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.

The real value of understanding vectors and how to manipulate them becomes clear when you realize that your knowledge allows you to solve a variety of problems that would be much more difficult without vectors. In this chapter, you'll find detailed explanations of four such problems: a mass sliding down an inclined plane, an object moving along a curved path, a charged particle in an electric field, and a charged particle in a magnetic field. To solve these problems, you'll need many of the vector concepts and operations described in Chapters 1 and 2.

Mass on an inclined plane

Consider the delivery woman pushing a heavy box up the ramp to her delivery truck, as illustrated in Figure 3.1. In this situation, there are a number of forces acting on the box, so if you want to determine how the box will move, you need to know how to work with vectors. Specifically, to solve problems such as this, you can use vector addition to find the total force acting on the box, and then you can use Newton's Second Law to relate that total force to the acceleration of the box.

To understand how this works, imagine that the delivery woman slips off the side of the ramp, leaving the box free to slide down the ramp under the influence of gravity.

Introduction

Every commercial kitchen must operate using a sound basis for costing and control of raw materials such as food, cleaning, wages and overheads. The principles are the same whether the food service is run at a hospital, aged-care home, institution, take-away shop or restaurant. The capacity of a restaurant to make a profit, make a fair financial return on the investment and reward the effort involved is very much dependent upon effective control of the day-to-day running costs. Likewise, hospitals, aged-care facilities and other food services need to operate within strict financial budgets.

Effective cost control has been described as the identification and regulation of operating costs. These will be influenced by the style and systems of service to be offered. Food and labour costs make up the largest proportion of operating costs. In this chapter we examine food cost.

An acceptable budget for food is calculated by costing menus and recipes. This involves choosing suppliers carefully after reviewing all the options. Suppliers need to know the purchasing standards that are required for the food. These will describe quality, size, colour, fat content, age and origin (organic or halal, for example). Is the supplier able to offer continuity of supply at an acceptable price? Does the supplier stock a range of food items needed for the menu? Buying from a large number of suppliers makes it difficult to build relationships, takes time attending to deliveries and increases accounting costs.

Introduction

A well-organised kitchen starts with the preparation of basic ingredients. Every chef knows that it is the quality of the first two hours’ work that sets the standard for the food service that follows. Things left undone or food poorly prepared during this preparation time can give rise to declining standards of cooking, poor presentation of dishes, frayed tempers at the busiest time of the service and sometimes lengthy delays between courses.

Getting things ready is referred to as mise en place (‘putting in place’). You can tell a well-run, happy kitchen by the way it organises its mise en place. It does not matter whether it is located in a restaurant, hospital or cafeteria – just before the time for service, the bustle of the kitchen will die down for 5 or 10 minutes. During this quiet lull you will see the work benches clean and tidy, dishes of food arranged, seasoning boxes filled and saucepans gently steaming or the food in the bain-marie neatly laid out, ready for plating. The hot cupboards will be hot and neatly stacked with plates and special dishes. Cold plates and dishes will also be ready for the serving of cold food.

This chapter describes the work involved in this preparation. The menu and type of service will determine what ingredients are selected and how they are prepared. In a well-run kitchen, preparation for the next meal will be started as soon as the current meal service is concluded.