The logic underlying the statistical tests described in this book is simple. A statistical test produces a test statistic of which the distribution is known. What we want to know is whether the test statistic has a value that is extreme, so extreme that it is unlikely to be attributable to chance. In the traditional terminology, we pit a null-hypothesis, actually a straw man, that the test statistic does not have an extreme value, against an alternative hypothesis according to which its value is indeed extreme. Whether a test statistic has an extreme value is evaluated by calculating how far out it is in one of the tails of the distribution. Functions like pt (), pf (), and pchisq () tell us how far out we are in a tail by means of p-values, which assess what proportion of the population has even more extreme values. The smaller this proportion is, the more reason we have for surprise that our test statistic is as extreme as it actually is.

However, the fuzzy notion of what counts as extreme needs to be made more precise. It is generally assumed that a probability begins to count as extreme by the time it drops below 0.05. However, opinions differ with respect to how significance should be assessed.

One tradition holds that the researcher should begin by defining what counts as extreme, before gathering and analyzing data.

Portions of this chapter are taken from Introduction to Chemical Engineering Analysis by Russell and Denn (1972) and are used with permission.

In Chapter 2, a constitutive equation for reaction rate was introduced, and the experimental means of verifying it was discussed for some simple systems. The use of the verified reaction-rate expression in some introductory design problems was illustrated in Chapter 2. Chapter 3 expanded on the analysis of reactors presented in Chapter 2 by dealing with heat exchangers and showing how the analysis is carried out for systems with two control volumes. A constitutive rate expression for heat transfer was presented, and experiments to verify it were discussed.

This chapter considers the analysis of mass contactors, devices in which there are at least two phases and in which some species are transferred between the phases. The analysis will produce a set of equations for two control volumes just as it did for heat exchangers. The rate expression for mass transfer is similar to that for heat transfer; both have a term to account for the area between the two control volumes. In heat exchangers this area is determined by the geometry of the exchanger and is readily obtained. In a mass contactor this area is determined by multiphase fluid mechanics, and its estimation requires more effort. In mass contactors in which transfer occurs across a membrane the nominal area determination is readily done just as for heat exchangers, but the actual area for transfer may be less well defined.

Figure 1.2 presents the logic leading to technically feasible analysis and design. In this chapter we illustrate the design process that follows from the analysis of existing equipment, experiment, and the development of model equations capable of predicting equipment performance. Design requests can come in the form of memos, but an ongoing dialogue between those requesting a design and those carrying out the design helps to properly define the problem. This is difficult to illustrate in a textbook but we will try to give some sense of the process in the case studies presented here.

Technically feasible heat exchanger and mass contactor design procedures were outlined in Sections 3.5 and 4.5. In this chapter we present case studies to illustrate how one can proceed to a technically feasible design. Recall that such a design must satisfy only the design criteria, i.e., the volume of a reactor that will produce the required amount of product, the heat exchanger configuration that will meet the heat load needed with the utilities available, or the mass contactor that will transfer the required amount of material from one phase to another given the flow rate of the material to be processed. Even for relatively simple situations, design is always an iterative process and requires one to make decisions that cannot be verified until more information is available and additional calculations are made.

The coefficients of heat and mass transfer rate expressions depend on any fluid flows in the system. Our personal experience with “wind-chill” factors on chilly winter days and in dissolving sugar or instant coffee in hot liquids by stirring suggests that the rate of heat and mass transfer can be greatly increased with increasing wind speed or mixing rates. The technically feasible design of heat and mass transfer equipment requires calculating the transport coefficients and their variation with the fluid flows in the device, which depend intimately on the design of the device. For example, the area for heat transfer calculated for a tubular–tubular heat exchanger can be achieved by an infinite combination of pipe diameters, lengths, and for shell-and-tube exchanges, the number of tubes. However, selecting a pipe diameter for a given volumetric flow rate sets the fluid velocity in the pipe and the type of flow (i.e., laminar versus turbulent), which sets the overall heat transfer coefficient. This is why the design of heat and mass transfer equipment is often an iterative process. This chapter presents methods for estimating transport coefficients in systems with fluid motion.

The central hypothesis for flowing systems is that the friction, resistance to heat transfer, and resistance to mass transfer are predominately located in a thin boundary layer at the interface between the bulk flowing fluid and either another fluid (liquid or gas) or a solid surface.

