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.

In many practical situations of interest to combustion, high-speed gas flow prevails. Examples are flame stabilization by bluff bodies within the combustion chamber of a gas turbine, accidental or intentional explosion of a combustible by a hot metal particle or projectile, thermal protection of reentry vehicles by ablative heat shields, and the burning of solid and liquid surfaces in an oxidizing gas stream.

When such a high-speed flow is adjacent to either a solid surface or another flow with slower velocity, a transition region exists. Across this region, the flow velocity, and possibly also temperature and concentration, will change from their respective freestream values to either satisfy the boundary conditions required at the solid surface or approach the freestream values of the slower flow. For fluids with small viscosity µ, the transition region is thin and the normal gradient across it, ∂u/∂y, is large such that despite the small µ, the shear stress, τ = µ∂u/∂y, may assume large values. Thus if the characteristic dimensions over which properties change appreciably in the x- and y-directions are ℓ and δ respectively, then the existence of a boundary layer is implied by the condition δ/ℓ ≪ 1. Furthermore, since for gases the diffusive transport processes of heat, mass, and momentum occur at comparable rates, we expect that the boundary-layer thicknesses for these three processes also should not differ too much from each other.

In 1974 Liñán published the first complete analysis of the asymptotic structure of nonpremixed flames based on the one-step second-order reaction. Although the analysis was conducted for the steady counterflow flame with unity Lewis number, it was subsequently recognized that the analysis is of such a canonical nature that the results are applicable to many laminar flame situations. Specifically, since one of the flame regimes identified has the characteristics of a premixed flame, the analysis is actually applicable to both nonpremixed and premixed flames. Furthermore, since the flame response is ultimately determined by that of the reaction zone, many subsequent analyses of other flame configurations invariably end up with the structure equations and, hence, solutions first derived by Liñán. We shall study these structures in this chapter.

We shall first present a separate analysis of the reaction zone structure of premixed flames, with the solution expressed in terms of some general boundary conditions that are to be supplied by the outer solutions for individual situations. From our study of the ignition of a combustible by a hot surface in Section 8.2, we anticipate that such a generalized formulation is possible for situations involving thin reaction zones embedded within flames. The concept of delta-function closure will also be introduced.

For nonpremixed flames, we have studied their combustion characteristics in the reaction-sheet limit in Chapter 6. It was demonstrated that much can be learned about the bulk combustion and flame behavior without any specification of the nature of the chemical reaction, except for the implicit assumption that they occur infinitely fast relative to diffusion.

This book is about combustion science and technology and, as such, covers not only the basic laws and phenomena related to the physics and chemistry of combustion, but also the implications of the fundamental understanding gained therein to the principles behind the practical combustion phenomena affecting our daily lives. It presents the diverse knowledge required of combustion scientists and engineers, the challenges they face, and the satisfaction they derive in providing the proper linkage between the fundamental and the practical.

In Section 0.1 we identify the major areas of practical combustion phenomena, illustrated by some specific problems of interest. In Section 0.2 we discuss the scientific disciplines comprising the study of combustion, and in Section 0.3 we present the classifications of fundamental combustion phenomena. An overview of the text is given in Section 0.4.

MAJOR AREAS OF COMBUSTION APPLICATION

It is fair to say that the ability to use fire is an important factor in ushering the dawn of civilization. Today our dependence on the service of fire is almost total, from heating and lighting our homes to powering the various modes of transportation vehicles. Useful as it is, fire can also be menacing and sometimes deadly. Wildland and urban fires cause tremendous loss of property and lives every year; the noxious pollutants from automotive and industrial power plants poison the very environment in which we live; and the use of chemical weapons continues to be an agent of destruction with ever greater efficiency.

In Chapter 2 we introduced the principles of the chemical kinetics of gas-phase reactions. In particular, we discussed the fundamental dependence of reaction rates on temperature, pressure, and reactant concentrations. We studied the concept of multistep reactions, and then categorized the reaction mechanisms and showed how they can affect the reaction rate in qualitatively different manners.

In this chapter, we discuss the oxidation mechanisms of specific fuel systems involving hydrogen, carbon monoxide, and various hydrocarbons. The formation of pollutants will also be covered. In particular, the reaction pathways leading to fuel consumption, the formation and destruction of intermediate species, and the final product formation are discussed in a qualitative manner. We shall see that numerous elementary chemical reactions are involved in the conversion of reactants to products, and that even for a given fuel these reactions often play different roles in different combustion environments. The intricate paths followed by hydrocarbon oxidation illustrate the complexity of chemical kinetics in combustion. Nevertheless, in spite of such apparent complexity, we shall also show that there appears to be only a finite number of reactions that exert significant influence in a combustion process, providing the possibility that the reaction mechanisms of fuel oxidation can be largely understood.

Since the mid-1970s there has been truly significant advancement in combustion science, spurred by the dual societal concerns for energy sufficiency and environmental quality, and enabled by the rapid increase in the sophistication of mathematical analysis, computational simulation, and experimental techniques. Consequently, we have witnessed the evolvement of combustion from a scientific discipline that was largely empirical to one that is quantitative and predictive, leading to its useful applications in combustion-related engineering devices and practices.

This text reflects my desire to incorporate these advances in my lectures on combustion. As a result, its preparation has been guided by the three distinguishing themes characterizing recent developments in combustion research, namely the canonical formulation of the theoretical foundation; the strong interplay between experiment, theory, and computation; and the description of combustion phenomena from the unified viewpoint of fluid mechanics and chemical kinetics.

The text also emphasizes analytical proficiency by presenting complete, albeit abbreviated, derivations that can be followed by the student with a modest effort. Alternate solutions are sometimes presented to demonstrate that a phenomenon can often be analyzed using different approaches and at different levels of rigor. I hope that through this gentle guidance the student can acquire the needed confidence to tackle more difficult problems on his or her own.

This text grew out of the lecture material prepared for a one-year graduate course that I have given at several academic institutions. No prerequisite in mathematics, fluid mechanics, and chemistry is expected apart from the usual undergraduate education in the physical sciences or mechanical, aerospace, or chemical engineering.