Introduction

Most chemical reactions are limited by equilibrium, and in many cases the equilibrium constant is such that the conversion to the desired product is too small to be economical. Doing “better than equilibrium” is one of the primary challenges in chemical engineering design. The general idea is obvious: The reaction should be carried out in the absence of product, so that the reverse reaction cannot proceed. Implementation is not so obvious, which is why this is a primary challenge.

The most interesting approach to overcoming equilibrium, which has been successful in many cases, is to design a system in which reaction and product separation take place simultaneously. In that way, the product is removed from the reactor and cannot participate in the reverse reaction. The inspiration for this idea may come from the biological cell, in which the reaction sites are enclosed within the cell membrane, which is permeable to reaction products that are intended for use outside the cell. The two most common manifestations outside nature are the membrane reactor, which mimics the cellular process, and reactive distillation, in which the reaction takes place in a distillation column. A proper treatment of distillation requires inclusion of the energy balance and the notion of vapor-liquid equilibrium, which is typically addressed in a subsequent course in thermodynamics, so we will not examine reactive distillation here. The membrane reactor is accessible to us now, however, and nicely illustrates the concept.

Introduction

Many physicochemical systems consist of two or more phases in intimate contact. We have already considered one such system in Chapter 8, where we noted that the bioreactor contains flocs of microorganisms suspended in an aqueous phase, together with air bubbles that provide oxygen. In that case, we made the approximation that we could treat the system as though it were one continuous phase, a process known as homogenization. We did the same for the CO oxidation reactor with finely dispersed catalyst. Homogenization works when all microstructural length scales are so small that the phenomena occurring in the various phases can be averaged together over a continuum length scale that is still very small relative to the macroscopic size of the system, in which case any time scales associated with transport within the microstructure are negligible relative to overall system scales. In many multiphase cases, however, the length scales are such that we must directly address the multiphase nature of the system and the concomitant transport of mass and energy between the phases.

The focus of this chapter will be on the foundations of interfacial mass transfer of a component species between the phases and the approach to an equilibrium distribution. The following chapter will address some of the processing issues that arise. As a preface to the analysis, however, it is useful to begin with a brief classification of various types of two-phase systems that may be encountered in practice.

Introduction

The chemical reactor is the heart of most industrial processes, although the reactor may represent only about 10 percent of the total capital cost; this is because the output from the reactor defines everything else that must be done downstream, particularly the separation processes. Similarly, if we wish to think about cellular rather than industrial processes, it is the chemical reactions that enable the cell or the organ to carry out its essential functions. We saw in Chapters 7 through 9 how a chemical reactor is integrated into simple processes, and we explored economic issues such as the trade-off between capital and operating costs. That discussion was limited, however, because we assumed in every case that the rate constants were fixed numbers. In doing so, we ignored one of the “handles” that the chemical engineer – or, in the case of an organism, evolution – has available to promote efficiency. Chemical reaction rates are highly temperature dependent, and precise temperature control can be critical in both the design and functioning of a reactor.

Chemical reaction engineering is a broad subject, and it typically occupies at least one full course in an undergraduate chemical engineering curriculum. We introduce some basic ideas here for completeness, but we are only touching on one of the foundations of the chemical engineering profession. We restrict ourselves throughout this chapter to liquid systems, as before, in order to simplify some of the analysis while retaining the essential features, and we address only well-mixed reactor configurations.

Design

Engineering is ultimately about making things for the benefit of society. We typically use the term design for the set of instructions by which a craftsperson can turn an idea into an object, and that meaning of the word carries over to the engineering activity of making tangible objects. Design has an additional meaning for a chemical engineer, however; it is the conceptualization of a process for manufacturing something – a chemical, perhaps. In this use of the word we address the problem of designing what pieces of equipment are needed, how they should be connected, how large they should be, and so on, but we do not address the question of how the equipment should be manufactured – materials of construction, locations of welds, precise geometry, and so forth. It is the second sense of the word that we employ in this chapter. Thus, we will exploit the understanding of reaction kinetics developed in Chapter 6 to determine reactor sizes and material flow rates, but we will view the reactor simply as a vessel of a given size, with no attention to the detail that would be required for actual fabrication.

Process design is a subject that is traditionally taught as a capstone course during the last year of a chemical engineering program, because a complete approach to process design obviously requires a broad base of understanding of chemical engineering fundamentals.

