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Atomization is a process of fast transition of a continuous bulk liquid into a dispersion of the liquid in an ambient gas phase. It involves the change in the partition energies of flowing fluids. The initial energies of the fluids entering an atomizer are partitioned mainly between the pressure-volume and kinetic energies. During atomization, some of the energies are converted into interfacial energy andother dissipative forms of energies such as turbulence, heat, and acoustic energy. Generally, atomization efficiency may be defined as the ratio of interfacial energy output to the total energy input. Thermodynamic analyses may be used to evaluate the energy partition that sets limits to the efficiency of producing interfacial energy, i.e., atomization efficiency.

Thermodynamic analyses of the conversion efficiency of pres- sure-volume and kinetic energies to interfacial energy during atomization have been conducted by Li and Tankin,[252] and Shellens and Brzustowski.[432] The pressure of the ambient gas and the entropy density of droplets are considered in the analyses. It is showed that increasing the pressure and entropy density leads to an increase in the proportion of bulk energy in the total energy of droplets and a corresponding decrease in atomization efficiency. Cohesive forces become dominant as droplet size decreases, suggesting that there exists a limit to the smallest size of droplets that can be generated by atomization.

Detailed descriptions of the practical aspects of atomizer performance, plant design, and operation for powder productionthrough atomization of liquid metals have been given by Yule and Dunkley.[5]

5.2.0MODELING OF DROPLET PROCESSES OF NORMAL LIQUIDS

Computer modeling and simulation are aimed at identifying control parameters, establishing detailed correlations between design parameters and spray characteristics, and providing better understanding of real phenomena in droplet processes and guidelines for optimization of processes and designs. The objective of spray combustion modeling, for example, is to develop computer codes that can provide sufficiently accurate descriptions of the physical and

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chemical processes during spray combustion, and to predict thermal, combustion, and energy efficiencies and emission of pollutants. Due to the highly transient and complex nature, atomization and spray are difficult to model and analyze. Theoretical models of liquid breakup processes from first principles are currently limited to predicting mean droplet size and velocity distributions.[252][432] These models need to be extended to predict local droplet size, and velocity distributions. In spray modeling, it is desired to predict and control the conditions at the initial plane of the spray for given atomizer geometry and input liquid and gas flow rates. The long-term goal is to design spray systems from first principles. This will require computations of nonlinear physical processes and coupling of different physical and chemical processes. The ultimate objective of theoretical efforts is to tailor droplet size distributions, and to generate specified distributions, such as mono-size, unimodal, multi-modal, skew, narrow-size, or other specific distributions.

Detailed modeling study of practical sprays requires solving the dynamics of interactions of droplets with surrounding gas. To calculate the mass, momentum, and energy exchanges between the droplets and gas, one must account for the distributions of droplet sizes, velocities, and temperatures. It is also necessary to consider the distortions, breakups, collisions, coalescences, and turbulent dispersions of the droplets, and in many cases the phase changes (vaporization or solidification) of the droplets. Modeling primary breakup of liquid requires information on the liquid and gas flows, such as velocity, shear stress, and turbulent kinetic energy distributions inside atomizer passages as initial conditions for liquid breakup region. In addition, disturbances on liquid surfaces need to be ascertained, along with their wavelengths, amplitude growth rates, and frequencies. To date, these phenomena have been the subjects of intensive theoretical analyses and numerical modeling studies. Comprehensive computer models and numerical methods have been developed and implemented with varied degrees of success. Currently, spray modeling is yet ineffective in dense sprays. An extensive review has been made by Reitz[525] on the current status and recent developments in computer

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modeling of sprays. A survey has been presented by Chigier[526] on the most recent developments in models and correlations of many droplet systems.

5.2.1Theoretical Analyses and Modeling of Liquid Jet and Sheet Breakup

The subject of liquid jet and sheet atomization has attracted considerable attention in theoretical studies and numerical modeling due to its practical importance.[527] The models and methods developed range from linear stability models to detailed nonlinear numerical models based on boundary-element methods[528][529] and Volume- Of-Fluid (VOF) method.[530]

The classical stability theories of liquid breakup developed by Taylor in 1940[205] and others are linear theories that involve wave amplitude growth and changes in wavelength. On the basis of this theory framework, Reitz and Bracco[229] conducted an analysis of liquid jet stability to address the droplet formation process by jet breakup. The analysis considered the growth of initial perturbations on the liquid surface and included the effects of liquid inertia, surface tension, viscous, and aerodynamic forces on the jet. A dispersion equation was derived to relate the growth rate of an initial perturbation of infinitesimal amplitude to its wavelength. The relationship also included the physical and dynamical parameters of the liquid jet and the surrounding gas. In this analysis, a column of liquid was assumed to be infinite in the axial direction. Governing equations for mass and momentum conservation were solved with a kinematic jump boundary condition at the interface, i.e., a normal stress balance and a tangential stress balance. The normal stress balance accounted for surface tension, dynamic pressure (inertia), viscous (normal) force, and recoil forces due to the vapor leaving the interface of an evaporating jet.[243] In the tangential stress balance, the gas was typically assumed to be inviscid, i.e., slip condition was presumed at the liquid-gas interface. This assumption may not allow for the presence of shear at the interface in the boundary layer of the gas flow.