Rate of Thermal Conduction

In Chapter 3 we presented model equations for heat exchangers with our mixed–mixed, mixed–plug, and plug–plug classifications. All these fluid motions generally require some degree of turbulence, and all heat exchangers, except for those for which there is direct contact between phases, require a solid surface dividing the two control volumes of the exchanger. To predict the overall heat transfer coefficient, denoted as U in the analyses in Part I, we must be able to determine how U is affected by the turbulent eddies in the fluids and the physical properties of the fluids and how the rate of heat transfer depends on the conduction of heat through the solid surface of the exchanger.

We begin our study of conductive transport by considering the transfer of heat in a uniform solid such as that employed as the boundary between the two control volumes of any exchanger. This requires a Level III analysis and verification of a constitutive equation for conduction. This is followed by a complementary analysis of molecular diffusion through solids and stagnant fluids.

Experimental Determination of Thermal Conductivity k and Verification of Fourier's Constitutive Equation

Consider an experiment whereby the heat flow through the wall between the tank and the jacket in Figure 3.7 is measured. For the purposes of this analysis, we consider the heat transfer to be essentially one dimensional in the y direction, with the barrier essentially infinite in the z–x plane.

This text is designed to teach you how to carry out quantitative analysis of physical phenomena important to chemical professionals. In the chemical engineering curriculum, this course is typically taught in the junior year. Students with adequate preparation in thermodynamics and reactor design should be successful at learning the material in this book. Students lacking a reactor design course, such as chemists and other professionals, will need to pay additional attention to the material in Chapter 2 and may need to carry out additional preparation by using the references contained in that chapter. This book uses the logic employed in the simple analysis of reacting systems for reactor design to develop the more complex analysis of mass and heat transfer systems.

Analysis is the process of developing a mathematical description (model) of a physical situation of interest, determining behavior of the model, comparing the behavior with data from experiment or other sources, and using the verified model for various practical purposes.

There are two parts in the analysis process that deserve special attention:

• developing the mathematical model, and

• comparing model behavior with data.

Our experience with teaching analysis for many years has shown that the model development step can be effectively taught by following well-developed logic. Just what constitutes agreement between model behavior and data is a much more complex matter and is part of the art of analysis.

In Part I of this text we developed the model equations for analyzing experiments and for the technically feasible design of laboratory-, pilot-, and commercial-scale processing equipment including reactors, heat exchangers, and mass contactors. Our organization in terms of the macroscale fluid motions in such equipment (Table 1.1) has broader applicability because many systems of interest in living organisms and in the natural environment can also be similarly analyzed.

The constitutive equations used in the model equations in Part I are summarized in Table 1.5. The overall heat transfer coefficient U and the mass transfer coefficient Km are engineering parameters defined by these constitutive equations. These transport coefficients depend on both the materials involved and the microscale and macroscale fluid motions of these materials, as well as their thermodynamic state (i.e., temperature and pressure). Our need to determine these parameters by experiment reflects our lack of understanding of the fluid mechanics affecting the transport of energy in a turbulent or laminar fluid to a solid surface, for example, or the transfer of a species at the interface between two phases with complex fluid motions. These boundary layers are critical regions at the fluid–fluid and fluid–solid interfaces where the dominant resistances to heat and mass transfer are located in flowing fluids. Transport coefficients deduced from analysis of existing equipment are accurate only if the model equations correctly describe the fluid motions in the experiment.

This book is designed to teach students how to become proficient in engineering analysis by studying mass and heat transfer, transport phenomena critical to chemical engineers and other chemical professionals. It is organized differently than traditional courses in mass and heat transfer in that more emphasis is placed on mass transfer and the importance of systematic analysis. The course in mass and heat transfer in the chemical engineering curriculum is typically taught in the junior year and is a prerequisite for the design course in the senior year and, in some curricula, also a prerequisite for a course in equilibrium stage design. An examination of most mass and heat transfer courses shows that the majority of the time is devoted to heat transfer and, in particular, conductive heat transfer in solids. This often leads to overemphasis of mathematical manipulation and solution of ordinary and partial differential equations at the expense of engineering analysis, which should stress the development of the model equations and study of model behavior. It has been the experience of the authors that the “traditional” approach to teaching undergraduate transport phenomena frequently neglects the more difficult problem of mass transfer, despite its being an area that is critical to chemical professionals.

