Duality symmetries and Hamiltonians

Duality symmetries are some of the most interesting symmetries in physics. The term “duality” is generally used by physicists to refer to the relationship between two systems that have very different descriptions but identical physics. The main subject of this chapter is one such situation that arises in closed string theory. You may think that a world where one dimension is curled up into a circle of radius R could easily be distinguished from a world in which the circle has radius α′/R (recall that α′ has units of length-squared), but in closed string theory these two worlds are indistinguishable for any value of R. There is a duality symmetry that relates them to each other. This symmetry is called T-duality, where the T stands for toroidal.

If a few crystals of a colored material like copper sulfate are placed at the bottom of a tall bottle filled with water, the color will slowly spread through the bottle. At first the color will be concentrated in the bottom of the bottle. After a day it will penetrate upward a few centimeters. After several years the solution will appear homogeneous.

The process responsible for the movement of the colored material is diffusion, the subject of this book. Diffusion is caused by random molecular motion that leads to complete mixing. It can be a slow process. In gases, diffusion progresses at a rate of about 5 cm/min; in liquids, its rate is about 0.05 cm/min; in solids, its rate may be only about 0.00001 cm/min. In general, it varies less with temperature than do many other phenomena.

This slow rate of diffusion is responsible for its importance. In many cases, diffusion occurs sequentially with other phenomena. When it is the slowest step in the sequence, it limits the overall rate of the process. For example, diffusion often limits the efficiency of commercial distillations and the rate of industrial reactions using porous catalysts. It limits the speed with which acid and base react and the speed with which the human intestine absorbs nutrients. It controls the growth of microorganisms producing penicillin, the rate of the corrosion of steel, and the release of flavor from food.

Diffusion causes convection. To be sure, convective flow can have many causes. For example, it can occur because of pressure gradients or temperature differences. However, even in isothermal and isobaric systems, diffusion will always produce convection. This was clearly stated by Maxwell in 1860: “Mass transfer is due partly to the motion of translation and partly to that of agitation.” In more modern terms, we would say that any mass flux may include both convection and diffusion.

This combination of convection and diffusion can complicate our analysis. The easier analyses occur in dilute solutions, in which the convection caused by diffusion is vanishingly small. The dilute limit provides the framework within which most people analyze diffusion. This is the framework presented in Chapter 2.

In some cases, however, our dilute-solution analyses do not successfully correlate our experimental observations. Consequently, we must use more elaborate equations. This elaboration is best initiated with the physically based examples given in Section 3.1. This is followed by a catalogue of flux equations in Section 3.2. These flux equations form the basis for the simple analyses of diffusion and convection in Section 3.3 that parallel those in the previous chapter.

After simple analyses, we move in Section 3.4 to general mass balances, sometimes called the general continuity equations. These equations involve the various coordinate systems introduced in Chapter 2. They allow solutions for the more difficult problems that arise from the more complicated physical situation.

Like its earlier editions, this book has two purposes. First, it presents a clear description of diffusion, the mixing process caused by molecular motion. Second, it explains mass transfer, which controls the cost of processes like chemical purification and environmental control. The first of these purposes is scientific, explaining how nature works. The second purpose is more practical, basic to the engineering of chemical processes.

While diffusion was well explained in earlier editions, this edition extends and clarifies this material. For example, the Maxwell–Stefan alternative to Fick's equation is now treated in more depth. Brownian motion and its relation to diffusion are explicitly described. Diffusion in composites, an active area of research, is reviewed. These topics are an evolution of and an improvement over the material in earlier editions.

Mass transfer is much better explained here than it was earlier. I believe that mass transfer is often poorly presented because it is described only as an analogue of heat transfer. While this analogue is true mathematically, its overemphasis can obscure the simpler physical meaning of mass transfer. In particular, this edition continues to emphasize dilute mass transfer. It gives a more complete description of differential distillation than is available in other introductory sources. This description is important because differential distillation is now more common than staged distillation, normally the only form covered. This edition gives a much better description of adsorption than has been available. It provides an introduction to mass transfer applied in biology and medicine.

The most common use of the mass transfer coefficients developed in Chapter 8 is the analytical description of large-scale separation processes like gas absorption and distillation. These mass transfer coefficients can describe the absorption of a solute vapor like SO2 or NH3 from air into water. They describe the distillation of olefins and alkanes, the extraction of waxes from lubricating oils, the leaching of copper from low-grade ores, and the speed of drug release.

