Drop-Size Distributions Produced by Flat-Spray Nozzles

This study deals with measurements of the drop-size distributions of different types of fan spray atomisers using a scattered-light particle size counting analyser. The drop-size distributions were determined at various locations in the spray cone. These local distributions change systematically from the fan's axis to its border. Superimposing these local distributions, adequately weighted, one acquires the entire distribution of all the drops. A comparison of the experimental results is made with those yielded by mathematical equations.     

Deconvolution with Maximum Entropy Solution to Determine Local Extinction Coefficient and Local Volume Concentration Values from Laser Diffraction Data

The traditional use of the laser diffraction technique provides line‐of‐sight liquid spray drop‐size distribution. However, deconvolution of the measurements can be performed for axisymmetric spray in order to determine local spray characteristics. In a previous publication, a new deconvolution technique making use of the maximum entropy principle was established and applied to determine the local drop‐size distributions. The entire approach was experimentally validated. In this work, the technique is employed to determine local extinction coefficient values. As in the previous investigation, the measurement procedure consists of scanning a laser beam through the spray cross‐section from the center to the edge of the spray. By use of the transmittance theory, the local extinction coefficients allow the local volume concentrations to be calculated. This theory introduces the mean scattering coefficient. The results show that this coefficient must be determined as a function of the Sauter mean diameter in order to avoid overestimation of the volume concentration. Although no proper validation is presented, the coherence of the overall approach is discussed in detail and solutions for improving the spatial resolution are presented. Finally, the local volume concentrations are combined with the local drop‐size distribution to provide local volume‐weighted, drop‐size distributions. These distributions provide information on the localization of the drops according to their diameter as well as on the spatial liquid distribution. This work illustrates applications and performances of laser diffraction technique that are rarely used.     

环状出口气泡雾化喷嘴研究

本文对环状出口气泡雾化喷嘴出口下游液膜的破碎过程进行了研究,发现气体的介入是促使液膜破碎的主要原因。利用DUALPDA对其下游流场的速度分布、颗粒直径分布以及通量分布进行了实验测量。发现喷嘴出口附近主流区域存在大量具有负向速度的颗粒,并且此处的颗粒平均直径显著减小,为气泡雾化机理提供了佐证;在喷嘴出口下游轴心处液雾呈现逆向流动趋势,证明此处存在负压回流;沿轴向的速度分布曲线与颗粒直径分布曲线的变化趋势说明气泡"爆炸"发生在出口下游5-15 mm距离内。     

低能He~(2+)入射He原子转移电离实验中出射电子成像研究

利用反应显微谱仪对70keV He2+-He转移电离过程中的出射电子进行了成像,研究了出射电子的空间速度分布特征.结果表明:电子主要集中在散射平面内;在散射平面内,电子速度分布介于零与入射离子速度Vp之间(即前向出射)且在散射离子和靶核核间轴处有一极小值,呈现出典型的双峰结构.出射电子的上述分布特征可由出射电子波函数σ振幅和π振幅的干涉进行定性解释,σ振幅和π振幅对出射电子波函数的贡献与碰撞参数相关.在小碰撞参数下,π振幅的贡献更加明显;而在大碰撞参数下,σ振幅的贡献更加显著.     

On the Capability of the Generalized Gamma Function to Represent Spray Drop‐Size Distribution

The three‐parameter, Generalized Gamma function solution of a recent MEF formulation used to derive liquid spray drop‐size distribution, is applied to sprays resulting from three different atomization processes. The objectives of these applications are to determine the sign of the parameters for which this function reports a more reliable fit and to further understand the parameter stability problem reported elsewhere. It is found that the lack of stability of the parameters is related to a characteristic feature of the mathematical function and appears for a series of spray drop‐size distributions with constant shape. For each situation analyzed in the present study, the Generalized Gamma function provides a very good fit with parameters that are either constant or correlated to the working conditions. As far as the sign of the parameters is concerned, the results show that the best formulation is a function of the spray and that it is impossible to know, a priori, which parameter sign will report the best fit. Finally, for one situation, it is found that the Generalized Gamma function allows extrapolation of drop sizes outside the measured values. All of the results converge to conclude that the three‐parameter Generalized Gamma function, which is identical to the well‐known Nukiyama‐Tanasawa distribution, accumulates valuable attributes to represent liquid spray drop‐size distributions.     

