Chapter 8 approaches the topics of phase equilibrium and phase transitions from a microscale perspective. Specifically, the roles of fluctuations and system stability in the onset of phase transitions are examined in detail. Using aspects of statistical thermodynamics theory developed in earlier chapters, the van der Waals model is used to demonstrate how fluctuations and system instability give rise to phase transitions in fluid systems. Binary fluid systems are considered, with pure fluid results being recovered when the mass of one species is set to zero. It is shown that critical exponents and the law of corresponding states can be deduced from the van der Waals model for pure fluids. Microscale aspects of solid–liquid transitions are also considered.

Fluctuations and Phase Stability

In the development of thermodynamics presented in the preceding chapters, we have identified different categories of substances according to the density of the substance and the nature of molecular interactions in the substance. Gases have low density and the molecules spend most of the time traveling through space with momentum and energy being exchanged between molecules only through brief collisions. In liquids, the molecules are free to roam about within the system but the density is much higher than in gases, with the mean distance between adjacent molecules being only one to two molecular diameters. Because the molecules are close to their neighbors, they continuously are subject to force interactions with nearby molecules, resulting in continuous exchange of momentum and energy.

This text differs from most statistical thermodynamics textbooks in that it does not deal exclusively with statistical aspects of thermodynamics. Instead, it attempts to weave together statistical and classical elements to develop the full theoretical framework of thermodynamics. Chapter 1 begins this development by establishing the basic features of energy storage at the atomic and molecular levels. It contains a very short introduction to basic aspects of quantum mechanics. The quantum models discussed in this chapter are models of energy storage modes found in common molecules. Conclusions regarding energy levels and their degeneracy for these modes of energy storage are cornerstones of the statistical thermodynamic theory developed in later chapters.

Microscale Energy Storage

Since this text is designed for graduate-level engineering instruction, it is likely that the reader has already encountered some elements of thermodynamics in previous courses and very likely that he or she has some idea of the usefulness of thermodynamic analysis for systems of scientific and technological interest. Mechanical or chemical engineers who are thoroughly versed in classical equilibrium thermodynamics may wonder what a statistical development of thermodynamics has to offer beyond the tools provided by classical thermodynamics.

In response to such an inquiry, we can identify two main benefits of developing a statistical thermodynamic theory. First, by design, statistical thermodynamics theory provides a link between macroscopic “classical” thermodynamic analysis of system behavior and the microscopic characteristics of the atoms, molecules, or subatomic particles that make up the system.

The ideal gases considered in Chapter 5 are arguably the simplest fluids encountered in real systems. The behavior of dense gases, liquids, and quantum fluids deviates strongly from that of an ideal gas. In this chapter, we examine how the thermodynamic framework developed in earlier chapters can be applied to such fluids. The van der Waals model for dense gases and liquids is explored in detail for pure and binary mixture systems. In doing so, we demonstrate that the statistical thermodynamic framework provides a link between microscopic characteristics of the molecules or particles and the macroscopic behavior of these fluids.

Behavior of Gases in the Classical Limit

In Chapter 5, we observed that if the temperature is high enough, we can replace the summation in the definition of the partition function with an integral to obtain the limiting form the partition function at high temperature. This reflects one of the fundamental characteristics of quantum theory, which is that classical behavior is attained in the limit of large quantum number. At high temperature, the average energy per molecule increases and this implies that the average quantum number for each energy storage mode is higher. Thus at higher temperatures, the mean behavior of the system is classical in nature.

In the previous sections of this text we have attacked the problem of determining the partition function by considering the problem from a quantum perspective.

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.