Introduction

Steady free-streamline flows of water when gravitational forces can be neglected have been discussed in §3.7. Most unsteady free-streamline problems are intractable except by numerical means and generally become more so when gravitational forces are important. However, flows involving gravity where the unsteady motion is a ‘small’ perturbation of a relatively simple mean state occur frequently in the form of surface waves. In the absence of motion the free surface of a liquid in equilibrium under gravity is often ‘horizontal’. A disturbance applied locally that distorts the surface brings into play gravitational restoring forces that cause the disturbance to spread out over the surface in the form of ‘waves’. The waves carry energy away from the source region, propagating parallel to the mean free surface. The agitation produced by a passing wave and the energy flux is generally in the form of a transient disturbance of the fluid particles (around approximately closed particle paths), which are not in themselves transported to any great extent by the wave, and the influence of the wave on fluid at depths exceeding a characteristic wavelength tends to be negligible. In this section these general properties of surface gravity waves are discussed and illustrated by simple examples.

Conditions at the free surface

Consider the simplest case of water whose free surface in equilibrium can be regarded as horizontal and in the plane z = 0 of the coordinate axes (x, y, z), where z increases vertically upwards (Figure 5.1.1).

Introduction

As discussed in the last part of Chapter 5, digital computer software capabilities have currently reached a point where numerical solutions to very large, linear, structural dynamics problems can be successfully achieved. As an indication of the growth in size of structural models being used in dynamic analyses, note that it is now not uncommon for structural dynamic analyses to employ the same detailed FEM models prepared for the purposes of static stress analyses. As a result of this marked increase in the number of DOF used in analyses, and just as importantly, as part of the clear trend toward automating everything, the integration of the equations of motion is rarely done by any means other than by digital computer-based numerical methods. Although these reasons are sufficient for looking at numerical integration techniques, there are still other important reasons. The foremost of these other reasons is that numerical integration is the only practical approach when material nonlinearities (e.g., plasticity) or geometric nonlinearities are part of the system's mathematical model.

Today, numerical integration is a well-developed field with many textbooks available to provide a comprehensive overview on both simplistic and sophisticated levels. See, for example, Refs. [9.1, 9.2]. Therefore it is appropriate for this textbook to provide only a brief introduction to the popular numerical integration techniques that are particularly suitable for the numerical integration of the ordinary differential equations that result from the modal transformation applied to a finite element model or are suitable for the direct integration of the matrix equation of motion in terms of the original generalized coordinates.

Introduction

The previous four chapters emphasized the advantages of using discrete mass mathematicaxsl models wherein both the structural mass and the nonstructural mass is “lumped” at selected (usually a relatively few) finite element nodes or at short distances from those finite element nodes. The alternative in mass modeling is the seemingly more realistic mathematical model where the mass is distributed throughout each structural element. Such distributed or continuous mass models are not nearly as useful as discrete mass models. However, continuous mass models do have enough instructional value and occasional engineering value that they cannot be wholly ignored. Their instructional value resides in (i) seeing the results of dealing with what is essentially an infinite DOF system; (ii) the reinforcement, and perhaps deeper understanding, obtained through repetition of the same analysis procedures used with discrete mass systems in a different context; and (iii) discovering the very few types of structures which can be usefully described by this much more concise type of modeling. Therefore the purpose of this chapter is to discuss some of those situations where the use of continuous mass models is of some, albeit small, value in the study of structural dynamics.

Again, continuous mass models are practical only in quite restricted circumstances. All cases examined here are limited to structures that are modeled as a single structural element (e.g., one beam or one plate).

The fluid state

Consider a fluid that can be regarded as continuous and locally homogeneous at all levels of subdivision. At any time t and position x = (x1, x2, x3) the state of the fluid is defined when the velocity v and any two thermodynamic variables are specified. A fluid in unsteady motion, in which temperature and pressure vary with position and time, cannot strictly be in thermodynamic equilibrium, and it will be necessary to discuss how to define the thermodynamic properties of the small individual fluid particles of which the fluid may be supposed to consist.

The distinctive fluid property possessed by both liquids and gases is that these fluid particles can move freely relative to one another under the influence of applied forces or other externally imposed changes at the boundaries of the fluid. Five scalar partial differential equations are required for determining these motions. They are statements of conservation of mass, momentum, and energy, and they are to be solved subject to appropriate boundary and initial conditions, dependent on the problem at hand. This book is concerned with the use of these equations to formulate and analyse a wide range of model problems whose solutions will help the reader to understand the intricacies of fluid motion.

The dynamics and thermodynamics of a chemically reacting flow are governed by the conservation laws of mass, momentum, energy, and the concentration of the individual species. In this chapter, we shall first present a derivation of these conservation equations based on control volume considerations. We shall then derive a simplified form of these equations describing only those effects which are of predominant importance in most of the subsonic combustion phenomena to be studied later. Some useful concepts and analytical techniques for combustion modeling will be discussed and several important nondimensional numbers will be introduced. Whenever possible, we will adhere to the nomenclature of Williams (1985) for consistency and ease of referencing. A summary of the symbols is given at the end of this chapter.

Further discussions emphasizing on the mathematical aspects of combustion theory can be found in Buckmaster and Ludford (1982) and Williams (1985).

CONTROL VOLUME DERIVATION

To derive the various conservation equations, we first take a control volume that is at rest with respect to an inertia reference frame. It has a volume V and a control surface S, with a unit normal vector n, as shown in Figure 5.1.1. Within this control volume a flow element of velocity v passes through. This bulk, mass-weighted velocity v is the resultant of the individual velocities vi of the various species.

