In this chapter, we turn to systems in which there are significant interactions between diffusing molecules. These interactions can strongly affect the apparent diffusion coefficients. In some cases, these effects produce unusual averages of the diffusion coefficients of different solutes; in others, they suggest a strong dependence of diffusion on concentration; in still others, they result in diffusion that is thousands of times slower than expected.

The discussion of these interactions involves a somewhat different strategy than that used earlier in this book. In Chapters 1–3, we treated the diffusion coefficient as an empirical parameter, an unknown constant that kept popping up in a variety of mathematical models. In more recent chapters, we have focused on the values of these coefficients measured experimentally. In the simplest cases, these values can be estimated from kinetic theory or from solute size; in more complicated cases, these values require experiments. In all these cases, the goal is to use our past experience to estimate the diffusion coefficients from which diffusion fluxes and the like can be calculated.

In this chapter, we consider the chemical interactions affecting diffusion much more explicity, rather than hiding them as part of the empirically measured diffusion coefficient. The interactions affecting diffusion are conveniently organized into three groups. As a first group, we consider in Section 6.1 solute–solute interactions, particularly in strong electrolytes. We want to discover how sodium chloride diffusion is an average of the diffusion of sodium ions and of chloride ions.

In this chapter, we consider the basic law that underlies diffusion and its application to several simple examples. The examples that will be given are restricted to dilute solutions. Results for concentrated solutions are deferred untilChapter 3.

This focus on the special case of dilute solutions may seem strange. Surely, it would seem more sensible to treat the general case of all solutions and then see mathematically what the dilute-solution limit is like. Most books use this approach. Indeed, because concentrated solutions are complex, these books often describe heat transfer or fluid mechanics first and then teach diffusion by analogy. The complexity of concentrated diffusion then becomes a mathematical cancer grafted onto equations of energy and momentum.

I have rejected this approach for two reasons. First, the most common diffusion problems do take place in dilute solutions. For example, diffusion in living tissue almost always involves the transport of small amounts of solutes like salts, antibodies, enzymes, or steroids. Thus many who are interested in diffusion need not worry about the complexities of concentrated solutions; they can work effectively and contentedly with the simpler concepts in this chapter.

Second and more important, diffusion in dilute solutions is easier to understand in physical terms. A diffusion flux is the rate per unit area at which mass moves. A concentration profile is simply the variation of the concentration versus time and position. These ideas are much more easily grasped than concepts like momentum flux, which is the momentum per area per time.

All thoughtful persons are justifiably concerned with the presence of chemicals in the environment. In some cases, chemicals like pesticides and perfumes are deliberately released; in other cases, chemicals like hydrogen sulfide and carbon dioxide are discharged as the result of manufacturing; in still others, chemicals like styrene and dioxin can be accidentally spilled. In all cases, everyone worries about the long-term effects of such chemical challenges.

Public concern has led to legislation at federal, state, and local levels. This legislation often is phrased in terms of regulation of chemical concentrations. These regulations take different forms. The maximum allowable concentration may be averaged over a day or over a year. The acid concentration (as pH) can be held within a particular range, or the number and size of particles going up a stack can be restricted. Those working with chemicals must be able to anticipate whether or not these chemicals can be adequately dispersed. They must consider the problems involved in locating a chemical plant on the shore of a lake or at the mouth of a river.

The theory for dispersion of these chemicals is introduced in this short chapter. As might be expected, dispersion is related to diffusion. The relation exists on two very different levels. First, dispersion is a form of mixing, and so on a molecular level it involves diffusion of molecules. This molecular dispersion is not understood in detail, but it takes place so rapidly that it is rarely the most important feature of the process.

Throughout this book, we have routinely assumed that diffusion takes place in binary systems. We have described these systems as containing a solute and a solvent, although such specific labels are arbitrary. We often have further assumed that the solute is present at low concentration, so that the solutions are always dilute. Such dilute systems can be analyzed much more easily than concentrated ones.

In addition to these binary systems, other diffusion processes include the transport of many solutes. One group of these processes occurs in the human body. Simultaneous diffusion of oxygen, sugars, and proteins takes place in the blood. Mass transfer of bile salts, fats, and amino acids occurs in the small intestine. Sodium and potassium ions cross many cell membranes by means of active transport. All these physiological processes involve simultaneous diffusion of many solutes.

This chapter describes diffusion for these and other multicomponent systems. The formalism of multicomponent diffusion, however, is of limited value. The more elaborate flux equations and the slick methods used to solve them are often unnecessary for an accurate description. There are two reasons for this. First, multicomponent effects are minor in dilute solutions, and most solutions are dilute. For example, the diffusion of sugars in blood is accurately described with the binary form of Fick's law. Second, some multicomponent effects are often more lucid if described without the cumbersome equations splattered through this chapter.

