Droplet Size Distribution Function Research Articles

Laws governing the fragmentation of a droplet in a high-speed stream

A common solution of the equations of ablation and motion of a droplet undergoing fragmentation in a homogeneous high-speed gas stream is obtained. Based on the approximate solution of the equation for the quantity of torn off droplets, an expression for the droplet size distribution function corresponding to these laws has been found. A comparison of the results of calculations carried out for the obtained distribution function with the same calculations for the earlier found function with the use of the empirical law of droplet motion points to their satisfactory agreement.

Dynamics of the droplet size distribution function during the gradual creation of a supersaturation state under different regimes of droplet growth

The method of the direct numerical integration of kinetic equations of droplet size distribution functions that was previously proposed by the authors is employed to solve the problem of condensation relaxation in a vapor-gas mixture during the creation of a supersaturation state at a finite rate. Two relaxation regimes are considered. In the static regime, the mixture is expanded at a constant rate until a preset supersaturation ratio is achieved; in the dynamic regime, the expansion is continued. Solutions are obtained for argon-cesium and argon-ethane mixtures, thus making it possible to study the dependence of the process character on the Knudsen number. The effects of the rate of the supersaturation creation and the relaxation regime on the droplet size distribution function are analyzed.

New method of simulation of volume condensation of supersaturated vapor

A method of direct numerical solution of kinetic equation for the droplet size distribution function is suggested for the investigation of the process of volume condensation of supersaturated vapor. Use is made of analogy with the corresponding method of solution of kinetic Boltzmann equation. An advantage of the suggested approach is the absence of limitations on the Knudsen number. The method is tested using the example of simulation of the behavior of vapor under conditions of rapid development of the state of supersaturation in a vapor-gas mixture as a result of adiabatic expansion. The results are compared in a wide range of Knudsen number with the results obtained using the moments method in a version of a set of moments equations. Investigation is performed of the effect of the droplet size dependence of saturation pressure on the dynamics of the process of condensation relaxation of supersaturated vapor.

Dynamics of the droplet size distribution function during bulk condensation in the case of finite-rate generation of supersaturation

Dynamics of the droplet size distribution function during bulk condensation in the case of finite-rate generation of supersaturation

Experiment and Computational Fluid Dynamics (CFD) Simulation of Urea-Based Selective Noncatalytic Reduction (SNCR) in a Pilot-Scale Flow Reactor

A turbulent reacting flow computational fluid dynamics (CFD) model involving a droplet size distribution function in the discrete droplet phase is first built for selective noncatalytic reduction (SNCR) processes using urea solution as a NOx removal reagent. The model is validated with the experimental data obtained from a pilot-scale urea-based SNCR reactor installed with a 150 kW gas burner. New kinetic parameters of seven chemical reactions for the urea-based NOx reduction are identified and incorporated into the three-dimensional turbulent flow CFD model. The two-phase droplet model with the non-uniform droplet size is also combined with the CFD model to predict the trajectory of the droplets and to examine the mixing between the flue gas and reagents. The maximum NO reduction efficiency of about 80%, experimentally measured at the reactor outlet, is obtained at 940 °C and a normalized stoichiometric ratio (NSR) = 2.0 under the conditions of 11% excess air and low CO concentration (10−15 ppm). At the reaction temperature of 940 °C, the difference of a maximum of 10% between experiments and simulations of the NO reduction percentage is observed for NSR = 1.0, 1.5, and 2.0. The ammonia slip is overestimated in CFD simulation at low temperatures, especially lower than 900 °C. However, the CFD simulation results above 900 °C show a reasonable agreement with the experimental data of NOx reduction and ammonia slip as a function of the NSR.

About use of a method of direct numerical solution for simulation of bulk condensation of supersaturated vapor

The results of direct numerical solution of the kinetic equation for the droplet size distribution function are presented. This method which is not restricted by the Knudsen number was developed using the analogy with a similar method of solution of the Boltzmann kinetic equation. The simulation of vapor behavior at fast creation of supersaturation state in vapor–gas mixture by means of adiabatic expansion was carried out for the verification of the method. The results obtained by this method were compared with those which were obtained by using the method of moments over a broad range of Knudsen number. The relevance of taking into account the dependence between saturation pressure and droplet size on the dynamics of condensational relaxation was studied.