In many combustion applications the fuel is originally present as either liquid or solid. In order to facilitate mixing and the overall burning rate, as pointed out in Section 6.4.1, the condensed fuel is frequently first atomized or pulverized, and then sprayed or dispersed in the combustion chamber. Consequently, in these devices combustion actually takes place in a two-phase medium, consisting of the dispersed fuel droplets or particles in a primarily oxidizing gas.

A description of two-phase combustion consists of three components, namely the gasification and dynamics of individual and groups of droplets (or particles); a statistical characterization of the spray; and the collective interaction of these droplets with the bulk gaseous medium through the description of the two-phase flow in terms of heat, mass, and momentum transfer. The first component was introduced in Section 6.4 through the d2-law of droplet vaporization and burning, and will be extensively studied in this chapter for both droplets and particles. Specifically, we shall first discuss the general phenomenology of droplet combustion with and without external convection, and the experimental methodologies commonly used in investigating droplet combustion. We shall then study the combustion of single-component droplets by relaxing the various assumptions associated with the d2-law, and hence examine effects of droplet heating, fuel vapor accumulation, and variable transport properties that were briefly mentioned in Section 6.4.4. We shall also relax the assumptions of gas-phase quasi-steadiness, stagnant environment, and solitary droplet by discussing effects due to gas-phase transient diffusion, external convection, and droplet interaction, respectively.

When the molecules in any region of a fluid medium possess an excess of energy, concentration, or momentum, such that gradients of these properties exist in the neighborhood of this region, the system will attempt to restore spatial uniformity by transporting the relevant property in the direction of the deficient region. The transport occurs even in the absence of any bulk motion in this direction. As an illustration, consider a body of stagnant gas situated between two parallel plates as shown in Figure 4.1.1a. If plate A is suddenly raised to a temperature TA > TB, then the region around plate B also will be heated soon. Therefore there can be a net transfer of heat from A to B despite the lack of any bulk fluid movement. Similarly if there is initially a higher concentration of a species at A relative to its concentration at B, then its molecules will slowly migrate from A to B, as shown in Figure 4.1.1b. Finally, if plate A is impulsively started to move in a direction parallel to itself (Figure 4.1.1c), then plane B will soon feel the motion and, if unrestrained, will tend to be dragged along. Note that in the last case even though there is a bulk flow in the direction parallel to the plates, the transport occurs in the direction normal to the plates in which there is no motion.

Either by nature or design, in most combustion systems fuel and oxidizer are initially spatially separated. If the subsequent mixing between them is not sufficiently fast before chemical reaction is initiated, then the mixing and reaction will take place only in thin reaction zones that separate them. Examples are a wood panel on fire, an oil spray burning in a furnace, a candle flame, and the sparks (i.e., burning metal particles) generated when a metal surface is abraded.

The structure of a nonpremixed flame therefore consists of three zones, with a reaction zone separating a fuel-rich zone and an oxidizer-rich zone. Figure 6.1.1a shows a typical configuration for the model problem to be studied in the next section. As the fuel and oxidizer are transported toward each other, through diffusion as well as whatever convective motion the system may have, they become heated and eventually meet and mix within the reaction zone. Reaction between them subsequently takes place rapidly. The combustion products together with the heat of combustion are then transported away from the reaction zone in both directions. Since reaction occurs at a finite rate and the reaction zone has a finite thickness, complete reaction cannot be accomplished. Small amounts of fuel and oxidizer invariably leak through the reaction zone, as shown in Figure 6.1.1b.

So far we have been concerned only with situations involving intense burning. In this chapter we shall study the transition between burning and nonburning states, namely phenomena involving ignition, extinction, flame stabilization and blowoff, and flammability.

There are many practical situations exhibiting ignition and extinction phenomena in our daily lives. For ignition, we can cite the striking of a match, turning on the gas stove by a pilot light or spark discharge, firing within an internal combustion engine through compressive heating or again by spark discharge, and the initiation of fires and explosions in mines, grain elevators, and upon the rupturing of fuel tanks by electric or frictional sparks. For extinction, we can cite firefighting through spraying of water and chemicals and the quenching of chemical reactions by the relatively cold wall of combustion chambers.

At the fundamental level, ignition can be achieved in one of two ways. One can either supply an amount of heat to part or all of a combustible mixture. The supply can be either momentary or continuous. The heated mixture responds Arrheniusly, reacts faster, and produces more heat. At the same time, however, being hotter it also tends to lose more heat to the walls and colder parts of the gas. Thus if the rate of heat generation exceeds that of cooling, then an accelerative, runaway process occurs that eventually leads to a state of intense burning.

Chemical thermodynamics is concerned with the description of the equilibrium states of reacting multicomponent systems. Compared to single-component systems in which only thermal equilibrium is required, we are now also interested in chemical equilibrium among all of the components. Since practical combustors are designed to ensure that fuel and air have sufficient residence time to mix, react, and attain thermodynamic equilibrium, global performance parameters such as the heat and power output can frequently be estimated by assuming thermodynamic equilibrium of the combustion products. Thus, the scientific elements of a large part of combustion engineering are covered by the subject of this chapter.

In Section 1.1, we introduce the concept of stoichiometry, which sensitively controls the temperature of a combustion process. In Section 1.2, the criterion for chemical equilibrium is derived and the methodology for calculating the equilibrium composition of a mixture, for given pressure and temperature, is discussed. We then apply this calculation procedure to hydrocarbon–air mixtures as an example in Section 1.3. In Section 1.4, energy conservation is considered, which enables the simultaneous determination of the final composition and temperature of a reactive mixture after equilibrium is established. This final temperature, called the adiabatic flame temperature, Tad, is perhaps the most important parameter of a reactive mixture, indicating not only its potential to deliver heat and power, but also the rates of progress of the various chemical reactions constituting the entire combustion process.