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.

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. […]

A substantial portion of this chapter is taken from Introduction to Chemical Engineering Analysis by Russell and Denn (1972) and is used with permission.

This short chapter is a review of the analysis of simple reacting systems for chemical engineers. It is also designed to teach the fundamentals of analysis to other chemical professionals.

Reactor analysis is the most straightforward issue that chemical professionals encounter because rates of reaction can be obtained experimentally. The analysis of experimental data for reacting systems with mathematical models and the subsequent use of the verified model equations for design provide a template for the analysis of mass and heat transfer. We begin with simple reacting systems because the laboratory-scale experiments enable determination of reaction-rate constants that, with reasonable assumptions, can be used in model equations for design and operation of pilot- or commercial-scale reactors.

The same general principles apply to the design of mass contactors and heat exchangers, although it is more difficult to get the necessary values for mass and heat transfer coefficients from experiment. As we will see, mass transfer analysis is further complicated by the need to determine interfacial areas.

All chemical reaction and reactor analysis begins with experiment. Most experiments are carried out in batch equipment in a laboratory, and efforts are made to ensure the vessel is well mixed. In batch experiments, the concentrations of critical reacting species and products are measured over a given time period, sometimes until equilibrium is reached.

In our study of mass transfer, we noted that both technically feasible analysis and design were complicated by the need to estimate two parameters in the rate expression: the mass transfer coefficient (Km) and the interfacial area (a). In Chapter 6 theoretical and correlative methods for estimating mass transfer coefficients were discussed for those cases in which transfer of some species was limited by a well-defined boundary layer, such as at a solid or liquid surface. Correlations were identified relating the Sherwood number to the Reynolds and Schmidt numbers, with the functionality specific to the geometry and type of flow (i.e., laminar or turbulent). Complications arise in applying these developments when estimating Km in liquid–liquid or gas–liquid mass contactors. We need estimates of bubble or drop size and some knowledge of the fluid motions in the vicinity of the bubbles or drops to calculate the Reynolds and Sherwood numbers. Furthermore, as shown in Section 6.5, resistances to mass transfer may occur in both phases, necessitating knowledge of the fluid motions inside the bubbles or drops. In this chapter we examine methods for estimating these quantities. This is an active research area requiring Level VI two-phase fluid motion modeling and experiment and is not understood nearly as well as its single-phase counterpart. For this reason, most handbooks and textbooks alike provide only empirical correlations of the product Kma specific to particular process equipment and do not address the problem of interfacial area determination independently from the prediction of mass transfer coefficients.

Heat exchanger analysis follows the same procedures as chemical reactor analysis with the added complication that we need to deal with two control volumes separated by a barrier of area a. As shown in Table 3.1 we need to consider both tank type and tubular systems and allow for mixed–mixed, mixed–plug, or plug–plug fluid motions.

A heat exchanger is any device in which energy in the form of heat is transferred. The types of heat exchangers in which we will be most interested are devices in which heat is transferred from a fluid at one temperature to another at a different temperature. This is most commonly done by the confinement of both fluids in some geometry in which they are separated by a conductive material. In such devices the area available for heat transfer is set by the device type and size, and is often the object of a design calculation. If the two fluids are immiscible it is possible to exchange heat by direct contact of one fluid with the other, but such direct contact heat exchange is somewhat unusual. It is, however, the configuration most commonly employed to transfer mass between two fluids, and as such, will be treated in detail in Chapter 4. Note that the area for heat (and mass) transfer is much more difficult to measure or calculate in direct fluid–fluid contacting.

A simple heat exchanger, easily constructed, is one in which the fluids are pumped through two pipes, one inside the other.