Processes involving coupled heat and mass transfer occur frequently in nature. They are central to the formation of fog, to cooling towers, and to the wet-bulb thermometer. They are important in the separation of uranium isotopes and in the respiration of water lilies. This chapter analyzes a few of these processes. Not unexpectedly, such processes are complex, for they involve equations for both diffusion and heat conduction. These equations are coupled, often in a nonlinear way. As a result, our descriptions will contain approximations to reduce the complexities involved.

We begin this chapter with a comparison of the mechanisms responsible for mass and heat transfer. The mathematical similarities suggested by these mechanisms are discussed in Section 21.1, and the physical parallels are explored in Section 21.2. The similar mechanisms of mass and heat transfer are the basis for the analysis of drying, both of solids and of sprayed suspensions. However, the detailed models differ, as shown by the examples in Section 21.3. In Section 21.4, we outline cooling-tower design as an example based on mass and heat transfer coefficients. Finally, in Section 21.5, we describe thermal diffusion and effusion.

Mathematical Analogies Among Mass, Heat, and Momentum Transfer

Analogies among mass, heat, and momentum transfer have their origin either in the mathematical description of the effects or in the physical parameters used for quantitative description. The mathematically based analogies are useful for two reasons.

Distillation is the process of heating a liquid solution to drive off a vapor and then collecting and condensing this vapor. It is the most common method of chemical separation, the workhorse of the chemical process industries. Distillation columns are ubiquitous; they are the brightly lighted towers that rise from chemical plants, and they are the stills used by moonshiners.

Distillation is carried out in two ways: differential distillation and staged distillation. The difference is what is inside of the distillation column. In differential distillation, the column contains packing like that used for gas absorption. In small laboratory columns, this packing is usually random, of Raschig rings or even glass beads. Such distillation aims to provide small amounts of very pure chemicals. In larger differential distillation columns, the column internals are usually structured packing. These distillations aim to produce large amounts of commodity chemicals at the lowest possible cost.

The second way to effect these separations is staged distillation. In staged distillation, the column internals are completely different than those normally used for gas absorption. Now these internals consist of a series of compartments or “trays,” where liquid and vapor are contacted intimately, in the hope that they will approach equilibrium. Now, the liquid and vapor concentrations in the column do not vary continuously, but discretely, jumping to new values on each tray. Staged distillation was an innovation for commodity chemicals a century ago, and was the standard during the rapid growth of the chemical industry.

In this chapter we want to connect mass transfer coefficients, diffusion coefficients, and fluid flow. In seeking these connections, we are combining the previous chapter, which deals with mass transfer, with the first two sections of the book, which dealt with diffusion.

To find these connections, we will develop theories of mass transfer. These theories are rarely predictive, but they clarify the chemistry and physics which are involved. They are less predictive because they are most often for fluid–fluid interfaces whose geometry is not well known. They are much more successful for solid–fluid interfaces, which are much better defined. Unfortunately, fluid–fluid interfaces are much more important for mass transfer than fluid–solid interfaces are.

Before reviewing the common theories, we should identify exactly what we want to predict. Almost always, we want to predict the mass transfer coefficient k as a function of the diffusion coefficient D and the fluid velocity v. In many cases, convection will be forced, i.e., the velocity will be caused by mechanical forces like pressure drop imposed from outside the system. In occasional cases, convection will be free, the consequence of gravity driven flows often caused by the mass transfer itself. While we will discuss both cases, we will stress forced convection because it is more important and more common in chemical processing.

In this chapter, we briefly describe fundamental concepts of heat transfer. We begin in Section 20.1 with a description of heat conduction. We base this description on three key points: Fourier's law for conduction, energy transport through a thin film, and energy transport in a semi-infinite slab. In Section 20.2, we discuss energy conservation equations that are general forms of the first law of thermodynamics. In Section 20.3, we analyze interfacial heat transfer in terms of heat transfer coefficients, and in Section 20.4, we discuss numerical values of thermal conductivities, thermal diffusivities, and heat transfer coefficients.

This material is closely parallel to the ideas about diffusion presented in the rest of this book. This parallelism is not unexpected, for heat transfer and mass transfer are described with equations that are very similar mathematically. The material in Section 20.1 is like that in Chapter 2, and the general equations in Section 20.2 are conceptually similar to those in Chapter 3. The material on heat transfer coefficients in Section 20.3 closely resembles the mass transfer material in Chapters 9 through 15, and the numerical values in Section 20.4 are parallel to those in Chapters 5 and 8.

Thus we are abstracting ideas of heat transfer in a few sections, whereas we detailed similar ideas of mass transfer over many chapters. This represents a tremendous abridgment. As those skilled in heat transfer recognize, the heat transfer literature is immense, of far greater size than the mass transfer literature.

