This extended chapter can be found on the website www.cambridge.org/nellisandklein. Mass transfer occurs whenever fluid flows; that is, some mass is transferred from one place to another. However, the focus in this chapter is on the transport of one chemical species (or component) within a mixture of chemical species that occurs as a direct result of a concentration gradient, independent of a pressure gradient. This type of mass transfer is called diffusion. Mass transfer, like momentum transfer, plays an important role in many important heat exchange processes and devices. For example, mass transfer is critical to the operation of cooling coils, cooling towers, and evaporative coolers and condensers that are commonly used in refrigeration and power systems. The energy transfer that occurs as a result of mass transfer can significantly improve the performance of these heat transfer devices. The processes of heat and mass transfer are analogous. The governing equations for heat and mass transfer are similar and therefore many of the relations and solution techniques that have been developed for heat transfer can be directly applied to mass transfer processes.

Chapter 9: Mass transfer

The website associated with this book www.cambridge.org/nellisandklein provides many more problems.

Introduction

Chapters 4 through 6 discuss convection involving single-phase fluids. The thermodynamic state of single-phase fluids is sufficiently far from their vapor dome so that even though temperature variations may be present, only one phase exists (vapor or liquid). In this chapter, two-phase convection processes are examined. Two-phase processes occur when the fluid is experiencing heat transfer near the vapor dome so that vapor and liquid are simultaneously present. If the fluid is being transformed from liquid to vapor through heat addition, then the process is referred to as boiling or evaporation. If vapor is being transformed to liquid by heat removal, then the process is referred to as condensation.

Chapter 6 showed that temperature-induced density variations in a single-phase fluid may have a substantial impact on a heat transfer problem because they drive buoyancy induced fluid motion. However, the temperature-induced density gradients that are present in a typical single-phase fluid are small and so the resulting buoyancy-induced fluid velocity is also small. As a result, the heat transfer coefficients that characterize natural convection processes are usually much lower than those encountered in forced convection processes. The density difference between a vapor and a liquid is typically quite large. For example, saturated liquid water at 1 atm has a density of 960 kg/m3 while saturated water vapor at 1 atm has a density of 0.60 kg/m3. Large differences in density lead to correspondingly large buoyancy-induced fluid velocities and heat transfer coefficients.

Internal flow concepts

Introduction

Chapter 4 discusses the behavior of the momentum and thermal boundary layers associated with an external flow. An external flow is broadly defined as one where the boundary layer can grow without bound; for the flat plate considered in Section 4.1, the boundary layer was never confined by the presence of another object. An internal flow is defined as one where the growth of the boundary layer is confined. Internal flows are often encountered in engineering applications (e.g., the flow through tubes or ducts) and this section discusses the qualitative behavior of internal flows. Many of the concepts that are discussed in Section 4.1 for an external flow can also be applied to internal flows in order to provide a physical understanding of their behavior.

Momentum considerations

Figure 5-1(a) illustrates laminar external flow over a plate and shows, qualitatively, the momentum boundary layer and velocity distribution that results. Figure 5-1(b) illustrates laminar flow through a passage that is formed between two parallel plates; notice that at some location, the momentum boundary layer becomes bounded.

The momentum boundary layers growing from the upper and lower plates in Figure 5-1(b) meet at some distance from the inlet; this distance is referred to as the hydrodynamic entry length, xfd, h . The momentum boundary layer thickness will remain constant as the fluid moves further down the flow passage (i.e., for x > xfd, h ).

Introduction to laminar boundary layers

Introduction

Chapters 1 through 3 consider conduction heat transfer in a stationary medium. Energy transport within the material of interest occurs entirely by conduction and is governed by Fourier's law. Convection is considered only as a boundary condition for the relatively simple ordinary or partial differential equations that govern conduction problems. Convection is the transfer of energy in a moving medium, most often a liquid or gas flowing through a duct or over an object. The transfer of energy in a flowing fluid is not only due to conduction (i.e., the interactions between micro-scale energy carriers) but also due to the enthalpy carried by the macro-scale flow. Enthalpy is the sum of the internal energy of the fluid and the product of its pressure and volume. The pressure-volume product is related to the work required to move the fluid across a boundary. You were likely introduced to this term in a thermodynamics course in the context of an energy balance on a system that includes flow across its boundary. The additional terms in the energy balance related to the fluid flow complicate convection problems substantially and link the heat transfer problem with an underlying fluid dynamics problem. The complete solution to many convection problems therefore requires sophisticated computational fluid dynamic (CFD) tools that are beyond the scope of this book.

