advantages and disadvantages are critically discussed. It is important to remind readers that most of the theory for quadrature formulas was developed for mono-dimensional integrals, namely integrals of a single independent variable. Therefore the discussion below starts from univariate distributions, for which the Gaussian quadrature theory applies exactly, and subsequently moves to bivariate and multivariate distributions. Although in the latter cases the quadrature is no longer strictly Gaussian, most of its interesting properties are still valid. In the univariate case, the weights and abscissas are used in the quadrature method of moments (QMOM) to solve moment-transport equations. Thus, we will refer to moment-inversion algorithms that use a full set of moments as QMOM, while other methods that use a reduced set will be referred to differently.