The purpose of this second edition is again a clear description of diffusion useful to engineers, chemists, and life scientists. Diffusion is a fascinating subject, as central to our daily lives as it is to the chemical industry. Diffusion equations describe the transport in living cells, the efficiency of distillation, and the dispersal of pollutants. Diffusion is responsible for gas absorption, for the fog formed by rain on snow, and for the dyeing of wool. Problems like these are easy to identify and fun to study.

Diffusion has the reputation of being a difficult subject, much harder than, say, fluid mechanics or solution thermodynamics. In fact, it is relatively simple. To prove this to yourself, try to explain a diffusion flux, a shear stress, and chemical potential to some friends who have little scientific training. I can easily explain a diffusion flux: It is how much diffuses per area per time. I have more trouble with a shear stress. Whether I say it is a momentum flux or the force in one direction caused by motion in a second direction, my friends look blank. I have never clearly explained chemical potentials to anyone.

However, past books on diffusion have enhanced its reputation as a difficult subject. These books fall into two distinct groups that are hard to read for different reasons. The first group is the traditional engineering text. Such texts are characterized by elaborate algebra, very complex examples, and turgid writing.

Extraction treats a feed with a liquid solvent to remove and concentrate a valuable solute. When the feed is a liquid, the process is called “liquid–liquid extraction,” or more commonly just “extraction.” When the feed is a solid, the process is called “solid–liquid extraction,” or more commonly “leaching.” In either case, the original solution is commonly called the feed; after the extraction, this stream is called the raffinate. Similarly, the second solvent is called the extract once it contains solute.

Extraction is almost never the first choice as a separation process. If the solute of interest is a gas, then we will first try gas absorption or stripping. If the solute of interest is volatile under convenient conditions, then we will attempt distillation. We will normally try extraction only after we fail at absorption and distillation. Still, we have included a separate chapter on extraction for two reasons. First, it is an important process, central to some petrochemical, pharmaceutical, and metallurgical processes. We discuss these in Section 14.1. Second and more importantly, extraction gives an extended example of the generalization of the analyses of absorption and distillation. When extraction is carried out in differential contactors like packed towers, its analysis is similar to gas absorption. When extraction is carried out in staged contactors, its analysis parallels staged distillation. Thus we can test our understanding of absorption and distillation by discussing extraction.

Diffusion is the process by which molecules, ions, or other small particles spontaneously mix, moving from regions of relatively high concentration into regions of lower concentration. This process can be analyzed in two ways. First, it can be described with Fick's law and a diffusion coefficient, a fundamental and scientific description used in the first two parts of this book. Second, it can be explained in terms of a mass transfer coefficient, an approximate engineering idea that often gives a simpler description. It is this simpler idea that is emphasized in this part of this book.

Analyzing diffusion with mass transfer coefficients requires assuming that changes in concentration are limited to that small part of the system's volume near its boundaries. For example, in the absorption of one gas into a liquid, we assume that gases and liquids are well mixed, except near the gas–liquid interface. In the leaching of metal by pouring acid over ore, we assume that the acid is homogeneous, except in a thin layer next to the solid ore particles. In studies of digestion, we assume that the contents of the small intestine are well mixed, except near the villi at the intestine's wall. Such an analysis is sometimes called a “lumped-parameter model” to distinguish it from the “distributed-parameter model” using diffusion coefficients. Both models are much simpler for dilute solutions.

If you are beginning a study of diffusion, you may have trouble deciding whether to organize your results as mass transfer coefficients or as diffusion coefficients.

Until now, we have treated the diffusion coefficient as a proportionality constant, the unknown parameter appearing in Fick's law. We have found mass fluxes and concentration profiles in a broad spectrum of situations using this law. Our answers have always contained the diffusion coefficient as an adjustable parameter.

Now we want to calculate values of the flux and the concentration profile. For this, we need to know the diffusion coefficients in these particular situations. We must depend largely on experimental measurements of these coefficients, because no universal theory permits their accurate a-priori calculation. Unfortunately, the experimental measurements are unusually difficult to make, and the quality of the results is variable. Accordingly, we must be able to evaluate how good these measurements are.

Before we begin, we should list the guidelines that tend to stick in everyone's mind. Diffusion coefficients in gases, which can be estimated theoretically, are about 0.1 cm2/sec. Diffusion coefficients in liquids, which cannot be as reliably estimated, cluster around 10–5 cm2/sec. Diffusion coefficients in solids are slower still, 10–30 cm2/sec, and they vary strongly with temperature. Diffusion coefficients in polymers and glasses lie between liquid and solid values, say about 10–8 cm2/sec, and these values can be strong functions of solute concentration.

The accuracy and origins of these guidelines are explored in this chapter. Gases, liquids, solids, and polymers are discussed in Sections 5.1 through 5.4, respectively. In these sections we give a selection of typical values, as well as one common method of estimating these values.

