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A model for multi-droplet interaction effects has been developed by Silverman and Sirignano[588] using a new approach for densespray modeling. The model was employed to study the effects of droplet interactions on the evaporation and motion of a dense spray in a hot gaseous environment. A statistical description of droplets in a cloud and correlation functions for the effects of interactions between neighboring droplets were used to extract correction factors for various parameters that are affected by droplet interactions (for example, drag coefficient, Nusselt number and Sherwood number). The correction factors enable the calculations of the drag coefficient, evaporation rate and heat transfer of a droplet in a cloud based on models for a single droplet. It was shown that the interaction effects are important during a large fraction of the droplet lifetime as the droplet size decreases. The multi-droplet interactions cause the drag coefficient of a droplet in a dense spray to be lower and hence its velocity higher than that for an isolated droplet. Thus, spray penetration is very much greater than that predicted for isolated droplets at the same initial conditions. For droplets of 100 µm in diameter, the evaporation rate decreases and the droplet lifetime increases due to the multi-droplet interactions. For droplets of 40 µm in diameter, however, the evaporation rate increases and the droplet lifetime decreases, despite slower heat transfer to an interacting droplet than to an isolated droplet under the same gas-phase conditions and relative velocity of the droplets. It is the higher relative velocity of a 40 µm interacting droplet that compensates for this effect, resulting in a shorter droplet lifetime.

5.2.5Modeling of Multiphase Flows and Heat and Mass Transfer in Sprays

A spray is a turbulent, two-phase, particle-laden jet with droplet collision, coalescence, evaporation (solidification), and dispersion, as well as heat, mass and momentum exchanges between droplets and gas. In spray modeling, the flow of gas phase is simulated typically by solving a series of conservation equations coupled with the equations for spray process. The governing equations for the gas phase include the equations of mass, momentum and energy

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conservation, as well as turbulence model equations. The simulation can be in the Lagrangian,[549][555] Eulerian,[543] or LagrangianEulerian[555] reference frame. Source terms are required for the coupling of the gas and droplet/particle phases. Detailed descriptions of the conservation equations, spray equation, exchange terms, and computational techniques have been given by O’Rourke,[550] Williams,[541] and Amsden et al.[555]

Two (or three) phase flow in a spray is typically turbulent with resultant dispersion of droplets, turbulence damping by small droplets,[589] and turbulence generation by large droplets.[590] Large-scale turbulent structures, with sizes of the order of spray diameter, influence liquid breakup, dispersion, and mixing. Dispersion of droplets in a spray is intimately related to turbulence.[591] Large droplets with Stokes numbers much larger than one are projected along their initial trajectories without (or with little) deflection by the turbulent gas flow. On the contrary, small droplets with Stokes numbers much smaller than one follow the gas flow closely without (or with small) slip velocity. These small droplets spiral and are centrifuged corresponding to their sizes when entering large-scale eddies. Droplets withStokes numbers of the order of one are partially rotated when entering eddies and before exiting from the eddies. Dispersion of droplets in a spray may become preferential, and in some cases may be higher than that of the gas phase alone. Thus, physical processes such as droplet clustering and Stokes-number dispersion[591] need to be considered in spray modeling. However, droplet dispersion cannot be predicted accurately without significant improvements in understanding and modeling of turbulence phenomenon. Currently, the most popular approach to predicting droplet dispersion in turbulence is the Monte-Carlo method in which the turbulence field is represented by a random number generator. The gas velocity is considered as the sum of a timeaveraged velocity and a fluctuating velocity. The fluctuating velocity is selected from a Gaussian distribution with a variance proportional to the turbulence kinetic energy. The droplet motion is then integrated with this velocity field until it passes through an eddy. Reitz and Diwakar[316] found that the effect of turbulence modulation is most

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important in the jet development region close to the nozzle in a highpressure spray. The calculated velocity decay is faster if turbulence modulation is not included, and the primary effect of the turbulence modulation is a shift in the virtual origin of the spray. The dispersion of droplets by turbulence and the gas phase turbulence modulation have been reviewed by Crowe et al.[591] and Faeth,[589] respectively.