Introduction to radiation

Radiation

From a thermodynamic perspective, thermal energy can be transferred across a boundary (i.e., heat transfer can occur) by only two mechanisms: conduction and radiation. Conduction is the process in which energy exchange occurs due to the interactions of molecular (or smaller) scale energy carriers within a material. The conduction process is intuitive; it is easy to imagine energy carriers having a higher level of energy (represented by their temperature) colliding with neighboring particles and thereby transferring some of their energy to them. Convection is the process in which the surface of a solid material exchanges thermal energy with a fluid. Although convection is commonly treated as a separate heat transfer mechanism, it is more properly viewed as conduction in a substance that is also undergoing motion. The energy transfer by conduction and fluid motion are coupled, making convection problems more difficult to solve than conduction problems. However, convection is still an intuitive process since it can be explained by interactions between neighboring molecules with different energy levels. Radiation is a very different heat transfer process because energy is transferred without the benefit of any molecular interactions. Indeed, radiation energy exchange can occur over long distances through a complete vacuum. For example, the energy that our planet receives from the sun is a result of radiation exchange. The process of radiation heat transfer is not intuitive to most engineers.

Introduction

A thin plate is a structural element, just as a long beam is a structural element. The thin plate also is characterized by its special geometry. While a long beam has two dimensions very much shorter than the length dimension, a thin plate, as is pictured in Fig. 1.11, has one dimension that is very much less than the other two dimensions. The thin plate's least dimension is, of course, called the plate thickness. The thickness is usually a constant in most vehicular structures, but the possibility of a gradually changing thickness can be incorporated into thin plate bending theory (Ref. [16]). The difference between a thin plate and a membrane is that a thin plate can resist both stretching and bending, while a membrane can only resist lateral or inplane loads by stretching.

There are two goals for this chapter. The first and lesser goal is to develop the equations of classical thin plate bending theory. More than an introductory discussion of the extensive body of classical thin plate bending solutions is outside the purposes of this textbook. The second and more important goal is to develop the equations necessary to describe a simple plate bending finite element comparable to the previously developed beam bending finite element. The greater importance assigned to the second, rather limited, goal not only suggests the present relative importance of the two topics but also suggests that certain simplifications like constant plate thickness are appropriate for the thin plate theory to be presented here.

Introduction

This chapter examines the standard engineering approaches to beam twisting used with those types of beam cross-sections that are commonly used in vehicular structures. These analytical approaches are far simpler than the theory of elasticity approach of Chapter 12, which requires the satisfaction of a second order partial differential equation and, in general, the satisfaction of a simple boundary condition on a difficult boundary. The basis of these approaches is the division of engineering beam cross-sections into two categories. The first category is that of thin-walled “open cross-sections” and the second category is that of thin walled “closed cross-sections.” A closed cross-section encloses one or more voids. A thin-walled pipe, or better yet, a thin-walled box beam built up from two oppositely facing channel cross-sections (i.e.,] [or []) connected by top and bottom plates so as to produce a rectangular or roughly rectangular interior void, are examples of closed cross-sections. On the other hand, an open cross-section does not have any interior voids. A single channel beam, or an H or I beam are examples of open cross-section beams.

Thin open section beams are generally more efficient than equally strong closed section beams when bending moments and shearing forces are the only significant loadings. That is, in those circumstances, thin open section beams generally weigh less than thin closed section beams. Open cross-section beams also have the advantage that they are easier to connect to other structural components, and inspect for damage.