Mass transfer coefficients are useful because they describe how fast these separations occur. They thus represent a step beyond thermodynamics, which establishes the maximum separations that are possible. They are a step short of analyses using diffusion coefficients, which have a more exact fundamental basis. Mass transfer coefficients are accurate enough to correlate experimental results from industrial separation equipment, and they provide the basis for designing new equipment.

All industrial processes are affected by mass transfer coefficients but to different degrees. Gas absorption, the focus of this chapter, is an example of what is called “differential contacting” and depends directly on mass transfer coefficients. Many mechanical devices, including blood oxygenators and kidney dialyzers, are analyzed similarly, as discussed in the next chapter. Distillation, the most important separation, is idealized in two ways. In the first, it is treated as “differential contracting” and analyzed in a parallel way to absorption, as described in Chapter 12. In the second idealization, distillation is approximated as a cascade of near equilibrium “stages.” Such “staged contacting,” is detailed in Chapter 13.

In the previous chapters, we have discussed how diffusion involves physical factors. We calculated the gas diffusion through a polymer film, or sized a packed absorption column, or found how diffusion coefficients were related to mass transfer coefficients. In every case, we were concerned with physical factors like the film's thickness, the area per volume of the column's packing, or the fluid flow in the mass transfer. We were rarely concerned with chemical change, except when this change reached equilibrium, as in solvation.

In this chapter, we begin to focus on chemical changes and their interaction with diffusion. We are particularly interested in cases in which diffusion and chemical reaction occur at roughly the same speed. When diffusion is much faster than chemical reaction, then only chemical factors influence the reaction rate; these cases are detailed in books on chemical kinetics. When diffusion is not much faster than reaction, then diffusion and kinetics interact to produce very different effects.

The interaction between diffusion and reaction can be a large, dramatic effect. It is the reason for stratified charge in automobile engines, where imperfect mixing in the combustion chamber can reduce pollution. It is the reason for the size of a human sperm. It can reduce the size needed for an absorption tower by 100 times. The interaction between diffusion and reaction can even produce diffusion across membranes from a region of low concentration into a region of high concentration.

The ideas of mass transfer covered in the previous three chapters provide a strong framework for describing mass transfer in a wide variety of situations. One obvious example is gas absorption, which is one route to controlling the carbon dioxide emissions that contribute to global warming. The ideas of mass transfer are also basic to many physiologic functions. For example, they are key to respiration, to digestion, and to drug metabolism.

The descriptions of mass transfer developed in this book are detailed and accurate because they were developed for well-defined problems in the chemical industry. In that highly competitive industry, small improvements in chemical processing can mean large increases in profits. This promise of higher profits led engineers to examine the details of mass transfer and to get highly quantitative results.

The descriptions of mass transfer developed in biology and medicine took place largely independently of those in chemistry and engineering. These biologically and medically based descriptions have often provided good insight into rate processes in living systems. However, these descriptions rarely take advantage of insights provided by the deeper engineering analyses. There are two reasons for this. First, the accuracy of data in living systems is often uncertain because living systems vary more widely. After all, the weight of a person varies more than the weight of a nitrogen molecule. As a result, details of mass transfer known from engineering may not be that useful.

Adsorption is very different from absorption, distillation, and extraction. These three processes, detailed in the five previous chapters, typically involve two fluids flowing steadily in opposite directions. In absorption, a gas mixture flows upward through a packed column while an absorbing liquid trickles down. In distillation, a liquid mixture is split into a more volatile liquid distillate and a less volatile bottoms stream. In extraction, two liquid streams move countercurrently to yield an extract and a raffinate. To be sure, in some cases, the contacting may involve near-equilibrium states, and in other cases it may be described with nonequilibrium ideas like mass transfer coefficients. Still, all three units operations involve two fluids at steady state.

In contrast, adsorption is almost always an unsteady process involving a fluid and a solid. The use of a solid is a major difference because solids are hard to move. They abrade pipes and pumps; they break into fine particles which are hard to retain. As a result, we usually pump the feed fluid through a stationary bed of solid particles to effect a separation by adsorption.

Thus adsorption asks a different kind of question than the questions asked in absorption, distillation, or extraction. In absorption or differential distillation, the basic question is how tall a tower is needed. This question is answered with a mass transfer analysis, including an operating line and an equilibrium line. The mass transfer analysis includes overall and individual coefficients summarized by dimensionless correlations.