Introduction

Most chemical engineering applications involve chemical reactions; this is true whether we are dealing with the manufacture of computer chips, the creation of scaffolding for cell growth in artificial organs, the design of a novel battery, or the conversion of biomass to synthetic fuel. Chemical reactions can take place in gas, liquid, or solid phases; in the bulk or at interfaces; and with or without catalysts. Many reactions of social, physiological, or industrial significance take place in multiphase environments, where reactive species and reaction products must cross phase boundaries. Some “cracking” reactions for the production of intermediate molecular weight hydrocarbons from heavy components of crude oil, for example, are carried out in “trickle-bed” reactors; here, liquid and gas phases flow together over a bed of catalyst particles. Design and operating considerations for trickle-bed reactors include ensuring the necessary contact between the phases so that chemical species can get to where they need to be for specific reactions to occur. Similarly, many biochemical reactions, including wastewater treatment, require that suspended microorganisms be able to access organic nutrients that are dissolved in the liquid phase, as well as oxygen that is supplied as a gas (perhaps in air) and must dissolve into the liquid.

The design and operating issues for complex reactors of the types mentioned here are addressed in advanced courses, but we can focus on some basic principles essential to our overall understanding that can be elucidated by considering single-phase reactors and simple geometries; many important reactions are, in fact, carried out in a single phase in relatively simple geometries.

Introduction

The implementation of a great many processes depends ultimately on the ability to separate various species from one another. In this chapter we will build on the basic principles of interphasemass transfer developed in Chapter 10 to explore some of the ideas involved in the design of a separation process. The essential component of the analysis is the equilibrium stage defined in Section 10.4.1. The design problem can be roughly broken down into the actual mechanical implementation of an equilibrium stage and the computation of the number of equilibrium stages needed to effect a desired degree of separation. We shall deal only with the latter; the former remains very much an art and beyond the scope of this introductory text. For simplicity we will restrict this chapter to separation by liquid-liquid extraction. The overall approach and the solution techniques have much greater generality and are applicable to nearly all separation processes, phase-change and nonphase-change alike.

Liquid-liquid extraction is a process in which a solute is transferred between two solvents. We will assume that the two solvents are absolutely immiscible in one another; this is a convenient approximation, although only sometimes realistic. It is traditional to do separation process calculations using mass fractions instead of concentrations. We shall follow this practice, and Section 11.2 is devoted to reformulating the description of an equilibrium stage in terms of mass fractions.

Equilibrium Stage

A mass transfer stage for extraction is shown in Figure 11.1. The two phases with dissolved solute A are fed to a well-stirred contactor, where the agitation is designed to create a large interfacial area and transfer of A across the interface occurs.

As noted in the first chapter, chemical engineering is a broad profession that is critical to addressing many of the issues facing modern society. The intent of this text has been to provide a fundamental understanding of the elements of chemical engineering and to provide a flavor of the challenges that a chemical engineer might face; the quantitative skills developed here are generalizable to problems of far greater complexity than those addressed in this introductory text. There is much more to come to complete a basic chemical engineering education; the core will normally include courses that cover thermodynamics, fluid mechanics, mass transfer, heat transfer, separations, and reactor analysis in depth, as well as a capstone course in design. Other courses in the curriculum will depend on the institution, but will include some selection of advanced courses in chemistry, materials, biology, and mathematics.

Most educational institutions offer undergraduate students an opportunity to do research, and this experience is invaluable for obtaining real insight into the scope of the profession – it is a truism that the research that chemical engineers do is rarely reflected in the courses in the undergraduate curriculum because of time limitations in a four-year professional program, and it is in the research laboratory that an undergraduate student is most likely to see the exciting topics in materials development, synthetic biology, nanotechnology, and so forth that were mentioned in Chapter 1, as well as to experience the intellectual excitement that comes with addressing real open-ended problems.

Introduction

Up to this point we have addressed physical situations for which mass is the only fundamental dependent variable, and by doing so we have been able to explore a wide range of chemical engineering applications. We have made some implicit assumptions about momentum and energy transport in doing so, however. When we assumed that vessels were perfectly mixed, with a consequent uniform concentration, we did not ask about the nature of mechanical agitation necessary to effect perfect mixing; to have done so would have required incorporation of momentum as a fundamental variable. Similarly, when we assumed that chemical reactors could be operated at a specified temperature and pressure, we did not consider the means by which temperature and pressure control could be effected, nor did we consider the possible impact of temperature transients or of the compressibility of a gas phase; to have done so would have required incorporation of energy as a fundamental variable.

Momentum transport is usually addressed in the chemical engineering curriculum in a course called Fluid Mechanics, or in the first part of a Transport Phenomena sequence. Energy is the subject of courses in Thermodynamics and Heat Transfer, where the latter may be incorporated in a subsequent part of a Transport Phenomena sequence, but an introduction to energy balances is often included in the first course for which this text is intended. We will therefore touch on energetics in order to illustrate the issues involved in incorporation of energy as a fundamental variable.