Size of the Detection Area of a Phase-Doppler Anemometer for reflecting and refracting particles

Measurements of particle size distributions in multi-phase flows with a phase-Doppler anemometer yield incorrect results if polydisperse particles are investigated. For weighting biased size distributions, different in situ methods, requiring the size of the detection area, are known, but all of these weighting procedures are restricted to very small measuring volumes if off-axis instrument configurations are considered. Moreover, the weighting functions have some disadvantages in the case of poor statistics in single size classes or the results are not suitable for determining the size of the detection area for particles which are larger than the beam waist. Therefore, the intention in this work was to measure the size of the detection area for different kinds of monodisperse particles, different instrument configurations and varied instrument sensitivities experimentally and to develop an improved weighting procedure that copes with the above difficulties. The application of the results obtained from the investigations with monodisperse particles to measured particle size distributions and volume flux densities of polydisperse water droplets in a spray cone of an atomizer confirms the applicability of this weighting procedure. It is still restricted to directed flows, perpendicular to the fringes.     

Absorption and Scattering of Light by Highly Concentrated Two‐phase Flows

The spray cone emerging during an extended metal atomization process (called spray forming) has been investigated in order to quantify the influence of highly concentrated multiphase flows on phase‐Doppler‐anemometry (PDA) measurements. Using this non‐intrusive, optical measurement technique not only the local particle size and velocity distributions of the spray can be obtained but also additional information about the mass flux in the multiphase flow. Since standard phase‐Doppler systems can be easily applied to low concentrated particle systems (spherical particles with smooth surfaces and an optical transparent continuous phase taken for granted) the application of this measurement technique to highly concentrated multiphase flows is more complex. Both the laser light propagating from the PDA device to the probe volume and the scattered one going backward to the PDA receiving system are disturbed by passing the highly concentrated multiphase flow. The resulting significant loss in signal quality especially concerns the measurement of the smaller particles of the spray because of their reduced silhouette (in comparison with the bigger ones). Thus, the detection of the smallest particles becomes partially impossible leading to measurement of a distorted diameter distribution of the entire particle collective. In this study the distortions of the measured distributions dependent on the particle number concentration as well as on the path length of the laser light are discussed.     

Mean Particle Diameters. Part VIII. Computer Program to Decompose Mixtures of (Truncated) Lognormal Particle Size Distributions Using Differential Evolution to Generate Starting Values for Nonlinear Least Squares

Multimodal size distributions can result from a mixing of two or more component distributions and arise in quite different application areas. Physical and statistical approaches are described for decomposition of a multimodal particle size distribution into a number of lognormal components. These approaches, incorporated in the Fortran computer program FitDist, use a nonlinear least‐squares (NLLS) optimization, requiring initial parameter estimates. A hybrid deconvolution method has been developed. Differential Evolution (DE), is used for generation of initial parameter values, followed by an NLLS optimization to derive precise parameter values at the local optimum. The DE algorithm is required to decide on the proper number of modes to be fitted. Mathematical relationships have been derived to convert the parameter values of one multimodal lognormal moment distribution, e.g., a number distribution, to those of another moment distribution, e.g., a volume distribution. Moreover, mathematical relationships have been derived to compute mean diameters (Moment‐Ratio notation) from the parameters of a multimodal lognormal size distribution. Fitting a 1.5th moment distribution, being just in between a number and a volume distribution, has been introduced as an instrument to balance inaccuracies in both tails of a distribution due to sampling inaccuracies or truncation of these tails. The program fits a truncated size distribution by fitting its frequency density distribution, whereas a complete size distribution is fitted by fitting the cumulative distribution. Some guidelines are given for fitting Number, Diameter, Surface area, and Volume distributions to measured size distributions. Although fitting of multimodal normal distributions is an option, higher moment distributions will not be fitted as these distributions are not normally distributed. Practical examples demonstrate the validity of the method to decompose multimodal particle size distributions by use of DE.     

Application of the Maximum Entropy Formalism on Sprays Produced by Ultrasonic Atomizers

A recent application of the Maximum Entropy Formalism on liquid atomization problems led to the development of a mathematical volume‐based drop‐size distribution. This function, which depends on three parameters, is a reduction of the four‐parameter generalized Gamma function. The aim of the present work is to investigate the relevance of the three parameters in the characterization of liquid atomization processes. To achieve this, a variety of experimental drop‐size distributions of ultrasonic sprays were analyzed with the mathematical function. Firstly, it is found that the mathematical drop‐size distribution is very suitable to represent the volume‐based drop‐size distribution of ultrasonic sprays. Furthermore, it is seen that when considering the three parameters introduced by the function, one of them is constant for all the situations investigated, and the other two are linked to a non‐dimensional group that includes the main parameters controlling the drop production. These results are very important, since they suggest a possible development of physical models of primary atomization based on the M.E.F., which would allow for the prediction of the spray drop‐size distribution. Thusfar, such a model does not exist.     

