When I was a very young boy I was enchanted by airplanes. The very idea that such a machine, with no apparent motions of its own – except, of course, for that tiny rotating thing at the front – could fly through the air was amazing. I could see that birds and insects could all fly with great dexterity, but that was because they could flap their wings, and thus support their weight as well as maneuver. And fish could even “fly” through water by motions of their body. How exciting it was then, to begin to learn something about how objects interact with the fluids surrounding them, and the useful consequences of those flows. That ultimately led, of course, to the broad study of fluid dynamics, with all of its wonderful manifestations.

There is hardly a single aspect of our daily lives, and indeed even of the entire universe in which we live, that is not in some way governed or described by fluid dynamics – from the locomotion of marine animals to the birth and death of distant galaxies. As a major field of technical and scientific knowledge, there are vast bodies of literature devoted to almost every facet of fluid behavior: laminar and turbulent flows, discontinuous (separated) flows, vortex flows, internal waves, free surface waves, compressible fluids and shock waves, multi-phase flows, and many, many others. With such a countless array of fluid phenomenon before us, what then leads to the focus of the present work?

Introduction

The dynamic analysis of a cylindrical shell experiencing elastic deformation that is comparable to its wall thickness cannot be described within the framework of the linear theory. The same is applied if the liquid free-surface amplitude is relatively large. In both cases, nonlinear analysis should be carried out. The presence of nonlinearities may result in nonlinear resonance conditions that cause complex response characteristics. One of the main difficulties in nonlinear problems of shell–liquid systems is that the boundary conditions are essentially nonlinear. This is in addition to the fact that the strain state of an elastic shell and the shape of the liquid free surface are not known a priori. The treatment of the nonlinear interaction of a liquid–shell system is a nonclassical boundary-value problem and relies on mechanics of deformable solids, fluid dynamics, and nonlinear mechanics.

With reference to nonlinear vibrations of cylindrical shells in vacuo, the literature is very rich and reports some controversies regarding the influence of nonlinearities on the shell dynamic behavior. The main results have been reviewed by Vol'mir (1972, 1979), Leissa (1973), Evensen (1974), Kubenko, et al. (1984), Amiro and Prokopenko (1997), and Amabili, et al. (1998b). Some attempts have been made to reconcile the reported discrepancies (see, e.g., Dowell, 1998, Evensen, 1999, and Amabili, et al. 1999c). It is believed that Reissner (1955) made the first attempt to study the influence of large-amplitude vibration for simply supported shells.

Sloshing means any motion of the free liquid surface inside its container. It is caused by any disturbance to partially filled liquid containers. Depending on the type of disturbance and container shape, the free liquid surface can experience different types of motion including simple planar, nonplanar, rotational, irregular beating, symmetric, asymmetric, quasi-periodic and chaotic. When interacting with its elastic container, or its support structure, the free liquid surface can exhibit fascinating types of motion in the form of energy exchange between interacting modes. Modulated free surface occurs when the free-liquid-surface motion interacts with the elastic support structural dynamics in the neighborhood of internal resonance conditions. Under low gravity field, the surface tension is dominant and the liquid may be oriented randomly within the tank depending essentially upon the wetting characteristics of the tank wall.

The basic problem of liquid sloshing involves the estimation of hydrodynamic pressure distribution, forces, moments and natural frequencies of the free-liquid surface. These parameters have a direct effect on the dynamic stability and performance of moving containers.

Generally, the hydrodynamic pressure of liquids in moving rigid containers has two distinct components. One component is directly proportional to the acceleration of the tank. This component is caused by the part of the fluid moving with the same tank velocity. The second is known as “convective” pressure and represents the free-surface-liquid motion. Mechanical models such as mass-spring-dashpot or pendulum systems are usually used to model the sloshing part.

Introduction

The problem of dynamic interaction of liquid sloshing with elastic structures may fall under one of the following categories:

1. Interaction of liquid sloshing dynamics with the container elastic modes in breathing and bending. This type is addressed in this chapter and Chapter 9.

2. Interaction of liquid sloshing dynamics with the supporting elastic structure. This type is treated in Chapter 10.

3. Liquid interaction with immersed elastic structures. This class will not be addressed in this book and the reader may consult Chen (1987), Paidoussis (1998) and Dzyuba and Kubenko (2002).

This chapter presents the linear problem of liquid interaction with its elastic container. Two limiting cases may occur where interaction disappears. The first case deals with the excitation of liquid surface modes where significant elastic modes of the container are not participating. In this case, the analysis of liquid dynamics in a rigid container will provide a satisfactory description of the overall behavior. The second case deals with the excitation of the container elastic modes where significant liquid motion does not occur. In this case, the presence of liquid will contribute to the mass distributed to the tank walls, and the analysis can be carried out without considering any interaction with liquid sloshing dynamics.

The first step in studying the interaction of liquid dynamics with elastic tank dynamics is to consider the linear eigenvalue problem and response to external excitations. The coupling may take place between the liquid-free-surface dynamics and with either the tank bending oscillations or breathing modes (or shell modes).

In previous chapters we have introduced many quantities, and we have developed many relations among those many quantities. We use this chapter to summarize the most important of those relations and to show you that we have consistently used a single approach in developing those relations. We start in § 6.1 by reminding you of the subtle distinctions between system states and constraints on interactions that may be in force when we change a state. Constraints are usually imposed in terms of measurables; for example, constant temperature or constant volume or no heat transfer. But such constraints can have profound effects on conceptuals and, in particular, on our choices for the most useful and economical expressions for relating measurables to conceptuals.