Introduction

Chapter 5 pointed out that the mechanical and thermal response of engineering structural materials is quite complex. Nevertheless, if a uniaxial stress value lies within the bounds of the compressive and tensile elastic/proportional limits, that complex behavior becomes relatively simple. Within those two limits, or as an approximation, slight extensions of those limits to the limits of the compressive and tensile yield stresses, there is very nearly a straight-line relation between stress and strain. Moreover, that straight-line relation is very nearly the same for both loading and unloading. That is, permanent set (plastic deformation) is negligible, and the relation between stress and strain is single-valued. In other words, the stress–strain relation no longer depends upon the previous load history. Under these circumstances materials are called Hookean, or linearly elastic. In summary, on the basis of extensive experimental evidence, engineering structural materials within the yield stress limits can be and are described by a mathematical model wherein the plot of stress versus strain is exactly a straight line, and this same straight line serves the dual purpose of being a loading line and an unloading line. Thus in this chapter the focus shifts from the complicated behavior of actual engineering materials to the justifications and implications of the relatively simple linearly elastic material model.

There are three important reasons to justify restricting almost all further studies within this text to stresses whose values are limited to being within the yield stress bounds of nearly linearly elastic behavior.

Introduction

Chapter 7 illustrated the direct approach to solving selected problems in structural mechanics. In that chapter solutions for structural displacements and stresses were obtained through the separate use of the four sets of equations that constitute the theory of elasticity. There were no major difficulties in solving the example problems of that chapter because the original three-dimensional problems were reduced to one-dimensional problems (i.e., problems involving only one independent spatial variable) by means of plane stress and symmetry concepts. More challenging problems are the theory of elasticity problems that involve two independent spatial variables, such as plane stress problems in general. The purpose of this chapter is to present two of the simplest of such problems and their solutions in order for the reader to obtain some familiarity with their characteristics. The selected example and exercise problem solutions of this chapter are sufficiently valuable to be referenced in succeeding chapters as proof of the accuracy of the approximations adopted in those chapters.

While the followinge theory of elasticity solutions provide valuable practice in all the essentials of structural engineering analysis, it is also important to know that from the point of view of everyday structural engineering practice, theory of elasticity solutions are rarely, if ever, referenced. The advent of modern digital computers and the development of modern numerical methods, particularly the finite element method that is explained in Part V of this textbook, have relegated theory of elasticity solutions to the role of mere curiosities.

Introduction

The use of Eq. (9.8), the strength of materials solution for the bending and extensional axial stress σxx(x, y, z) in a long, straight beam, requires a knowledge of the internal axial force N(x) and the internal bending moments My(x) and Mz(x). The difficulty is that a free body diagram (FBD) of a beam under study is often insufficient by itself to determine the axial force or bending moments at any point along the beam x-axis. As previously mentioned, whenever the equilibrium equations alone are insufficient to determine the internal stress resultants, the structure is called indeterminate. Most beams, or beam elements, that are parts of aerospace structures are indeterminate because these beams are mostly elements of beam grids and frames with, as much as possible, rigid connections. The grids and frames are often covered by thin sheeting, referred to as the vehicle skin. Thus there are many internal unknown reations. One purpose of this type of construction is to make the structure, and hence individual beams, as stiff as possible within the constraint of least weight. Increased stiffness has many advantages in a beam or a structure. For example, the stiffer the beam or structure, the higher the load required to buckle the beam or structure, and the lesser the chance of aeroelastic instabilities such as those discussed in Chapter 9. The stiffer the beam or structure, the higher its natural frequencies and the lesser the chance that gusts or control motions will stress the structure.

Introduction

The Euler beam buckling type of elastic instability, where an entire beam axis moves laterally, is discussed in Sections 11.6 and 11.7. Plate buckling and local flange and web type beam buckling are discussed in Section 22.8. These previous elastic instability discussions centered upon the solution of appropriate differential equations. In the first part of this chapter the focus is upon the use of the finite element method to calculate elastic buckling loads. The use of the finite element method makes practical the elastic buckling analysis of structures as opposed to the buckling analysis of one or two isolated structural elements. These FEM beam buckling solutions have the same limits of applicability as the Euler beam buckling solutions.