A numerical study of water mist mitigation of tunnel fires

A Eulerian–Eulerian two-phase model is developed to numerically investigate the efficiency of water mist systems in mitigating thermoplastic fires in a tunnel. The polydisperse nature of the spray is modeled using the moments of the droplet size distribution function. The system of Favre-averaged Navier–Stokes equations for the gas phase is closed using the k – ε RNG turbulence model, the Eddy-Break-Up–Arrhenius model for turbulent combustion, and a multiphase radiative transfer equation including the contributions of soot, combustion products, and water droplets. A parametric study is carried out to assess the influence of the size distribution and injection conditions of the spray, and of the ventilation conditions on fire mitigation in a 25 m × 3 m × 5 m tunnel. Ventilation velocity is in the range 1–2 m/s and water flow rate is in the range 0.01–0.3 kg/s. Model results show that spray efficiency is mainly related to droplet dynamics and that optimum spray characteristics for mitigation and/or extinction are highly dependent on the configuration. It is found that fire extinguishment occurs during the transient stage at relatively high water flow rates as a consequence of a sharp decrease in the fuel pyrolysis rate. If extinction does not occur, then the two-phase flow resulting from the fire/water spray interaction tends rapidly toward a quasi-steady state. With an increasing water flow rate, fire mitigation exhibits an asymptotic behavior at steady state.

Intensity and efficiency of spray fuel-fed well-mixed adiabatic combustors

Motivated by the insights it can provide, we revisit the classical problem of liquid fuel-fed idealized steady-flow combustors. New quadrature-based results are presented for the theoretical combustion intensity and corresponding efficiency for well-stirred adiabatic vessels fed with a prescribed polydispersed spray. Each droplet of the spray is assumed to evaporate according to a non-quasi-steady (non-QS) gas-phase energy/mass diffusion-controlled rate for the pseudo-single-component fuel. As a byproduct, we calculate the complete droplet size distribution (DSD) function exiting the chamber, of interest for the design of downstream components. We explicitly assume that the volumetric rate of chemical energy release in such “primary” combustion chambers is controlled by the liquid fuel physical vaporization process (with negligible lags due to propellant droplet heat-up or vapor-phase ignition). In this instructive asymptotic limit, two decisive non-dimensional parameters are shown to be: (1) a vaporization Damköhler number (defined by the ratio of the mean residence time of the chemically reacting vapor mixture in the combustion space, to the reference value of the vaporization lifetime of a droplet with the injector-Sauter-mean diameter, and (2) a single dimensionless non-QS parameter. Illustrative numerical results for a kerosene-like fuel burning in air at pressures up to 24 atm are displayed for the case of a log-normal feedstream DSD with a range of spreads. Our results reveal the existence of an optimum vaporization Damköhler number which maximizes the combustion intensity—with maximum intensities, occurring well before nearly complete fuel evaporation, being quite sensitive to the non-QS parameter at high pressures. These deliberately idealized mathematical model results, spanning more than a 1000 combinations of operational parameters, set instructive bounds to the achievable performance of “real” spray combustors. Even without tractable enhancements (see Section 5.2), this approach can be used to economically map the sensitivity of spray combustor performance to a large number of important design and control parameters.

Antioxidant dispersions in emulsified olive oils

Abstract In this paper water droplets dispersed in olive oils were used as containers of l -ascorbic acid and its inhibitory effect on lipid oxidation was studied. In detail, olive oil samples were emulsified by adding a certain quantity of vitamin C aqueous solutions and then oxidized by UV light. Data collected from sample analysis by optical microscopy were used to calculate the average droplets parameters and to characterize the droplet size distribution function. The results revealed a strong correlation between the initial rate of hydroperoxide formation and the specific surface area of aqueous dispersed phase. Such correlation has been explained as the consequence of a synergistic interaction between the ascorbic acid contained in the droplets and the natural tocopherols present in olive oil. A key role is played by the water–olive oil interface which is the boundary where the synergisms take place.

