The spray equations have been studied and solved for many applications: singlecomponent and multicomponent liquids, high-temperature and low-temperature gas environments, monodisperse and polydisperse droplet-size distributions, steady and unsteady flows, one-dimensional and multidimensional flows, laminar and turbulent regimes, subcritical and supercritical thermodynamic regimes, and recirculating (strongly elliptical) and nonrecirculating (hyperbolic, parabolic, or weakly elliptic) flows. The analyses discussed here will not be totally inclusive of all of the interesting analyses that have been performed; rather, only a selection is presented.

Spray flows can be classified in various ways. One important issue concerns whether the gas is turbulent or laminar. In this chapter, only laminar flows are considered; the turbulent situation is discussed in Chapter 8. Another issue concerns whether thermodynamic conditions are subcritical, on the one hand, or near critical to supercritical, on the other hand. In this chapter, only subcritical situations are discussed. Near-critical and supercritical behavior will be discussed in Chapter 9.

In the most general spray case, the gas and the droplets are not in thermal and kinematic equilibria, that is, the droplet temperature and the droplet velocity differ from those properties of the surrounding gas. Of course, heat transfer and drag forces result in the tendency to move toward equilibrium. The equilibrium case is sometimes described as a locally homogeneous flow. It is possible to have thermal equilibrium or kinematic equilibrium without the other.

The purpose of this appendix is to summarize the algorithms available to predict in an approximate but reasonable manner droplet heating, vaporization, and trajectory in a userfriendly manner. We consider single-component and multicomponent droplets, isolated droplets and droplets interacting with droplets, and stagnant and moving droplets. We do not consider turbulence or critical thermodynamic conditions in this section. The underlying theories are detailed in Chapters 2 and 3.

The description of the quasi-steady gas film can be summarized by the use of drag coefficient CD, lift coefficient CL, Nusselt number Nu, Sherwood number Sh, and a nondimensional vaporization rate ṁ/4πρDR. These quantities can be prescribed as functions of the transfer number B, the droplet Reynolds number Re, and a nondimensional spacing between neighboring droplets.

The vaporization rate and the other parameters will depend (through the parameters B and Re) on the ambient conditions outside the gas film and on the droplet-surface conditions. To determine the vaporization rate, Nusselt number, and Sherwood number quantitatively, we must simultaneously solve for the temperature and composition at the droplet surface. Typically the solution of diffusion equations for heat and mass in the droplet interior must be solved. These solutions of the diffusion equations require some determination of the internal droplet-velocity field.

For the single-component, isolated droplet, there are three approximate models that have been considered in Chapter 2. Table B.I presents the summary for the spherically symmetric case; Tables B.2 and B.3 present the vortex model and the effectiveconductivity model, respectively.

The fluid dynamics and transport of sprays is a rapidly developing field of broad importance. There are many interesting applications of spray theory related to power, propulsion, heat exchange, and materials processing. Spray phenomena also have natural occurrences. Spray and droplet behaviors have a strong impact on vital economic and military issues. Examples include the diesel engine and gas-turbine engine for automotive, power-generation, and aerospace applications. Manufacturing technologies including droplet-based net form processing, coating, and painting are important applications. Applications involving medication, pesticides and insecticides, and other consumer uses add to the impressive list of important industries that use spray and droplet technologies. These industries involve annual production certainly measured in tens of billions of dollars and possibly higher. The potentials for improved performance, improved market shares, reduced costs, and new products and applications are immense. An effort is needed to optimize the designs of spray and droplet applications and to develop strategies and technologies for active control of sprays in order to achieve the huge potential in this answer.

In this book, I have attempted to provide some scientific foundation for movement toward the goals of optimal design and effective application of active controls. The book, however, will not focus on design and controls. Rather, I discuss the fluid mechanics and transport phenomena that govern the behavior of sprays and droplets in the many important applications. Various theoretical and computational aspects of the fluid dynamics and transport of sprays and droplets are reviewed in detail.

OVERVIEW

A spray is one type of two-phase flow. It involves a liquid as the dispersed or discrete phase in the form of droplets or ligaments and a gas as the continuous phase. A dusty flow is very similar to a spray except that the discrete phase is solid rather than liquid. Bubbly flow is the opposite kind of two-phase flow wherein the gas forms the discrete phase and the liquid is the continuous phase. Generally, the liquid density is considerably larger than the gas density, so bubble motion involves lower kinematic inertia, higher drag force (for a given size and relative velocity), and different behavior under gravity force than droplet motion.