At this point we have developed two principal ways for relating conceptuals to measurables: one based on the ideal gas (Chapter 4) and the other based on the ideal solution (Chapter 5). Both routes use the same strategy—determine deviations from a well-defined ideality—with the deviations computed either as differences or as ratios. Since both routes are based on the same underlying strategy, a certain amount of symmetry pertains to the two; for example, the forms for the difference measures—the residual properties and excess properties—are functionally analogous.

We use § 6.2 to emphasize the symmetries that exist among difference measures and among ratio measures. Difference measures are commonly used to compute thermodynamic properties of single homogeneous phases, while ratio measures are most often used in phase and reaction equilibrium calculations.

In this chapter we review elementary concepts that are used to describe Nature. These concepts are so basic that we call them primitives, for everything in later chapters builds on these ideas. You have probably encountered this material before, but our presentation may be new to you. The chapter is divided into primitive things (§ 1.1), primitive quantities (§ 1.2), primitive changes (§ 1.3), and primitive analyses (§ 1.4).

PRIMITIVE THINGS

Every thermodynamic analysis focuses on a system—what you're talking about. The system occupies a definite region in space: it may be composed of one homogeneous phase or many disparate parts. When we start an analysis, we must properly and explicitly identify the system; otherwise, our analysis will be vague and perhaps misleading. In some situations there is only one correct identification of the system; in other situations, several correct choices are possible, but some may simplify an analysis more than others.

A system can be described at either of two levels: a macroscopic description pertains to a system sufficiently large to be perceived by human senses; a microscopic description pertains to individual molecules and how those molecules interact with one another. Thermodynamics applies to macroscopic entities; nevertheless, we will occasionally appeal to microscopic descriptions to interpret macroscopic phenomena. Both levels contain primitive things.

Macroscopic Things

Beyond the system lies the rest of the universe, which we call the surroundings. Actually, the surroundings include only that part of the universe close enough to affect the system in some way.

Much of thermodynamics concerns the causes and consequences of changing the state of a system. For example, you may be confronted with a polymerization process that converts esters to polyesters for the textile industry, or you may need a process that removes heat from a chemical reactor to control the reaction temperature and thereby control the rate of reaction. You may need a process that pressurizes a petroleum feed to a flash distillation unit, or you may need a process that recycles plastic bottles into garbage bags. In these and a multitude of other such situations, a system is to be subjected to a process that converts an initial state into some final state.

Changes of state are achieved by processes that force the system and its surroundings to exchange material or energy or both. Energy may be exchanged directly as heat and work; energy is also carried by any material that enters or leaves a system. A change of state may involve not only changes in measurables, such as T and P, but it may also involve phase changes and chemical reactions. To design and operate such processes we must be able to predict and control material and energy transfers.

Thermodynamics helps us determine energy transfers that accompany a change of state. To compute those energetic effects, we can choose from two basic strategies, as illustrated in Figure 2.1. In the first strategy we directly compute the heat and work that accompany a process.

Multiphase systems and chemical reactions pervade the chemical processing industries. For example, we routinely force the creation of a new phase to exploit the accompanying change in composition; consequently, phase changes are used in many separation processes, including distillation, crystallization, and solvent extraction. In addition, we routinely suppress the creation of a new phase, for example, to maintain inventory of liquids by controlling loss due to evaporation and to meet health and safety standards by controlling evaporation of flammable, hazardous, and toxic substances. Likewise, we often promote chemical reactions to convert inexpensive raw materials into valuable products. But we also try to prevent other reactions that convert valuable materials into costly wastes, and we try to prevent reactions that convert benign substances into hazardous or toxic chemicals. In all such situations, the design and operation of appropriate processes may hinge upon computing proper solutions to phase-equilibrium problems or reaction-equilibrium problems or both.

In previous chapters we developed the thermodynamics of phase and reaction equilibria, and we illustrated certain principles using straight forward computational procedures. We used only simple procedures so as not to detract from thermodynamic issues. In this chapter we consider more complex situations and therefore give more attention to computational techniques. No new thermodynamics is introduced in this chapter; instead, we try to show how the thermodynamics already developed can be used in multicomponent phase and reaction-equilibrium situations.

You are part of a development group assigned to determine the properties and phase behavior of certain mixtures that are to be used in a new process for your company. Your supervisor is relying on the group to provide a quick and thorough assessment of the proposed process: each day of production delay costs the company one million dollars.

You begin by asking how the information will be used: Is it for exploratory research, conceptual design, process development, equipment sizing, troubleshooting? You next ask what processing steps are involved: reactions, separations, heating, cooling, pumping, expansions, recycles? And which steps could affect business decisions for commercialization: Are reaction yields limited by rates or by equilibrium conversions? Are separations hindered by formation of azeotropes or solutropes? If additional solvents are introduced, how will they be removed, so the product is not contaminated? Can any solvents be recycled to avoid disposal and waste? Finally, you ask precisely what properties are being requested. Are they compositions of phases in equilibrium? Densities and enthalpies of single phase liquids, gases, or solids? Reaction rate constants? In short, you must decide what properties are to be quantified and then decide how those values will be used: in appropriate hand calculations or in a process simulator.

At this preliminary stage, you may be tempted to skimp on the quality of property data, but then you remember that inadequate thermodynamic information can lead to improper designs and process failures.