Predicting Fire Suppression Efficiency Using Polydisperse Water Sprays

Thermoplastic fire suppression by water sprays is numerically investigated using an Eulerian-Eulerian two-phase approach. The polydisperse spray model is based on the moments of the droplet size distribution function. Turbulent combustion is approached using the Arrhenius/Eddy-Break-up model coupled with the RNG k − ϵ turbulence model. A multiphase radiative transfer equation including the contributions of soot, combustion products and water droplets is used to describe radiation. Pyrolyzate mass flow rate is predicted from a thermal degradation model for poly-methyl-methacrylate. The influence that the main parameters of the water spray system, located directly above the fire source, have on the fire suppression efficiency is examined. Over the wide range of spray conditions tested, it is found that, first, polydisperse sprays are generally more efficient than monodisperse sprays, and, second, for a polydisperse spray, there exists an optimal reference Sauter mean radius, which corresponds to the shortest time for fire extinguishment. Results indicate that the time for fire suppression using polydisperse sprays decreases as the water flow rate increases, but with an asymptotic behavior. Finally, the model is used to determine the minimal water flow rate required for extinction.

Study of the volume condensation process in supersaturated vapor by the direct numerical solution of the kinetic equation for the droplet size distribution function

The volume condensation of supersaturated vapor is investigated by the direct numerical solution of the basic kinetic equation for the droplet size distribution function by analogy with the corresponding solution of the Boltzmann kinetic equation. The proposed consideration of the condensation growth of droplets is applicable at any Knudsen number. The method is tested by the example of vapor condensation under the conditions of the rapid development of supersaturation in a vapor-gas mixture as a result of its adiabatic expansion. In a wide range of Knudsen numbers, the results of the modeling are compared with those obtained by the moment method.

Effect of the droplet-size dependence of saturated-vapor pressure on the dynamics of supersaturated-vapor condensation relaxation

409 In this paper, we analyze the variation of the dropletsize distribution functions, of the fractional droplet composition for a condensation aerosol, and of certain integral characteristics (temperature, supersaturation, and the droplet-number density) that occur in the course of condensation-relaxation process. The consideration is restricted to the period after the rapid creation of the supersaturated state in a vapor‐gas mixture as a result of adiabatic expansion. In order to describe the dynamics of the droplet-size distribution function, we make use of the corresponding kinetic equation. This equation is solved by the direct numerical method. The results are obtained for growing droplets in the cases of large and intermediate Knudsen numbers with and without allowance for the dependence of the saturatedvapor pressure on the droplet radius. The analysis of the results has allowed us to reach a conclusion about how the droplet-size dependence of the saturated-vapor pressure affects the dynamics of the condensationrelaxation process.

On phase transition in compressible flows: modelling and validation

A physical model for compressible flows with phase transition is described, in which all the processes of phase transition, i.e. nucleation, droplet growth, droplet evaporation and de-nucleation, are incorporated. The model is focused on dilute mixtures of vapour and droplets in a carrier gas with typical maximum liquid mass fraction smaller than 0.02. The new model is based on a reinterpretation of Hill's method of moments of the droplet size distribution function. Starting from the general dynamic equation, it is emphasized that nucleation or de-nucleation correspond to the rates at which droplets enter or leave droplet size space, respectively. Nucleation and de-nucleation have to be treated differently in agreement with their differences in physical nature. Attention is given to the droplet growth model that takes into account Knudsen effects and temperature differences between droplets and gas. The new phase transition model is then combined with the Euler equations and results in a new numerical method: ASCE2D. The numerical method is first applied to the problem of shock/expansion wave formation in a closed shock tube with humid nitrogen as a driver gas. Nucleation and droplet growth are induced by the expansion wave, and in turn affect the structure of the expansion wave. When the main shock, reflected from the end wall of the low-pressure section, passes the condensation zone, evaporation and de-nucleation occur. As a second example, the problem of the flow of humid nitrogen in a pulse-expansion wave tube, designed to study nucleation and droplet growth in monodisperse clouds, is investigated experimentally and numerically.