Most current multidimensional spray simulations have adopted the thin or very thin spray assumptions,[550] i.e., the volume occupied by the dispersed phase is assumed to be small. This can be justified if a simulation starts some distance downstream of the nozzle exit, where the gas volume fraction is large enough, or if the computational cells are relatively large. Accordingly, two major classes of models have been used in spray modeling: locally homogeneous flow (LHF) models and two-phase-flow or separated-flow (SF) models.

In the LHF models, it is assumed that droplets are in dynamic and thermodynamic equilibrium with gas in a spray. This means that the droplets have the same velocity and temperature as those of the gas everywhere in the spray, so that slip between the phases can be neglected. The assumptions in this class of models correspond to the conditions in very thin (dilute) sprays. Under such conditions, the spray equation is not needed and the source terms in the gas equations for the coupling of the two phases can be neglected. The gas equations, however, need to be modified by introducing a mixture density that includes the partial density of species in the liquid and gas phases based on their mass fractions. Details of the LHF models have been discussed by Faeth.[589]

In the SF models, all of the terms in the droplet and gas conservation equations are retained. Therefore, the SF models are the more general models for spray calculations. The models account for mass, momentum and energy exchanges between droplets and gas. To formulate the exchange terms, the nature of the conditions at droplet-gas interface is of importance. The exchange processes are typically modeled by means of semi-empirical correlations.

To date, spray modeling has been largely dependent on solving the governing equations of multiphase flows and specifying initial and

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boundary conditions. This typically involves solving the governing equations for 2-D, turbulent, particle-laden flows with simultaneous chemical reactions and geometrically complex boundary conditions. Chemical reactions for individual species are treated by Arrhenius-type kinetics[555] and vaporization is calculated using the Frossling correlation[1] or the comprehensive model of Priem et al.[586] Detailed reviews of advanced droplet vaporization models have been given by Sirignano.[14][15] For specifying initial and boundary conditions, an initial plane needs to be designated where liquid breakup is complete and droplets are spherical. The typical outputs of spray modeling calculations include detailed spatial and temporal distributions of droplet sizes, velocities, angles of flight, number densities, temperatures, and mass flux, along with turbulent characteristics of gas flow, such as distributions of mean and fluctuating velocities, shear stresses, and kinetic energy of turbulence. Numerical computations have focused on seeking steady state solutions of 2-D, time-averaged, turbulent flow equations. New models/methods that allow computations of 3-D, time-dependent flows with resolution of large scales of turbulence include those for large eddy[592] and vortex dynamics[593] simulations. These models/methods have been applied to single phase jet flows, and need adaptation to two-phase jet spray flows.

The state-of-the-art in computations of sprays is the relation of the Reynolds-averaged Navier-Stokes equations with k-ε turbulence closure (where k is turbulent kinetic energy and ε is its dissipation rate) and particle-source-in-cell coupling to a Lagrangian condensed phase for droplets.[548] The particle-source-in-cell method proposed by Crowe et al.[548] relates the liquid phase to gas phase and is very sensitive to accurate determination of the joint distribution functions of droplet sizes, 3-D components of velocities, temperatures, and evaporation rates in sprays. The method is most suited for steady-state spray computations and is currently used in many commercial spray codes. The developments of new computer programs in massively parallel processing will allow more coupling of the full Reynolds-averaged Navier-Stokes equations with droplet dynamics.

Young[594] derived a set of fundamental equations for gasdroplet multiphase flows in which small liquid droplets polydisperse

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in a continuous gas-phase consisting of either a pure vapor of the same chemical species as the droplets or a mixture of the pure vapor and an inert gas. Some problems associated with previous formulations were resolved by more judicious definitions of interphase transfer terms. A consistent model was included to represent the surface energy and entropy of the droplets in order to maintain consistency with droplet growth models in which the droplet temperature depends on its radius due to capillary effect. An equation for the rate of entropy creation due to departures from equilibrium was also derived on the basis of linear irreversible thermodynamics. It was indicated that some commonly used mathematical models of droplet growth may not be physically realistic in certain circumstances.