Droplet size and velocity distributions for spray modelling

Methods for constructing droplet size distributions and droplet velocity profiles are examined as a basis for the Eulerian spray model proposed in Beck and Watkins (2002,2003) [5], [6]. Within the spray model, both distributions must be calculated at every control volume at every time-step where the spray is present and valid distributions must be guaranteed. Results show that the Maximum Entropy formalism combined with the Gamma distribution satisfy these conditions for the droplet size distributions. Approximating the droplet velocity profile is shown to be considerably more difficult due to the fact that it does not have compact support. An exponential model with a constrained exponent offers plausible profiles.     

Ligament-mediated spray formation

The spray formed when a fast gas stream blows over a liquid volume presents a wide distribution of fragment sizes. The process involves a succession of changes of the liquid topology, the last being the elongation and capillary breakup of ligaments torn off from the liquid surface. The coalescence of the liquid volumes constitutive of a ligament at the very moment it detaches from the liquid bulk produces larger drops. This aggregation process has its counterpart on the shape of the size distribution associated with the ligament breakup, found to be very well represented by gamma distributions. The exponential shape of the overall distribution in the spray coincides with the large excursion wing of these elementary distributions, underlying the crucial role played by the ligament dynamics in building up the broad statistics of sprays.     

气泡雾化喷嘴喷雾平均直径在下游流场中的分布

文利用激光衍射粒度仪对气泡雾化喷嘴下游流场进行了实验研究,主要分析了雾化颗粒直径随径向和轴向距离变化的趋势．由于喷嘴出口处气液两相流型和颗粒自身重量的影响,液雾颗粒沿径向呈现非轴对称分布;而液雾颗粒直径随着轴向距离的增加呈现先减小、后增大的趋势,颗粒直径的减小是大量气泡爆炸的结果,而后的增加则是由于颗粒之间的相互粘结造成的。     

Influence of Raindrop-Size Distribution on the Differential Reflectivity up to Submillimeter Wavelength of 0.96 mm

The rain attenuation was calculated by using the Marshall and Palmer, the gamma and the Weibull raindrop-size distributions. The microwave experimental measurement data from 8.4 GHz (3.75 mm) to 312.5 GHz (0.96 mm) were compared with our calculations. The Weibull distribution has the best agreement with the measurements, especially above 100 GHz. The differential reflectivity Z _DR was also calculated using these three distributions. To this end, we used the horizontal and vertical radar cross sections which were computed from Waterman's T-matrix method. It is shown that Z _DR greatly depends on raindrop-size distributions and has potential for determing drop-size distributions with high accuracy.     

Determination of Particle Size Distribution Parameters using a Laser-Scattered Light Spectrometer

A method based on the measurement of scattered light intensity distributions is demonstrated to be able to determine directly the particle size of monodisperse supermicron-size particles. In all other cases of a particle cloud, information about the size distribution can be acquired from comparison of measured and calculated intensities as a function of scattering angle. This indirect method is only applicable if the assumptions made in the theory used for comparison are fulfilled. Therefore, the method is limited to spherical particles with known refractive index. The type of size distribution also has to be known. In the cases considered a log-normal size distribution was assumed. The uncertainty of the result increases with increase in the number of parameters that have to be determined. The method seems to be limited to unimodal distributions described with two parameters.     

Shape and Size Determination by Laser Diffraction: Parametric Density Estimation by Neural Networks

The feasibility of the inversion of laser diffraction data for size and shape distribution by neural networks has been investigated by computer simulation. The size and shape density distributions are represented by only four parameters: the peak positions and the full width at half maximum. Compared to the approach whereby the distributions are represented by a histogram with 30 grid points, the results are an order of magnitude less accurate.     

A Predictive Model for the Initial Droplet Size and Velocity Distributions in Sprays and Comparison with Experiments

The distribution of sizes and velocities of droplets initially formed in sprays is an important piece of information needed in the spray modelling, because it defines the initial condition of the spray droplets in the predictive calculations of the downstream two‐phase flow fields. A predictive model for the initial droplet size and velocity distributions in sprays is formulated in this study. The present model incorporates both the deterministic and the stochastic aspect of spray formation process. The deterministic aspect takes into account of the unstable wave motion before the liquid bulk breakup through the linear and nonlinear instability analysis, which provides information for the liquid bulk breakup length, the mass‐mean diameter and a prior distribution for the droplet sizes corresponding to the unstable wave growth of various wavelengths. The stochastic aspect deals with the final stage of droplet formation after the liquid bulk breakup by statistical means through the maximum entropy principle based on Bayesian entropy. The two sub‐models are coupled together by the various source terms signifying the liquid‐gas interaction, the mass mean diameter and the prior distribution based on the instability analysis. The initial droplet size and velocity distributions are measured experimentally by phase‐Doppler interferometry for sprays generated by a planar research nozzle and a practical gas turbine airblast nozzle. For the two nozzles, the liquid bulk sheet is formed before its breakup in a coflowing air stream. It is found that the model predictions are in satisfactory agreement with the experimental data for all the cases measured. Hence the present model may be applied to a variety of practical sprays to specify the initial conditions for the spray droplets formed in practical spray systems.     