Stochastic diffusion interactions and coarsening in a system of droplets growing from a supersaturated gas mixture

In this work we study diffusion interactions among liquid droplets growing in stochastic population by condensation from supersaturated binary gas mixture. During the postnucleation transient regime collective growth of liquid droplets competing for the available water vapor decreases local supersaturation leading to the increase of critical radius and the onset of coarsening process. In coarsening regime the growth of larger droplets is prevailing noticeably broadening the droplet size-distribution function when the condensation process becomes more intensive than the supersaturation yield. Modifications in the kinetic equation are discussed and formulated for a stochastic population of liquid droplets when diffusional interactions among droplets become noteworthy. The kinetic equation for the droplet size-distribution function is solved together with field equations for the mass fraction of disperse liquid phase, mass fraction of water vapor component of moist air, and temperature during diffusion-dominated regime of droplet coarsening. The droplet size and mass distributions are found as functions of the liquid volume fraction, showing considerable broadening of droplet spectra. It is demonstrated that the effect of latent heat of condensation considerably changes coarsening process. The coarsening rate constant, the droplet density (number of droplets per unit volume), the screening length, the mean droplet size, and mass are determined as functions of the temperature, pressure, and liquid volume fraction.

Equivalence of two different integral representations of droplet distribution moments in condensing flow

It is proved that two different and independently derived integral representations of droplet size distribution moments encountered in the literature are equivalent and, moreover, consistent with the general dynamic equation that governs the droplet size distribution function. One of these representations consists of an integral over the droplet radius while the other representation consists of an integral over time. The proof is based on analytical solution of the general dynamic equation in the absence of coagulation but in the presence of both growth and nucleation. The solution derived is explicit in the droplet radius, which is in contrast with the literature where solutions are presented along characteristics. This difference is essential for the equivalence proof. Both the case of nonconvected vapor as well as the case of convected vapor are presented.

On the development of a spray model based on drop-size moments

To date, all polydisperse spray models have involved discretizing the liquid flow field into groups of equally sized droplets. This paper describes the design and implementation of a spray model able to capture the full polydisperse nature of the spray flow without using droplet–size classes. Instead, the moments of the droplet–size distribution function are used to describe the distribution of droplet sizes. Transport equations are written for the two moments, which represent the liquid mass and surface area, and two more moments representing total radius and droplet number are approximated via use of a presumed distribution function allowed to vary in space and time. The velocities to be used in the two transport equations are obtained by defining moment–average quantities and constructing further transport equations for the relevant moment–average velocities. An equation for the energy of the liquid–phase and standard gas–phase equations, including a k – e turbulence model, are also solved. All the equations are solved in an Eulerian framework using the finite–volume approach, and the phases are coupled through source terms. Effects such as droplet break–up and droplet–droplet collisions are also captured through the use of source terms, and all the source terms are derived in terms of the four moments of the droplet–size distribution in order to find the net effect on the whole spray flow field. The model is here applied to a wide variety of different sprays, including high–pressure diesel sprays, wide–angle solid –cone water sprays and hollow–cone sprays. Tests show the model behaves consistently, and performs well in comparisons with experimental data. The inter–phase drag model produces excellent agreement in the spray–penetration results, and the moment–average velocity approach gives good radial distributions of droplet size, showing the capability of the model to predict polydisperse behaviour.