Creismeas[595] calculated the evolution of a water droplet spray inside a complex geometry by means of an Eulerian-Lagrangian model. In this modeling, the transient, incompressible, turbulent Navier-Stokes equations were solved using a projection method. To accommodate the complex geometry, a rectangular mesh was defined and each mesh node was assigned to a node function whose value depends on the location of the node. Droplets were regarded as individual entities and treated using the Lagrangian approach. Mass and heat transfer coefficients were introduced for individual droplets to model mass and heat transfer phenomena. The modeling results may be used to determine the spray characteristics in the vicinity of obstacles. Landau-Levich model of a diffusion boundary layer was suggested for modeling of mass transfer in continuous and dispersed phases of two immiscible liquids during droplet motion in extractors.[64]

Aerosol production and transport over the oceans are of interest in studies concerning cloud physics, air pollution, atmospheric optics, and air-sea interactions. However, the contribution of sea spray droplets to the transfer of moisture and latent heat from the sea to the atmosphere is not well known. In an effort to investigate these phenomena, Edson et al.[12] used an interactive Eulerian-Lagrangian approach to simulate the generation, turbulent transport and evaporation of droplets. The k-ε turbulence closure model was incorporated in the Eulerian-Lagrangian model to accurately simulate

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stable, near-neutral, and unstable boundary layers of the air-sea interactions. The Langevin equation was modified to account for the effects of the gravity and inertia of the droplets. It was showed that although the influence of the droplets was small under the conditions in this modeling, the potential for substantial modification of the surface energy budget exists for high-wind conditions over the oceans. The model system may be used to investigate interactions between evaporating droplets and turbulent fields of temperature and humidity in general.

In spray combustion processes, the multi-phase flows in internal combustion piston engines are unsteady, compressible (low Mach number), turbulent, cyclic and non-stationary, both spatially and temporally. The distribution of liquid and vapor fuel in an engine greatly influences the combustion characteristics. Fuel injection introduces additional complexity to the process physics and consequently to the modeling of the dense multi-phase flows. In recent years, the use of computer models to simulate fuel injection has become common practice. The models range from highly empirical equations[90] to detailed multidimensional models requiring the solution of basic conservation equations.[596] Intermediate approaches have also been adopted using integral continuity and momentum equations and relying heavily on experimental data. These models have been used to characterize and predict fuel spray penetration, trajectory, mixing and combustion within an engine combustion chamber.[438] The predictions are then used for engine design and analysis of data acquired from engines in operation. Comparisons of model predictions and experiments[597] have shown that the model predictions can respond properly to changes in nozzle geometry, injection pressure and direction, and air properties with reasonable accuracy.

Some early spray models were based on the combination of a discrete droplet model with a multidimensional gas flow model for the prediction of turbulent combustion of liquid fuels in steady flow combustors and in direct injection engines. In an improved spray model,[438] the full Reynolds-averaged Navier-Stokes equations were

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solved along with energy equation and k-ε turbulence model including compressibility effect for air flow velocity, pressure, density and enthalpy. Fuel droplet trajectories, size and temperature histories were calculated by solving the equations of motion of discrete droplets as well as heat and mass transfer equations. In the equations of motion, the pressure gradient and Basset terms were neglected due to their very small magnitudes. The Saffman lift and Magnus forces were also neglected because the droplets are not in a high shear region of the air flow under the flow conditions considered. The model included the effects of droplet size distribution and turbulence mixing. The interaction between a droplet and air was modeled by averaging the overflow processes on a scale smaller than the droplet size and using typical correlations for droplet drag, heat and mass transfer. The drag coefficient correlation for a non-evaporating droplet and the heat transfer coefficient correlation were multiplied by a correction factor to account for the decrease of droplet mass during the evaporation period. The injection nozzle was divided into five introduction locations and the injection period was discretized into a number of equal intervals of 1° crank angle. The quantity of fuel injected during each period was calculated from a detailed atomization model. The number of droplets for each size class was determined by dividing the total number of droplets equally on the five introduction locations and using the droplet size distribution. The initial droplet velocity varied corresponding to variations in fuel and combustion chamber pressures. Each droplet was assigned with a different initial location, size and velocity. A size increment of 10 µm was used to cover the entire range of droplet size. The flow field in the combustion chamber was subdivided into grid elements and each element was treated as a control volume. The net efflux rate of droplet mass from a computational element, which is a source term in the continuity equation, was obtained by assuming overall trajectories traversing that element. At any instant of time, the flow field was first assumed and predictions for a time increment were then obtained by solving the gas flow equations using a marching integration algorithm. From the solutions of the gas flow equations at each time step, the fuel concentration was calculated to determine spray contours. The solutions

