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. More volatile material still exists in the droplet interior and will tend to diffuse toward the surface because of concentration gradients created by the prior vaporization.

Consider the gas phase that surrounds or adjoins liquid droplets, films, pools, and/or streams. Figure B.1 presents a diagram that portrays a wide variety of two-phase reacting flow situations included here. Laminar multicomponent flow with viscosity, Fourier heat conduction, and Fickian mass diffusion will be analyzed. Diffusivities for all species will be assumed identical; the Lewis number value will be unity; radiation will be neglected; and kinetic energy will be neglected in comparison with thermal energy so that, with regard to the energetics, pressure will be considered uniform over the space although the pressure gradient can be significant in the momentum balance (small Mach number). We will analyze, as a general case, the situation in which a one-step exothermic reaction occurs between the ambient gas and the vapor from the bulk liquid. We will refer to the liquid as the fuel and the ambient gas as the oxidizer, although the opposite situation, e.g., oxygen droplet in hydrogen gas, can be analyzed in exactly the same fashion. The nonreacting case can be the special subcase in which the reaction rate becomes zero.

Both steady and unsteady scenarios will be discussed. The roles of the gas-phase energy equation and the species equations will be emphasized. Of course, they must be coupled with the continuity and momentum equations and with a gas equation of state. In addition, the solution of conservation equations for the liquid phase might be required.

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 is 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 because 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. The governing partial differential equations reflecting the conservation laws are presented in Appendix A. Several coordinate systems are considered. Formulations with primitive velocity variables and formulations with stream functions are discussed. Appendix B discusses some conserved scalar variables whose analytical use can be very convenient and powerful under certain ideal conditions.

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).

Introduction

It is intended herein to present the current status of the fundamental understanding about the liquid-atomization processes for various injection configurations. The limitations of the theory and the need for future work will be made apparent. This chapter is not intended to be a guide for the practicing engineer; the current state of the art is based on empirical approaches that are discussed in Chapter 1. Rather, this chapter reviews theoretical research that should eventually lead to improved design methodology and design tools for liquid-atomization systems. Other overviews of the theory can be found in Lefebvre (1989), Bayvel and Orzechowski (1993), Sirignano and Mehring (2000, 2005), and Lin (2003).

The atomization problem could be divided according to three subdomains of the fluid mechanical field. The upstream subdomain lies within the liquid-supply piping, plenum chamber, and orifice (nozzle) of the injector hardware. More than the mass flow and average velocity from the orifice into the combustion chamber are important here; velocity and pressure fluctuations in the liquid that are due to turbulence, collapse of bubbles formed through cavitation, supply-pressure unsteadiness, and/or active-control devices are critical in affecting the temporal and spatial variation of the liquid flow over the orifice exit cross section. A small amount of research has been performed on this subject.

The second subdomain involves the liquid stream from the orifice exit to the downstream point where disintegration of the stream begins. The neighboring gas flow (or gas and droplet flow) is part of this subdomain.

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 to the mean kinetic energy of the liquid flow. Therefore the turbulent field is much more likely in this situation to receive kinetic energy transferred from the mean gas flow than kinetic energy transferred from the mean liquid flow.

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; Delplanque and Sirignano, 1993; Poplow, 1994; 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, and influence of convection (including secondary atomization); d2-law behavior at supercritical conditions (Daou et al., 1995); droplet-lifetime predictions (Yang et al., 1992; Delplanque and Sirignano, 1993, 1994; 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.

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 does 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 that is due to vaporization or condensation, flow modifications that are 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.

To this point in this book, 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 9.

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.

Introduction

Although the emphasis in this book is on the dynamics of vaporization of liquids in the form of drops and sprays, it is important to note when liquids might better be applied in a form other a spray. Such a situation can develop when miniature devices are of interest. The use of a wall film rather than a spray might provide sufficient surface area of liquid to vaporize at desired rates. Also, other benefits might arise. In this chapter, we discuss a concept of liquid-film combustors that are superior for miniaturization.

Combustion has the potential to provide simultaneously high-power density and high-energy density; these parameters make it more attractive than batteries and fuel cells for applications for which weight is an issue, e.g., flight or mobile power sources. So it is important to study this method of power generation on a small scale. The microgas turbine (combustor volume 0.04 cc), the mini (0.078-cc displacement) and micro (0.0017-cc displacement) rotary engine, the microrocket (0.1-cc combustion chamber), and the micro Swiss-roll burner are examples of such studies. See Dunn-Rankin et al. (2006), Waitz et al. (1998), Fu et al. (2001), Micci and Ketsdever (2000), Lindsay et al. (2001), and Sitzki et al. (2001). These devices are not yet sufficiently efficient to compete with the best batteries; however, the feasibility of internal combustion as a miniature power source has been shown. The major challenge for all miniature-combustor designs is the increasing surface-to-volume (S/V) ratio with decreasing size.

In this chapter, we examine the effects of droplet motion relative to the surrounding gas on the vaporization, heating, and acceleration of the droplet. The fluid velocity and scalar properties are examined for both the gas film surrounding the droplet and the liquid interior of the droplet. We use a frame of reference instantaneously travelling at the velocity of the center of the droplet; so the droplet appears stationary while the gas flows around the droplet. Still, liquid motion can occur because of internal circulation.

The relative motion between a droplet and the immediate surrounding gas results in an increase of heat and mass transfer rates in the gas film surrounding the droplets; a thin boundary layer forms over the forward section of the droplet. This boundary layer also extends over a portion of the aft section. At a sufficiently high Reynolds number (based on relative velocity, droplet radius, and gas properties), separation of the gas flow occurs at the liquid interface. Because the liquid surface moves under shear, the separation phenomenon is not identical to separation on a solid sphere; for example, the zero-stress point and the separation point are not identical on a liquid sphere, as they are on the solid sphere. The zero-stress points on the liquid sphere and the solid sphere occur at approximately the same point (110°–130° measured from the forward stagnation point), but the separation point on the liquid sphere is well aft of that.