Effects of heterogeneous adoption thresholds on contact-limited social contagions

Limited contact capacity and heterogeneous adoption thresholds have been proven to be two essential characteristics of individuals in natural complex social systems, and their impacts on social contagions exhibit complex nature. With this in mind, a heterogeneous contact-limited threshold model is proposed, which adopts one of four threshold distributions, namely Gaussian distribution, log-normal distribution, exponential distribution and power-law distribution. The heterogeneous edge-based compartmental theory is developed for theoretical analysis, and the calculation methods of the final adoption size and outbreak threshold are given theoretically. Many numerical simulations are performed on the Erdös-Rényi and scale-free networks to study the impact of different forms of the threshold distribution on hierarchical spreading process, the final adoption size, the outbreak threshold and the phase transition in contact-limited propagation networks. We find that the spreading process of social contagions is divided into three distinct stages. Moreover, different threshold distributions cause different spreading processes, especially for some threshold distributions, there is a change from a discontinuous first-order phase transition to a continuous second-order phase transition. Further, we find that changing the standard deviation of different threshold distributions will cause the final adoption size and outbreak threshold to change, and finally tend to be stable with the increase of standard deviation.     

A Monte Carlo study of size and angular properties of a three-dimensional Poisson-Delaunay cell

On the basis of simulation of 1.2×10⁶ three-dimensional Poisson-Delaunay cells, the statistical properties of their size and angular parameters have been studied. The moments of the volume, face area, and edge length distributions are found to be equal to those obtained from the exact expressions of Miles and of Moller. The volume, surface area, and face area distributions can be described by the two-parameter gamma distribution. The normal distribution can be used to describe the distributions of the total edge length of a cell and the perimeter of a face. The edge length distribution has also been studied. The distribution of the angle in a face is found to be in accordance with its theoretical distribution.     

Bubble Velocity and Size Measurement with a Four‐Point Optical Fiber Probe

The possibility to measure the velocity and size of individual bubbles in a high‐void fraction bubbly flow is investigated by using a four‐point optical fiber probe. The air bubbles have an initial spherical equivalent diameter ranging from 4 to 10 mm and the void fraction is up to 0.3. Firstly, single bubble experiments show that intrusiveness effects, i.e. bubble deformations due to the probe, are negligible provided that the bubble approaches the probe at the axis of the central fiber. A selection criterion is utilized for multiple bubble experiments. A good compromise can be found between the required accuracy, the duration of the measurements and the number of validated bubbles required for reliable statistical averaging. In an air‐water high‐void fraction vertical bubbly pipe flow, the void fraction obtained with the instrument is found to be in good agreement with both local single‐fiber probe measurements, and with the volume average void fraction obtained from pressure gradient measurements. The area average volumetric gas flow rate, based on the bubble velocity and void fraction as measured with the four‐point probe, agree with the measured gas flow rate. Also, the liquid velocity is measured by means of a laser‐Doppler anemometer, to investigate the slip velocity. The results show that reliable and interesting measurements can be obtained by using a four‐point optical fiber probe in high void fraction flows.     

Stochastic modeling of atomizing spray in a complex swirl injector using large eddy simulation

Large-eddy simulation of an atomizing spray issuing from a gas-turbine injector is performed. The filtered Navier–Stokes equations with dynamic subgrid scale model are solved on unstructured grids to compute the swirling turbulent flow through complex passages of the injector. The collocated grid, incompressible flow algorithm on arbitrary shaped unstructured grids developed by Mahesh et al. (J. Comp. Phys. 197 (2004) 215–240) is used in this work. A Lagrangian point-particle formulation with a stochastic model for droplet breakup is used for the liquid phase. Following Kolmogorov’s concept of viewing solid particle-breakup as a discrete random process, the droplet breakup is considered in the framework of uncorrelated breakup events, independent of the initial droplet size. The size and number density of the newly produced droplets is governed by the Fokker–Planck equation for the evolution of the pdf of droplet radii. The parameters of the model are obtained dynamically by relating them to the local Weber number and resolved scale turbulence properties. A hybrid particle-parcel is used to represent the large number of spray droplets. The predictive capability of the LES together with Lagrangian droplet dynamics models to capture the droplet dispersion characteristics, size distributions, and the spray evolution is examined in detail by comparing it with the spray patternation study for the gas-turbine injector. The present approach is computationally efficient and captures the global features of the fragmentary process of liquid atomization in complex configurations.