At the University of Delaware, chemical engineering students take this course in mass and heat transfer the spring semester of their junior year, after having courses in thermodynamics, kinetics and reactor design, and fluid mechanics.

Hypothesis . The dramatic hypotheses (‘prefaces’ or ‘plot summaries’) which are preserved in the surviving medieval manuscripts and ancient papyri fall into three general categories (Zuntz (1955a) 129–52; cf. also van Rossum-Steenbeek (1998) 1–39; for the papyri, see Haslam (1975) 150–6, Diggle (2005)).

The first and most valuable are those derived from the Alexandrian edition of Aristophanes of Byzantium (cf. Pfeiffer (1968) 192–6, Mastronarde (1994) 168 n. 2). Although only fragments of these survive (occasionally combined with the two other types of hypothesis), we have enough material to be able to piece together the scope and typical features of Aristophanes’ introductory notes (for Eur., cf. Alc., Med., Hipp., Andr., Hec., Phoen., Or., Bacch., Rhes.). They combine basic points of scenography (e.g. where the play is set, the identity of the chorus and prologue speaker) with more scholarly information regarding such topics as the treatment of the myth by the other two tragedians, the date of the play, the titles of companion plays, and the results of the contest (Alc. second prize, Med. third prize, Hipp. first prize, Phoen. second prize).

The second type of hypothesis is uniquely Euripidean, stemming from a collection of Euripidean plot summaries, arranged alphabetically by the first letter of the title, which was composed in the first or second century ad To gain scholarly respectability, the collection was ascribed to Aristotle's fourth-century bc pupil Dicaearchus of Messene. These Tales from Euripides (as Zuntz (1982) 358 called them) were, however, written for a popular audience, and their narratives were intended to be simple summaries of the plot.

All physical situations of interest to engineers and scientists are complex enough that a mathematical model of some sort is essential to describe them in sufficient detail for useful analysis and interpretation. Mathematical expressions provide a common language so different disciplines can communicate among each other more effectively. Models are very critical to chemical engineers, chemists, biochemists, and other chemical professionals because most situations of interest are molecular in nature and take place in equipment that does not allow for direct observation. Experiments are needed to extract fundamental knowledge and to obtain critical information for the design and operation of equipment. To do this effectively, one must be able to quantitatively analyze mass, energy, and momentum transfer (transport phenomena) at some level of complexity. In this text we define six levels of complexity, which characterize the level of detail needed in model development. The various levels are summarized in Table 1.1.

Level I, Conservation of Mass and/or Energy. At this level of analysis the control volume is considered a black box. A control volume is some region of space, often a piece of equipment, that is designated for “accounting” purposes in analysis. Only the laws of conservation of mass and/or energy are applied to yield the model equations; there is no consideration of molecular or transport phenomena within the control volume. It is a valuable approach for the analysis of existing manmade or natural systems and is widely employed.

Chemical engineers educated in the undergraduate programs of departments of chemical engineering have received an education that has been proven highly effective. Chemical engineering educational programs have accomplished this by managing to teach a methodology for solving a wide range of problems. They first did so by using case studies from the chemical process industries. They began case studies in the early part of the 20th century by considering the complete processes for the manufacture of certain chemicals and how they were designed, operated, and controlled. This approach was made much more effective when it was recognized that all chemical processes contained elements that had the same characteristics, and the education was then organized around various unit operations. Great progress was made during the 1940s and 1950s in experimental studies that quantified the analysis and design of heat exchangers and equilibrium stage operations such as distillation. The 1960s saw the introduction of reaction and reactor analysis into the curriculum, which emphasized the critical relationship between experiment and mathematical modeling and use of the verified models for practical design. We have built upon this approach, coupled with the tools of transport phenomena, to develop this text.

Our approach to teaching mass and heat transfer has the following goals:

1. Teach students a methodology for rational, engineering analysis of problems in mass and heat transport, i.e., to develop model equations to describe mass and heat transfer based on the relationship between experimental data and model.

2. Using these model equations, teach students to design and interpret laboratory experiments in mass and heat transfer and then to effectively translate this knowledge to the operation and design of mass and heat transfer equipment.

3. […]