Important and intellectually challenging fluid-dynamic and -transport phenomena can occur in many different ways with sprays. On the scale of an individual droplet size in a spray, boundary layers and wakes develop because of relative motion between the droplet center and the ambient gas. Other complicated and coupled fluid-dynamic factors are abundant: shear-driven internal circulation of the liquid in the droplet, Stefan flow due to vaporization or condensation, flow modifications due to closely neighboring droplets in the spray, hydrodynamic interfacial instabilities leading to droplet-shape distortion and perhaps droplet shattering, and droplet interactions with vortical structures in the gas flow (e.g., turbulence).

On a much larger and coarser scale, we have the complexities of the integrated exchanges of mass, momentum, and energy of many droplets in some subvolume of interest with the gas flow in the same subvolume.

GENERAL COMMENTS

There is interest in the droplet-vaporization problem from two different aspects. First, we wish to understand the fluid-dynamic and -transport phenomena associated with the transient heating and vaporization of a droplet. Second, but just as important, we must develop models for droplet heating, vaporization, and acceleration that are sufficiently accurate and simple to use in a spray analysis involving so many droplets that each droplet's behavior cannot be distinguished; rather an average behavior of droplets in a vicinity are described. We can meet the first goal by examining both approximate analyses and finite-difference analyses of the governing Navier–Stokes equations. The second goal can be addressed at this time with only approximate analyses since the Navier–Stokes resolution for the detailed flow field around each droplet is too costly in a practical spray problem. However, correlations from Navier–Stokes solutions provide useful inputs into approximate analyses. The models discussed herein apply to droplet vaporization, heating, and acceleration and to droplet condensation, cooling, and deceleration for a droplet isolated from other droplets.

Introductory descriptions of vaporizing droplet behavior can be found in the works of Chigier (1981), Clift et al. (1978), Glassman (1987), Kanury (1975), Kuo (1986), Lefebvre (1989), and Williams (1985). Useful research reviews are given by Faeth (1983), Law (1982), and Sirignano (1983, 1993a, 1993b). The monograph by Sadhal et al. (1997) is also noteworthy.

The vaporizing-droplet problem is a challenging, multidisciplinary issue. It can involve heat and mass transport, fluid dynamics, and chemical kinetics.

There are various complications that occur when a multicomponent liquid is considered [Landis and Mills (1974) and Sirignano and Law (1978)]. Different components vaporize at different rates, creating concentration gradients in the liquid phase and causing liquid-phase mass diffusion. The theory requires the coupled solutions of liquid-phase species-continuity equations, multicomponent phase-equilibrium relations (typically Raoult's Law), and gas-phase multicomponent energy and species-continuity equations. Liquid-phase mass diffusion is commonly much slower than liquid-phase heat diffusion so that thin diffusion layers can occur near the surface, especially at high ambient temperatures at which the surface-regression rate is large. The more volatile substances tend to vaporize faster at first until their surface-concentration values are diminished and further vaporization of those quantities becomes liquid-phase mass diffusion controlled.

Mass diffusion in the liquid phase is very slow compared with heat diffusion in the liquid and extremely slow compared with momentum, heat, or mass diffusion in the gas film or compared with momentum diffusion in the liquid. In fact, the characteristic time for the liquid-phase mass diffusion based on droplet radius is typically longer than the droplet lifetime. Nevertheless, this mass diffusion is of primary importance in the vaporization process for a multicomponent fuel. At first, early in the droplet lifetime, the more volatile substances in the fuel at the droplet surface will vaporize, leaving only the less volatile material that vaporizes more slowly.

To this point in this treatise, we have discussed only isolated droplets. In a practical situation, of course, many droplets are present in a spray and the average distance between droplets can become as low as a few droplet diameters. A typical droplet therefore will not behave as an isolated droplet; rather, it will be strongly influenced by immediately neighboring droplets and, to some extent, by all droplets in the spray.

