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.

This chapter provides examples of how to apply the tensor concepts contained in Chapters 4 and 5, just as Chapter 3 provided examples of how to apply the vector concepts presented in Chapters 1 and 2. As in Chapter 3, the intent for this chapter is to include more detail about a small number of selected applications than can be included in the chapters in which tensor concepts are first presented.

The examples in this chapter come from the fields of Mechanics, Electromagnetics, and General Relativity. Of course, there's no way to comprehensively cover any significant portion of those fields in one chapter; these examples were chosen only to serve as representatives of the types of tensor application you're likely to encounter in those fields.

The inertia tensor

A very useful way to think of mass is this: mass is the characteristic of matter that resists acceleration. This means that it takes a force to change the velocity of any object with mass. You may find it helpful to think of moment of inertia as the rotational analog of mass. That is, moment of inertia is the characteristic of matter that resists angular acceleration, so it takes a torque to change the angular velocity of an object.

Many students find that rotational motion is easier to understand by keeping the relationships between translational and rotational quantities in mind.

The vector concepts and techniques described in the previous chapters are important for two reasons: they allow you to solve a wide range of problems in physics and engineering, and they provide a foundation on which you can build an understanding of tensors (the “facts of the universe”). To achieve that understanding, you'll have to move beyond the simple definition of vectors as objects with magnitude and direction. Instead, you'll have to think of vectors as objects with components that transform between coordinate systems in specific and predictable ways. It's also important for you to realize that vectors can have more than one kind of component, and that those different types of component are defined by their behavior under coordinate transformations.

So this chapter is largely about the different types of vector component, and those components will be a lot easier to understand if you have a solid foundation in the mathematics of coordinate-system transformation.

Coordinate-system transformations

In taking the step from vectors to tensors, a good place to begin is to consider this question: “What happens to a vector when you change the coordinate system in which you're representing that vector?” The short answer is that nothing at all happens to the vector itself, but the vector's components may be different in the new coordinate system. The purpose of this section is to help you understand how those components change.