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.