Introduction

Most processes, both physicochemical and biological, involve one or more separation process. Human physiology, for example, requires the transfer of oxygen from air in the lungs to the blood stream, and the simultaneous transfer of CO2 from the blood stream to the lungs for removal. The function of the kidney is to process a continuous flow of liquid in order to separate waste products for removal from the body. The production of ethanol by the fermentation of sugars produced from natural products, whether for energy applications or for whiskey, requires that the ethanol be separated from an aqueous stream in which ethanol comprises less than 20 percent. The manufacture of polyethylene terephthalate for textile fibers requires the removal of ethylene glycol that is produced during the condensation polymerization process. A DNA analysis requires the separation of DNA fragments with different lengths and base pairs by gel electrophoresis. The production of oxygen for industrial or medical applications requires that the oxygen be separated from an air stream.

There are a variety of separation methodologies, and the analysis of separation processes has historically held a prominent place in the chemical engineering curriculum. Some, such as distillation and extraction, are familiar. (Brewing coffee or tea is an extraction process that is carried out on a very small scale.) Most physiological separations are membrane processes, in which a thin membrane keeps fluid (liquid or gas) streams apart while permitting certain species to move across the membrane.

Introduction

The problems studied in Chapter 2 illustrate the use of the overall mass balance. Most chemical engineering applications, whether in biotechnology, chemical or materials processing, or environmental control, involve a number of distinct mass species that might or might not react chemically with one another. In this chapter we will consider mass balances for multicomponent systems in which there is a single phase (i.e., we exclude immiscible oil-water systems, solid-liquid suspensions, etc.) and the component species are nonreactive. With this foundation we can go on to the far more interesting and relevant reactive and multiphase systems in subsequent chapters.

Well-Stirred Systems

We consider again the flow system shown in Figure 2.4, but we now presume that there are two components; for specificity we will take these to be dissolved table salt (NaCl) and water, but they could be any two completely miscible, nonreacting materials. The flow diagram is shown in Figure 4.1, where to each stream we associate a density and concentration. The concentration of the salt, in mass units (e.g., kg/m3), is denoted c, while subscripts f and e again denote feed and effluent streams, respectively. Only one mass concentration, together with the density, is required to define this binary system, since the mass concentration of water is simply ρ – c (total mass/unit volume less mass of salt/unit volume). Salt concentrations are easily measured; the most elementary way is to evaporate the water from a known volume and weigh the residual salt, but a better way is to measure the electrical conductivity, which can easily be correlated with the ionic concentration.

Introduction

Most systems of interest to chemical engineers are multicomponent, and a bit of reflection tells us that the internal energy of a multicomponent system must depend on the composition as well as on the temperature and pressure. We know, for example, that the temperature increases without adding any heat when sulfuric acid and water at the same temperature are mixed together. Writing energy balances for multicomponent systems is straightforward, but it is delicate and requires a bit of care. The engineering literature, including textbooks and basic handbooks, abounds with incorrect energy balances, often because of unwarranted shortcuts. I have published a brief catalog of incorrect energy balances elsewhere. Perhaps the most unsettling example on that list is a computer program offered for sale by a leading corporation to model a chemical reactor for converting coal to CO and H2. The most important consideration in operating such a reactor is getting the location and magnitude of the highest temperature (the hot spot) right, because too high a temperature or an incorrect location will affect the structural integrity of the reactor. The model predicted steady-state and transient profiles of solid- and gas-phase temperatures, coal conversion, and the concentrations of many gaseous species, but it contained an error that guaranteed that the hot spot would be computed incorrectly!

“Chemical engineering is the field of applied science that employs physical, chemical, and biochemical rate processes for the betterment of humanity.” This opening sentence of Chapter 1 has been the underlying paradigm of chemical engineering for at least a century, through the development of modern chemical and petrochemical, biochemical, and materials processing, and into the twenty-first century as chemical engineers have applied their skills to fundamental problems in pharmaceuticals, medical devices and drug-delivery systems, semiconductor manufacturing, nanoscale technology, renewable energy, environmental control, and so on. The role of the introductory course in chemical engineering is to develop a framework that enables the student to move effortlessly from basic science and mathematics courses into the engineering science and technology courses that form the core of a professional chemical engineering education, as well as to provide the student with a comprehensive overview of the scope and practice of the profession. An effective introductory course should therefore be constructed around the utilization of rate processes in a context that relates to actual practice.

Chemical engineering as an academic discipline has always suffered from the fact that the things that chemical engineers do as professionals are not easily demonstrated in a way that conveys understanding to the general public, or even to engineering students who are just starting to pursue their technical courses. (Every secondary school student can relate to robots, bridges, computers, or heart-lung machines, but how do you easily convey the beauty and societal importance of an optimally designed pharmaceutical process or the exponential cost of improved separation?)