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were marched forward in time until the desired period had been covered. The calculated results suggested that the characteristics of the mixing and the ensuing combustion processes in diesel engines are mainly influenced by droplet size distribution, physical and chemical properties of fuel, and medium conditions. The droplet size distribution is one of the important factors affecting the characteristics of spray combustion and subsequent emission formation.

A detailed review has been made by Reitz[525] on recent advances in the development of a comprehensive analytical model of spray combustion as a design tool for industrial applications. The model is based on the KIVA code.[555] In the model, the conservation equations are solved for the transient dynamics of vaporizing fuel sprays interacting with flowing multi-component gases and undergoing mixing, ignition, chemical reactions and heat transfer. The model encompasses sub-models for those processes that occur on the time and length scales much smaller than can be resolved by a single macro-model. The modifications made to the KIVA code include sub-models for atomization[232] and droplet drag,[572][598] dropletdroplet and droplet-wall impingement, heat transfer, ignition, combustion, soot formation and emissions.

As described previously, in the atomization sub-model,[232] droplet parcels are injected with a size equal to the nozzle exit diameter. The subsequent breakups of the parcels and the resultant droplets are calculated with a breakup model that assumes that droplet breakup times and sizes are proportional to wave growth rates and wavelengths obtained from the liquid jet stability analysis. Other breakup mechanisms considered in the sub-model include the Kelvin-Helmholtz instability, Rayleigh-Taylor instability,[206] and boundary layer stripping mechanisms. The TAB model[310] is also included for modeling liquid breakup.

In the droplet drag sub-model,[572][598] the effects of droplet distortion and oscillation due to droplet-gas relative motion on droplet drag coefficient are taken into account. Dynamical changes of the drag coefficient with flow conditions can be calculated with this submodel. Applications of the sub-models to diesel sprays showed that

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spray tip penetration is relatively insensitive to droplet drag coefficients whereas droplet size distribution is greatly influenced by droplet drag.

The spray model in the KIVA code[555] could not assess the physical phenomena in the dispersion process of spray impingement on a wall since all impinging droplets, called parcels, are treated as sticking on the wall.[422] To address this problem, Naber and Reitz[599] developed droplet-wall impingement sub-models for droplet-wall interactions during spray impingement on a wall, an important phenomenon in direct injection engines. The models include a reflect sub-model for droplet rebound and a liquid jet analogy sub-model for wall jet. Impinging droplets may rebound or form liquid films on a wall. During cold starting of engines, droplet impingement velocities are relatively low and the models predict droplet rebound as the dominant mode. The droplet rebound may be a mechanism for the back penetration of fuel vapor into the central region of a combustion chamber where gas temperature is high enough for ignition.

During the ignition delay period, the injected fuel is atomized and mixes with air after evaporation. Kong and Reitz[598] applied the multistep Shell ignition model, developed by Halstead et al.[600] for the autoignition of hydrocarbon fuels at high pressures and temperatures, to diesel combustion processes. The Shell model gives good predictions of diesel ignition.

Up to the time point of ignition,combustion may be controlled by the chemical kinetic rates. The combustion rate turns to be controlled by turbulent mixing after ignition. Several turbulent combustion models, such as the one-step reaction model,[601] the coherent flame model,[602] and the laminar-and-turbulent characteristic-time model,[229] have been developed that account for the effects of turbulence on mean reaction rates. In these models, the combustion time scale was related to the turbulent kinetic energy and its dissipation rate on the basis of the k-ε turbulence model. The model of Reitz and Bracco[229] adopted the eddy-breakup concept and was demonstrated to perform well along with the Shell ignition model in various engine applications. Details of the combustion model have been described by Kong and Reitz.[598]