Diffusion rates can be tremendously altered by chemical reactions. Indeed, these alterations are among the largest effects discussed in this book, routinely changing the mass fluxes by orders of magnitude. The effects of a chemical reaction depend on whether the reaction is homogeneous or heterogeneous. This question can be difficult to answer. In well-mixed systems, the reaction is heterogeneous if it takes place at an interface and homogeneous if it takes place in solution. In systems that are not well mixed, diffusion clouds this simple distinction, as detailed in Section 16.1.

The effects of chemical reactions are exemplified by the data for ammonia adsorption in water summarized in Fig. 17.0-1. The overall mass transfer coefficient, in cm/sec, is based on a liquid side driving force given in mol/cm3. The specific values shown are for a hollow-fiber membrane contactor, though similar values would be obtained in a packed tower or other more conventional apparatus.

The different mass transfer coefficients shown in Fig. 17.0-1 represent different forms of ammonia and different rate-controlling steps for mass transfer. At pH above 5, the mass transfer coefficient is small, somewhat less than typical values for liquids. Below pH 4, the mass transfer coefficient rises.

As outlined in the previous chapter, distillation is a separation based on volatility differences. It is by far the most important separation. In North America alone, distillation consumes over one million barrels of oil per day, or about four percent of the continent's energy consumption. It is estimated to be only eleven percent efficient, and so offers an opportunity for increased energy efficiency.

Distillation normally involves three pieces of equipment: a column, a condensor on top of the column, and a reboiler at the bottom of the column. The reboiler, often a steam-jacketed kettle, is heated so that much of its contents evaporate and flow upwards through the column. The vapors passing out of the column are liquefied in the condensor and much of this condensate is sent back downwards through the column. This countercurrent flow of vapor and condensate is common to all forms of distillation.

The internals of the column itself can differ dramatically. In many columns, including almost all found in the laboratory, the columns' internals are random or structured packing. These packed columns were described in detail in Chapter 12. In many other columns, especially older, large-scale equipment, the column's internals are “stages,” volumes providing close contact between vapor and condensate. In most cases, the stages are designed so that liquid and vapor leave to each stage nearly in equilibrium with each other.

Diffusion across thin membranes can sometimes produce chemical and physical separations at low cost. These low costs have spurred rapid development of membrane separations, especially during the last 20 years. This rapid development has sought both high fluxes and high selectivities. It has included the separation of gases, of sea water, and of azeotropic mixtures. It has used hollow fibers and spiral wound modules; it has centered on asymmetric membranes with selective layers as thin as 10 nm. This rapid development is a sharp contrast to other diffusion-based separations like absorption, where the basic ideas have been well established for 50 years.

We will describe membrane separations in this chapter. In this description, we must recognize that membrane separations usually employ a somewhat different vocabulary than that used on other separations. The stream that flows into any membrane module is sensibly called the feed. That part of the feed retained by the membrane is called the retentate rather than the raffinate. That part of the feed that crosses the membrane is called the permeate. Any stream added to improve permeate flow is usually called a sweep.

Our description of membrane separations also involves two overlapping traditions: filtration and diffusion. These traditions use a vocabulary that is confusing. The biggest confusion comes from the term permeability. In ultrafiltration, the solvent's permeability relates the solvent flow through the membrane's pores to the pressure drop across the membrane. As such, it is like a Darcy's law permeability for flow through porous media.

Controlled-release technologies are used to supply compounds like drugs, pesticides, or fragrances at prescribed rates. The prescribed rates offer improved efficacy, safety, and convenience. The most commonly cited example is that of a drug dosed either by periodic pills or by a controlled-release technology. The concentration of the drug in the blood is shown schematically in Fig. 19.0-1. When the drug is given in a pill form, its concentration rises abruptly right after the pill is taken. This rise can carry the drug concentration past the effective level and briefly above the toxic level. The concentration then drops below the effective level. In contrast, when the drug is delivered by controlled release, its concentration rises above the level required to be effective and stays there, without sudden excursions to toxic or ineffective levels. Such delivery is often called zero-order release.

Typical products using controlled release are listed in Table 19.0-1. In the case of drugs, we normally want to release a single solid species, typically with a molecular weight greater than 600 daltons. The water solubility of these molecules is often strongly pH dependent because of pendant carboxylic acid or amino groups. The molecules normally will have several chiral centers. While this species is normally nonvolatile, it is usually unstable if it is warmed. It is often crystalline but may be a polymorph. As the previous paragraph suggests, we most often will seek to release the drug at a constant rate, although in some cases, we may want a periodic discharge.

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.