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Typical solutions of the dispersion equation predicted wave growth rates versus wave number as a function of the Weber and Ohnesorge numbers.[232] The analysis results showed that there is a maximum wave growth rate at a particular wavelength, and the fastest growing (or most probable) waves on the liquid surface are responsible for the breakup. This maximum wave growth rate increases and the corresponding wavelength decreases rapidly with increasing Weber number. Liquid viscosity reduces the wave growth rate, but has a secondary stabilizing effect. Gas compressibility and liquid vaporization destabilize low-speed liquid jets, whereas vaporization stabilizes high-speed liquid jets.[243] Surface evaporation slows the breakup of surface waves and produces larger droplets.

This jet stability theory has been found to provide a reasonably complete description of the mechanisms of low-speed jet breakup and a framework for organizing jet breakup regimes. For high-speed liquid jets, the initial state of the jets appears to be progressively more important and less understood. Reitz[232] suggested that the surface wave mechanism of the second wind-induced regime could still apply to a jet in the atomization regime. However, direct verification of this mechanism by means of experiments has not yet been possible because the dense spray surrounding the jet obscures the breakup details.

The jet stability theory has also been used to model liquid jet breakup by considering the unstable growth of waves on a distorting liquid surface.[232] In this model, surface waves grow and break up when a breakup time is reached, producing droplets with sizes that are related to the wavelength of the most unstable wave. The breakup time is related to the wave growth rate. The model is implemented by injecting parcels of liquid, called blobs or bulk liquid masses, which have a characteristic size equal to the nozzle orifice diameter. The blobs distort and new blobs are formed from the liquid that is stripped from the parent blobs as a result of the wave breakup process. It is assumed that the sizes of small droplets are proportional to the wavelength of the fastest-growing or most unstable surface wave, while the sizes of large droplets are determined from the volume of the liquid contained in one surface wave, assuming one droplet is formed each period with a presumed disturbance frequency.

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The most comprehensive theoretical treatment of jet atomization is perhaps that by Ponstein.[531] However, the number of published studies addressing the effect of swirl on liquid jet atomization has been very limited. Kang and Lin[532] studied the spatial instability of a swirling liquid jet including the effect of nonaxisymmetric disturbances. Lian and Lin[242] investigated the convective instability of a viscous liquid jet interacting with a swirling inviscid gas and found that gas swirl hinders jet atomization. Methods for solving the problems of secondary atomization, droplet size distribution and properties of multiple jets have been suggested by different researchers, but none of these problems has been solved satisfactorily.[220]

All these previous studies on liquid jet atomization were mainly based on the linear theory of hydrodynamic stability. The problem associated with the linear stability analyses lies in the assumption that the perturbations are infinitesimal. In reality, unstable perturbations will grow to finite quantities after a finite interval of time. Liquid breakup, droplet formation and secondary breakup are all nonlinear phenomena that can no longer be treated by the linearizing method of small disturbances since the amplitude of the disturbance at breakup is of the same order as the jet radius.[220] Such nonlinear phenomena may be analyzed by, for example, bifurcation theory that predicts bursting and chaotic formation of droplets, as suggested by Chigier.[211] In addition, liquid breakup may also be analyzed on the basis of thermodynamic theories that relate the input kinetic energies of gas and liquid to the final state of droplets generated.[252][432]

Alternative methods of analysis have been examined and evaluated. Shokoohi and Elrod[533] solved the Navier-Stokes equations numerically in the axisymmetric form. Bogy[527] used the Cosserat theory developed by Green.[534] Ibrahim and Lin[535] conducted a weakly nonlinear instability analysis. The method of strained coordinates was also examined. In spite of the mathematical or computational elegance, all of these methods suffer from inherent complexity. Lee[536] developed a 1-D, nonlinear direct-simulation technique that proved to be a simple and practical method for investigating the nonlinear instability of a liquid jet. Lee’s direct-simulation approach formed the

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basis of a comprehensive treatment of the jet instability and atomization presented by Chuech et al.[537] Ibrahim[241] further extended the direct-simulation analysis of Chuech et al. to include the effect of swirlon jet atomization. It was found that liquid swirl enhances jet atomization.

Under most practical conditions, relatively small satellite droplets may form, interspersed among the main droplets, during the disintegration of a liquid jet. Because the formation of the satellite droplets is a nonlinear phenomenon, a linear theory can not adequately describe the formation mechanism of the satellite droplets. Ibrahim[538] studied the modification of the interfacial profile of a liquid jet caused by swirl on the basis of the nonlinear directsimulation technique of Chuech et al.[537] The simulation results suggested that the effect of gas swirl contradicts that of liquid jet swirl. Thus, swirling gas would hinder liquid jet atomization. In addition, liquid swirl causes the satellite droplets to merge forward with the main droplets.

Recently, Spangler, Hilbing and Heister[528][529] performed nonlinear modeling of liquid jet breakup in the wind-induced regimes. They developed a boundary element method to solve for the nonlinear evolution of a liquid jet at unsteady inflow conditions and under the influence of both surface tension and aerodynamic interactions with surrounding gas. For longer waves, the aerodynamic effects were shown to cause a “swelling” of the liquid surface in the trough region. The model predicted the formation of main (parent) and satellite (product) droplets for fixed wavelength perturbations in the first windinduced regime, and the evolution of a “spiked” surface at the periphery of the jet for conditions corresponding to the second wind-induced regime. These researchers further examined the effects of disturbance wave number, liquid Weber number, and liquid-gas density ratio on the jet breakup. They also identified transition conditions for various flow regimes. It was shown that the size of droplets is influenced by changes in the perturbation wavelength, perturbation magnitude and Weber number. Satellite droplet velocities are lower than main droplet velocities due to the sequential shedding of droplets from the orifice. Based on these modeling results, one can predict the