Introduction

The use of a separate chapter for truss problems is prompted by the convenience of using the simpler bar element, as opposed to the beam element, as a means of illustrating the following topics that are less vital to an overall understanding of the basics of the FEM: (i) bars (or beams, or whatever) that are neither horizontally nor vertically oriented so that the element DOF are rotated relative to the global DOF; (ii) the equivalent loads that arise from enforced boundary deflections; and (iii) the equivalent loads that arise as a result of temperature changes or other initial strains. For the sake of simplicity, the developments of this chapter will be kept within the confines of planar trusses. The only slightly more complicated geometry of three-dimensional structures, which is available in all major commercial programs, is not necessary to the development of the reader's understanding of the concepts to be explored here. However, the bar element in three dimensions is adequately introduced in Exercise 18.10(b).

Bar elements do not differ much from beam elements. A distinction is usually made in a commercial FEM program between a beam and a bar in order to use the simpler bar element formulation wherever possible. It can be expected that any such program offers a beam element that includes the axial deformation properties of a bar. That is, there is always available a beam element with six DOF at each beam end, for a total of 12 DOF.

Introduction

In the course of a career, an engineer can expect to be called upon to deal with the behavior of a great many different types of materials from which an object with a structural purpose is to be, or has been, fashioned. Metals, plastics, woods, and man-made composites of all kinds are plentiful in structural engineering practice. The paragraphs below provide a brief overview of these materials, starting with the metals.

The material that is used most extensively in modern engineering practice is steel. In comparison with other metals and composites, steels are cheap, and have the advantages of being stiff, strong, and hard. Thus steels are often the material of choice for vehicular applications such as automobile frames and aircraft landing gear. The superior high-temperature properties of certain alloy steels often results in their choice as a material for the structure of fuel-burning engines such as rockets. The principal drawback of steel for vehicular use is its relatively high weight density of approximately 0.28–0.29 lb/in3. There is presently a great variety of types of steel and steel alloys (Refs. [10–14]). There is also a great variety of designations of steels. Lists of specification equivalents can be found, for example, in Ref. [13]. The American Iron and Steel Institute (AISI) and the Society of Automotive Engineers (SAE) have agreed upon an AISI–SAE four-digit numbering system for steels that is widely, but not exclusively, used. It is outlined on p. 143 of Ref. [10].

Introduction

The finite element method introduced in this chapter is the routine choice for the analysis of structures in government and industry, large companies and small. The finite element method is especially useful in the aerospace sciences and all related fields. For engineering applications, the finite element method was introduced to the engineering community by aerospace structural dynamicists in Ref. [49]. It is not the only useful, or necessary-to-know, method (one other method requires discussion), but it dominates because it is the only method suitable for large, complicated structures such as airplanes, helicopters, ships, and land vehicles, as well as the small redundant structures discussed here. It is particularly broad in scope. As a mathematical concept the finite element method is applicable to a wide variety of problems including such diverse problems as those of fluid dynamics, heat transfer, electromagnetic fields, and electrical circuits. As a mathematical concept, it is an adaptation of the Rayleigh–Ritz method. In this adaptation, instead of the requirement for choosing approximation functions for each new analysis, a finite element analysis can be computer programed because the same functions are repeatedly used for the same type of structural element. Instead of using the complicated functions that are sometimes necessary to meet boundary condition requirements of complicated structures, this adaptation uses simple functions, almost exclusively polynomial terms for quick computer processing.

Introduction

Routine analyses of structures whose mathematical models contain more than two or three structural elements (e.g., beams, plates, springs) are accomplished today using one or another of the commercially available, large or small capacity, standard structural analysis digital computer software packages. The larger commercially available structural analysis programs have become so inclusive, that few types of analyses are beyond their reach. In addition to producing what is called a numerical analysis of the structure's mathematical model, such software packages, perhaps in combination with auxiliary programs, are also commonly used in creating a structure's mathematical model, and in simplifying the interpretation of the analysis results. The basis for the large majority, if not all, of these general use software packages is a numerical analysis technique called the finite element method, which is introduced in Chapter 17.

The essence of any numerical analysis is the replacement of a collection of partial or ordinary differential equations (or sometimes integral equations) by a much larger set of simultaneous, algebraic equations. The set of algebraic equations involve an equal number of algebraic variables that, in one sense or another, are used to approximate the continuous, unknown functions of the differential equations being replaced by the algebaric equations. In almost every case, the differential equations are those obtained from strength of materials. If the differential equations are nonlinear, so too are the algebraic equations.

Introduction

While the finite element method, with occasional assistance from the unit load method or some such application of the principle of complementary virtual work, is a completely satisfactory approach to all linear structural analysis tasks, there are still many topics that need to be touched upon in this introductory textbook in order to provide a reasonably complete overview of aerospace structural analysis. The first of these topics is the mechanics of two-dimensional structural elements other than merely the equations for the out-of-plane stretching of thin membranes and the finite elements that can be used to determine the in-plane deflections and stresses of thin skins. Chapter 22 considers the strength of materials approach to thin plate bending and buckling. Again, the objective is to provide a sufficient understanding of the mechanics and the mathematics of thin plate bending so that a simple, but representative, thin plate finite element can be introduced. The next step in complexity after thin plates is thin shells. A treatment of thin shells of general geometry is complicated. Hence the analysis of thin shells is left entirely to textbooks that are specific to that topic.

Chapter 23 returns to the essential topic of elastic stability by discussing a simple finite element for beam buckling. Just as any finite element analysis is much superior to a differential equation–based analysis in its ability to address geometric and material complexities, a beam buckling finite element greatly expands the range of beam and beam frame buckling problems that can be so analyzed.

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.