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.

In Chapter 6 we discussed the fundamentals of turbulence and reviewed the mixing length and eddy diffusivity models. As was mentioned there, these classical models do not treat turbulence as a transported property, and as a result they are best applicable to equilibrium turbulent fields. In an equilibrium turbulent field at any particular location there is a balance among the generation, dissipation, and transported turbulent energy for the entire eddy size spectrum, and as a result turbulence characteristics at each point only depend on the local parameters at that point.

Our daily experience, however, shows that turbulence is in general a transported property, and turbulence generated at one location in a flow field affects the flow field downstream from that location. One can see this by simply disturbing the surface of a stream and noting that the vortices resulting from the disturbance move downstream.

In this chapter, turbulence models that treat turbulence as a transported property are discussed. Turbulence models based on Reynolds–averaged Navier–Stokes [(RANS)-type] models are first discussed. These models, as their title suggests, avoid the difficulty of dealing with turbulent fluctuations entirely. We then discuss two methods that actually attempt to resolve these turbulent fluctuations, either over the entire range of eddy sizes [direct numerical simulation (DNS) method] or over the range of eddies that are large enough to have nonuniversal behavior [largeeddy simulation (LES) method].