Chapter preview

This chapter explores a range of issues surrounding the study of “world musics.” It begins by examining the various uses and histories of the term World Music (“world music”), arguing that this category influences the ways we think about musical repertoires and relate to particular musical cultures. The next section considers a few historical musical encounters, the impact of the concepts of “evolution” and “culture,” going on to chart how the study of world music and the discipline of ethnomusicology developed. The role of ethnography and of methods that focus on performance, event, and orality are highlighted as distinctive aspects of the study of world music and ethnomusicology. Drawing on several case studies, the discussion then turns to the relationship between music and place. Both the critical relationship between music and place and the dangers of uncritically mapping music onto place are stressed, leading to a discussion of the relationship between identity, place, and authenticity. The final part of the chapter focuses on issues surrounding the reception of unfamiliar musics. It is an exercise in “ear cleaning,” which aims to help us to recognize how power and cultural conditioning shape the ways we hear. It contrasts so-called “listening” and “doing” musics, examines how sounds that might challenge hegemonic modes of hearing are often avoided in the World Music market, and questions our perceptions of rhythm and harmony.

Key issues

• World Music(s): exclusions and inclusions.

• Who studies world musics?

• Does music have a place?

• Can world music be mapped?

• Sounding authentic?

• Can we trust our ears?

• […]

Chapter preview

Whatever one's prejudices about “opera,” there is only one way to categorize it: as a branch of lyric drama, or music theater. This is a huge and interesting subject, whether as entertainment, as study, or as material to perform. Many students will get the chance to sing, or play in, or direct, musicals and operas. Others will perhaps help back-stage or front-of-house. Theaters demand many different activities, as we saw in chapter 3. Studying opera also exists as part of music history; equally, writing music theater is still a flourishing activity, and thus part of the composer's curriculum. Performing and writing about opera often entails what W. B. Yeats (in a poem about staging drama) called “the fascination of what's difficult.” There is no universal theory; opera studies borrow methods from whichever discipline is useful. This is seen as we progress through the sections below, each of which explores a separate area. Perhaps because of its obvious challenges, opera studies (within “musicology”) is a relative newcomer.

Key issues

• How do opera and music theater fit into history?

• How far is opera relevant to society now?

• What systems are there for analyzing music theater?

• How did opera cope with the twentieth century?

• Singing as a means of persuasion.

• Some ways that music theater conveys morals and messages.

• Production issues: interpreting musical stage works.

Opera as entertainment and ritual

Opera today can still be thought of like a ritual, a human activity bringing people together for a common cultural purpose.

Chapter preview

Music exists in performance. It may seem obvious to say this, but it is not. To say that music exists in performance is to focus on a particular way of thinking about what music is: i.e., that it is a practice. Viewing music as a performing art emphasizes the experiential dimensions of music, and its immediacy. Experiencing music in performance highlights music as an interactive process. People respond to musical performances emotionally, bodily, and critically. They may dance to music, fall into trance, or feel a sense of community. This chapter asks questions such as “What are the meanings of musical performances?” and “What are their ritual, social, or political significances?” Music is performed in a wide variety of contexts including concert settings, family celebrations, healing ceremonies, rituals, and competitions. Musical performance features in the realms of everyday experience, from singing lullabies to listening to music while you shop. Yet there is also something special about performance: it is often understood as standing apart from everyday life and it involves presentation to an “audience.” Performers may display virtuosic musical skills or they may take on important social roles, for example, commenting on socio-political trends or mediating between supernatural and natural forces. This chapter explores musical performances from a global perspective, addressing questions about the nature, function, and processes of performance, as well as about the social roles and training of performers.

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This chapter demonstrates why aesthetics, the branch of philosophy concerned with art and beauty, should be a part of the study of music. It outlines the emergence of aesthetics from the eighteenth century onwards in terms of the changing relationship between what is considered to be “subjective” and what is considered to be “objective.” This relationship has important implications for both the historical and the analytical study of music. In the modern period, ideas about objectivity are changed by the growing sense in many areas of Western society that there is no divine order of things, and that objectivity is therefore in some way dependent upon human subjectivity. Music is a form of art that is both objective, in the sense that it relies on rules of harmony, acoustics, etc., some of which can be formulated mathematically, and subjective, because it addresses human feelings and is judged in part on the basis of feelings. Music becomes important in the modern period because its meaning can be interpreted in very different ways, which are often influenced by issues in the society in which it is located. Aesthetic questions lie at the heart of debates in the contemporary study of music over whether music should be looked at in formal, analytical terms, or whether it should be connected to social and political issues.

Key issues

• What is music aesthetics?

• Is aesthetic evaluation merely subjective, or can it be objective?

• Are aesthetic problems purely philosophical, or are they also historical?

• What does music mean?

• […]