On the Development of Spray Submodels Based on Droplet Size Moments

Hitherto, all polydisperse spray models have been based on discretising the liquid flow field into groups of equally sized droplets. The authors have recently developed a spray model that captures the full polydisperse nature of the spray flow without using droplet size classes (Beck, 2000, Ph.D thesis, UMIST; Beck and Watkins, 2001, Proc. R. Soc. London A). The parameters used to describe the distribution of droplet sizes are the moments of the droplet size distribution function. Transport equations are written for the two moments which represent the liquid mass and surface area, and two more moments representing the sum of drop radii and droplet number are approximated via use of a presumed distribution function, which is allowed to vary in space and time. The velocities to be used in the two transport equations are obtained by defining moment-average quantities and constructing further transport equations for the relevant moment-average velocities. An equation for the energy of the liquid phase and standard gas phase equations, including a k–ε turbulence model, are also solved. All the equations are solved in an Eulerian framework using the finite-volume approach, and the phases are coupled through source terms. Effects such as interphase drag, droplet breakup, and droplet–droplet collisions are also captured through the use of source terms. The development of the submodels to describe these effects is the subject of this paper. All the source terms for the hydrodynamics of the spray are derived in this paper in terms of the four moments of the droplet size distribution in order to find the net effect on the whole spray flow field. The development of similar submodels to describe heat and mass transfer effects between the phases is the subject of a further paper (Beck and Watkins, 2001, J. Heat Fluid Flow). The model has been applied to a wide variety of different sprays, including high-pressure diesel sprays, wide-angle solid-cone water sprays, hollow-cone spray s, and evaporating sprays. The comparisons of the results with experimental data show that the model performs well. The interphase drag model, along with the model for the turbulent dispersion of the liquid, produces excellent agreement in the spray penetration results, and the moment-average velocity approach gives good radial distributions of droplet size, showing the capability of the model to predict polydisperse behaviour. Good submodel performance results in droplet breakup, collisions, and evaporation effects (see (Beck and Watkins, 2001, J. Heat Fluid Flow)) also being captured successfully.

Rheology of spray‐dried egg yolk‐stabilized emulsions

This paper deals with the influences that concentrations of oil fraction (65–77.5% w/w) and egg yolk (1–5% w/w), as well as temperature (5–35 °C), exerted on the droplet size distribution and rheological functions of concentrated oil‐in‐water food emulsions that were stabilized by a spray‐dried egg yolk product. This work must be considered as a preliminary study concerning the use of low‐in‐cholesterol egg yolk as emulsifier. In order to achieve this aim, steady‐state flow and dynamic viscoelasticity tests were done with emulsions processed with a rotor‐stator turbine. The same processing conditions were always maintained. All of the emulsions prepared showed a high stability. An increase in oil fraction yielded higher values for the rheological functions, and larger droplet diameters. The influence of processed egg yolk concentration was more complex. Thus, an increase in egg yolk content yielded lower values for the rheological functions, but, after a particular concentration, a further increase in emulsifier content yielded higher values for the rheological functions. The experimental results have been discussed taking into account the close relationship between rheology and emulsion structural parameters.

On the Theoretical Prediction of Fuel Droplet Size Distribution in Nonreactive Diesel Sprays

Droplet size distribution function and mean diameter formulas are derived using information theory. The effects of fuel droplet evaporation and coalescence within combustion chamber on the droplet size are emphasized in nonreactive diesel sprays. The size distribution function expressions at various spray axial cross sections are also formulated. The computations are compared with experimental data and KIVA-II code. A good agreement is obtained between numerical and experimental results. Droplet size distribution and mean diameter at various locations from injector exit and at various temperature conditions are predicted. The decreases of droplet number and variations of mean diameter are computed at downstream and higher temperature.

Water Spray Pattern Discharged from High Headroom Atrium Sprinkler

The water spray patterns discharged from an atrium sprinkler are studied in this paper. Equations for calculating the size distribution and the trajectory of the droplets in the water spray are reviewed. The trajectories of water droplets are studied by solving the equation of motion, with the velocity components of droplets expressed in analytical form. A new droplet size distribution function for the water spray derived earlier from information theory was used. The shape of the sprinkler water sprays was calculated. Results are useful in studying the interaction between sprinkler and free-induced airflow. Furthermore, dividing the spray radius into different parts, the volume of water received at the floor level was calculated to determine an important parameter: the water distribution function.