The second part of this chapter examines certain instabilities of structures that arise from fluid–structure interactions. These instabilities have much in common with the purely elastic instabilities. Whereas the object of an elastic instability analysis is to discover the critical magnitude of a particular type of load that will cause a sudden and sizable lateral deflection of the elastic structure, the object of the aeroelastic analysis is to discover the critical value of the airspeed (or Reynolds number or Mach number) beyond which the airloads cause an ever increasing deflection or vibration amplitude. A representative static instability (divergence) and a representative dynamic instability (wing flutter) are studied in some detail. Again the FEM is useful for describing the linear elastic properties of the structure being studied.

An Overview of Part I

Vehicular weight, particularly that of aircraft and spacecraft, has a strong effect on the performance or economics of all such vehicles. Thus it is well worth spending many engineering man-hours on their design and analysis so as to make those vehicles as light-weight as possible. To make those many engineering hours of analysis as effective as possible, it is important that all the different types of loads that the vehicle will bear be well estimated, and then the structural response to those loads be carefully calculated. To carefully calculate the response of structures to estimated or measured loadings, it is important to use structural analysis techniques to which considerable confidence can be assigned. High degrees of confidence are achieved through experience and through thorough understanding of any approximations that are incorporated within the derivations of the selected structural analysis techniques. Thus it would seem that, in general, the fewer and the smaller the approximations, the more useful the structural analysis technique. This surmise is only partially true. As will be seen as the material of this textbook unfolds, the use of structural analysis techniques that contain essentially no approximations for many circumstances can be much too expensive and time consuming. Hence a compromise between cost and accuracy is necessary for good engineering practice. To understand how that compromise is found, this introduction to aerospace structures begins with the fundamentals of structural mechanics where the approximations are few in number and small in impact.

Introduction

Chapters 1–6 developed (i) the general equilibrium equations from a free body diagram (FBD) of a differential rectangular parallelepiped taken from a structural body of any shape and material; (ii) the strain–displacement equations and the equivalent compatibility equations from the geometry of the deformations of the same parallelepiped; and finally (iii) the constitutive equations for the isotropic and orthotropic linearly elastic material models. Again, these three sets of equations, which apply over the interior of the structural body, are called the field or domain equations. The Cauchy equations, which relate the tractions and the stresses at the boundary, were also discussed at length, and mention was made of prescribed displacement equations that specify displacements at the boundary. Again, the Cauchy and the prescribed displacement equations are collectively called the boundary condition equations. Together, the three sets of field equations and the boundary condition equations form the four sets of equations that are the basis for what is called the theory of elasticity. This chapter demonstrates what can be done with these four sets of equations.

There are six unknown stresses, six unknown strains, and three unknown displacements throughout the domain (interior) of the structural body of interest. There are three equilibrium equations, six stress–strain equations, and six strain–displacement equations. Thus there are a total of 15 unknown stresses, strains, and displacements, and 15 independent equations relating those quantities.

Introduction

This chapter introduces three topics that expand the usefulness of the Bernoulli–Euler beam bending and extension equations developed in the previous chapter. The first topic is elastic beam end supports. The use of elastic end supports begins the process, developed further in Part V, of modeling beams that are parts (elements) of larger elastic structures. The second topic is partial span distributed loads, and concentrated loads acting at points other than the beam ends. Then, both as another form of loading, and as a prelude to the third topic, combined lateral and axial loading cases are also examined. The third topic is beam buckling. This chapter provides only a brief introduction to beam buckling theory. However, some of the complexities of the topic are mentioned without being explored mathematically. The mathematical differences between the one standard type of buckling analysis introduced here and all the other beam analyses of this chapter and Chapter 10 are underscored. Additional aspects of beam and plate buckling theory are provided in Part VI.

Before proceeding to these three topics, it is worthwhile mentioning again a limitation on the scope of the beam bending theory developed in Chapter 10 which is retained in this chapter. That limitation is that the bending deflections are small. Thus it is possible to confine the axial and bending interactions to the bending equations, and to deal with the bending and twisting deflections separately, without regard for any interaction between them.