Introduction

Chemical engineering design, operation, and discovery generally require the analysis of complex physicochemical processes. The quantitative treatment of such systems is frequently called modeling, which is a process by which we employ the principles of chemistry, biochemistry, and physics to obtain mathematical equations describing the process. These equations can then be manipulated to predict what will happen under given circumstances. Thus, if it is a chemical reactor that we are modeling, we will know, for example, the effect on the final product of changing the temperature at which the reactor operates. If it is an artificial kidney that we are modeling, we will know the time required for treatment in terms of the flow rate of the dialysis fluid. The analysis process is straightforward and systematic. In this chapter we will examine the approach, see how a model of one simple process unit can be obtained and applied, and get a preview of the things to look for in more complex situations.

The Analysis Process

The specific goals of analysis are as follows:

1. Describe the physical situation through equations (obtain the model).

2. Use the model equations to predict behavior.

3. Compare the prediction with the actual behavior of the real system.

4. Evaluate the limitations of the model, and revise if necessary.

5. Use the model for prediction and design.

The logical sequence of the analysis process is shown in Figure 2.1. This is a manifestation of what is often called the scientific method.

Introduction

In Chapter 2 we introduced the concept of a balance equation to account for the total mass in the control volume. Mass is a conserved quantity that is neither created nor destroyed, so the concept of a balance equation is straightforward. We can and often do write balance equations for quantities that are not conserved, and it is appropriate to digress briefly to consider this point. One important example of a quantity that is not conserved is the mass of a reactive species. Suppose, for example, we wish to model the distribution and metabolism of the anticancer drug methotrexate in the human body. Methotrexate is not conserved: It enters the body and then disappears because of metabolism. Nevertheless, we are able to write a balance equation for this chemical species. (Note that the number of atoms of each of the elements making up the drug is a conserved quantity.)

Most people gain experience in the use of balances through personal finance. Wealth, be it personal, national, or global, is not a conserved quantity. Nevertheless, we can and do account for wealth, typically through balancing a checkbook or analyzing monthly statements from the bank. In this chapter we will illustrate the application of balance equations to the problem of determining the true cost of future expenditures. This is a problem of inherent interest to most of the population, but the net present worth accounting principle outlined here is of particular relevance to engineers involved in project planning and design.

This book has one purpose: to help you understand vectors and tensors so that you can use them to solve problems. If you're like most students, you first encountered vectors when you took a course dealing with mechanics in high school or college. At that level, you almost certainly learned that vectors are mathematical representations of quantities that have both magnitude and direction, such as velocity and force. You may also have learned how to add vectors graphically and by using their components in the x-, y- and z-directions.

That's a fine place to start, but it turns out that such treatments only scratch the surface of the power of vectors. You can harness that power and make it work for you if you're willing to delve a bit deeper – to see vectors not just as objects with magnitude and direction, but rather as objects that behave in very predictable ways when viewed from different reference frames. That's because vectors are a subset of a larger class of objects called “tensors,” which most students encounter much later in their academic careers, and which have been called “the facts of the Universe.” It is no exaggeration to say that our understanding of the fundamental structure of the universe was changed forever when Albert Einstein succeeded in expressing his theory of gravity in terms of tensors.

If you were tracking the main ideas of Chapter 1, you should realize that vectors are representations of physical quantities – they're mathematical tools that help you visualize and describe a physical situation. In this chapter, you can read about a variety of ways to use those tools to solve problems. You've already seen how to add vectors and how to multiply vectors by a scalar (and why such operations are useful); this chapter contains many other “vector operations” through which you can combine and manipulate vectors. Some of these operations are simple and some are more complex, but each will prove useful in solving problems in physics and engineering. The first section of this chapter explains the simplest form of vector multiplication: the scalar product.

Scalar product

Why is it worth your time to understand the form of vector multiplication called the scalar or “dot” product? For one thing, forming the dot product between two vectors is very useful when you're trying to find the projection of one vector onto another. And why might you want to do that? Well, you may be interested in knowing how much work is done by a force acting on an object. The first instinct of many students is to think of work as “force times distance” (which is a reasonable starting point).

The previous chapter contains several ideas that are important to a full understanding of tensors. The first is that any vector may be represented by components that transform between coordinate systems in one of two ways. “Covariant” components transform in the same manner as the original basis vectors pointing along the coordinate axes, and “contravariant” components transform in the inverse manner of those basis vectors. The second main idea is that coordinate basis vectors are tangent to the coordinate axes, and that there also exist reciprocal or dual basis vectors that are perpendicular to the coordinate axes; these dual basis vectors transform inversely to the coordinate basis vectors. The third idea is that combining contravariant components with original basis vectors and combining covariant components with dual basis vectors produces a result that is invariant under coordinate transformation. That result is the vector itself, and the vector is the same no matter which coordinate system you use for its components.

This chapter extends the concepts of covariance and contravariance beyond vectors and makes it clear that scalars and vectors are members of the class of objects called “tensors.”

Definitions (advanced)

In the basic definitions of Chapter 1, scalars, vectors, and tensors were defined by the number of directions involved: zero for scalars, one for vectors, and more than one for tensors.