There are three levels of interaction among neighboring droplets in a spray. If droplets are sufficiently far apart, the only impact is that neighboring droplets (through their exchanges of mass, momentum, and energy with the surrounding gas) will affect the ambient conditions of the gas field surrounding a given droplet. As the distance between droplets becomes larger, the influence of neighboring droplets becomes smaller and tends toward zero ultimately. At this first level of interaction, the geometrical configuration of the (mass, momentum, and energy) exchanges between a droplet and its surrounding gas is not affected by the neighboring droplets. In particular, the Nusselt number, Sherwood number, and lift and drag coefficients are identical in values to those for an isolated droplet. This type of interaction will be fully discussed in Chapter 5.

At the next level of interaction, droplets are closer to each other, on average, and the geometrical configurations of the exchanges with the surrounding gas are modified.

High pressures and supercritical conditions in liquid-fueled diesel engines, jet engines, and liquid rocket engines present a challenge to the modelling and the fundamental understanding of the mechanisms controlling the mixing and combustion behavior of these devices. Accordingly, there has been a reemergence of investigations to provide a detailed description of the fundamental phenomena inherent in these conditions. Unresolved and controversial topics of interest include prediction of phase equilibria at high and supercritical pressures (Curtis and Farrell, 1988; Litchford and Jeng, 1990; Hsieh et al., 1991; Poplow, 1994; Delplanque and Sirignano, 1993; Yang and Lin, 1994; Delplanque and Potier, 1995; Haldenwang et al., 1996), including the choice of a proper equation of state, definition of the critical interface, importance of liquid diffusion, significance of transport property singularities in the neighborhood of the critical mixing conditions, influence of convection (including secondary atomization); d2 law behavior at supercritical conditions (Daou et al., 1995); droplet-lifetime predictions (Delplanque and Sirignano, 1993, 1994; Yang et al., 1992; Yang and Lin, 1994; Delplanque and Potier, 1995; Haldenwang et al., 1996); dense spray behavior (Delplanque and Sirignano, 1995; Jiang and Chiang, 1994a, 1994b, 1996); combustion-product condensation (Litchford and Jeng, 1990; Litchford et al., 1992; Delplanque and Sirignano, 1994; Daou et al., 1995); and flame structures at high and supercritical pressures (Daou et al., 1995). The actual combustion process is characterized by the supercritical combustion of relatively dense sprays in a highly convective environment.

The interactions of a spray with a turbulent gas flow is important in many applications (e.g., most power and propulsion applications). Two general types of studies exist. In one type, the global and statistical properties associated with a cloud or spray within a turbulent field are considered. In the other type, detailed attention is given to how individual particles behave in a turbulent or vortical field. Some studies consider both perspectives. Most of the research work in the field has been performed on the former type of study. Faeth (1987), Crowe et al. (1988), and Crowe et al. (1996) give helpful reviews of this type of research.

The interactive turbulent fields can be separated into homogeneous turbulent fields and free-shear flows (e.g., jets and mixing layers). In some theoretical studies, two-dimensional vortical structures interacting with a spray have been examined. Most of the studies deal with situations in which the contribution of the spray to the generation of the turbulence field is secondary, that is, there is a forced gas flow whose mass flux and kinetic-energy flux substantially exceed the flux values for the liquid component of the dilute flow. The turbulent kinetic-energy flux of the gas flow is much less than the mean kinetic-energy flux of the gas flow and is comparable with the mean kinetic energy of the liquid flow.

THE IDEAL OTTO CYCLE

A cycle is an idealization of what goes on in one of the devices that thermodynamicists call heat engines: that is, a gasoline or diesel engine, a jet engine, a steam engine, and so forth. All of these take some energy source and convert some of that energy into useful work. In the spark-ignition engine the energy source is a chemical fuel, usually gasoline, which is combined with oxygen from the air by burning to release heat. Expansion of the heated gases does the mechanical work.

For the spark-ignition engine the idealization is called the Otto cycle, after Dr. N. A. Otto who, in 1876, patented a stationary gas engine using approximately this cycle. In order to understand this ideal cycle, we must imagine a piston in a cylinder. The piston is connected to a crank by a connecting rod – see Figure 1.1. The crank rotates, and the piston travels up and down. There are two valves, an inlet and an exhaust valve, and an arrangement to open and close them. The idealized cycle is illustrated in Figure 1.2.

In Figure 1.2 we plot the pressure in the cylinder against the volume in the cylinder. Notice that the piston does not go quite all the way to the top of the cylinder; the piston is at the top of its travel at 0, 2 and 3, and there is a small space still above it, the combustion chamber.