Drop Size Distribution Function Research Articles

Dropwise condensation mechanisms when varying vapor velocity

The promotion of dropwise condensation (DWC) has been identified as an effective strategy to significantly improve the heat transfer coefficient (HTC) as compared to filmwise condensation (FWC). Understanding the mechanisms governing dropwise condensation on modified wettability surfaces is crucial for a wide range of energy applications. In the literature, most of the experimental data are collected during DWC with quiescent vapor. On the other hand, in industrial applications, the vapor to be condensed can have a non-negligible velocity which is expected to affect the droplet population on the condensing surface and the heat transfer. However, measurements of heat transfer coefficient and droplet population with flowing steam are rare and the effect of vapor velocity on the drop-size distribution, which is a key parameter in DWC modeling, needs to be investigated.In the present work, the effect of steam velocity during DWC is experimentally studied on two different specimens: a sol–gel coated aluminum sample and a reduced graphene oxide coated copper sample. HTC, droplet departing radius and drop-size distribution measurements are performed at constant saturation temperature and heat flux, while varying the inlet vapor velocity in the range between 3 and 15.5 m s−1. Due to the increase of the vapor drag force on the droplets, a reduction of the droplet departing radius is observed along with an increase of the condensation HTC. The vapor flow is found to affect the droplet population and, in particular for the largest droplets radii, the drop-size distribution function is lowered when increasing vapor velocity. The experimental data are used to assess a model for the estimation of the droplet departing radius in presence of vapor velocity previously proposed by the present authors. As a second step, the equation for the calculation of the droplet departing radius is coupled with available models for droplet population and heat transfer through a single droplet to model the whole DWC process. The proposed calculation method is able to predict the effect of vapor velocity on the DWC heat transfer coefficient.

Turbulent droplet breakage in a von Kármán flow cell

Droplet dispersion in liquid–liquid systems is a crucial step in many unit operations throughout the chemical, food, and pharmaceutic industries, where improper operation causes billions of dollars of loss annually. A theoretical background for the description of droplet breakup has been established, but many assumptions are still unconfirmed by experimental observations. In this investigation, a von Kármán swirling flow device was used to produce homogeneous, low-intensity turbulence suitable for carrying out droplet breakage experiments using optical image analysis. Individual droplets of known, adjustable, and repeatable sizes were introduced into an isotropic turbulent flow field providing novel control over two of the most important factors impacting droplet breakage: turbulence dissipation rate and parent droplet size. Introducing droplets one at a time, large data sets were gathered using canola, safflower, and sesame oils for the droplet phase and water as the continuous phase. Automated image analysis was used to determine breakage time, breakage probability, and child droplet size distribution for various turbulence intensities. Breakage time and breakage probability were observed to increase with increasing parent droplet size, consistent with the classic and widely used Coulaloglou–Tavlarides breakage model (C–T model). The shape of the child drop size distribution function was found to depend upon the size of the parent droplet.

Drop sizes and its distribution in jet-driven liquid–liquid mixing column: substantial application for the liquid–liquid extraction process

The present work reports the study of drop size distribution, drop aspect ratio, and interfacial area in a liquid–liquid jet-driven mixing column. Four different liquids, i.e., kerosene, diesel, 1-decanol, and paraffin, are used as dispersed phases and water as the continuous phase. The effect of the jet velocity, the volume fraction of the dispersed phase, and the physical properties of a liquid mixture such as density, viscosity, and surface tension are studied on the drop size distribution and drop aspect ratio. The column diameter of 0.37 m and nozzle diameter of 0.005 m were used. The velocity of the jet is varied between 0.68 m/s–1.71 m/s. It is observed that the drop size decreases as the volume fraction of the dispersed phase and jet velocity increases. Drop size distribution is found to follow the Log-logistic drop size distribution function. The aspect ratio is close to unity for a small drop, and it decreases as the drop size increases. The increase in the Eötvös number leads to a decrease in the aspect ratio. The correlation for the drop aspect ratio is proposed in terms of the Reynolds and Eötvös numbers in the ranges of 0.061 <Re < 342.59, 0.004 <Eo < 1.166, and (−4.756) < log10 (Mo) < (−9.171), respectively. Interfacial area of drop enhances with an increase in the dispersed volume phase and jet velocity. Empirical correlation is developed to predict Sauter mean drop diameter, drop aspect ratio, distribution function parameters, and interfacial area based on operating variables and physical properties of the system.

Invariance of the Double-Moment Normalized Raindrop Size Distribution through 3D Spatial Displacement in Stratiform Rain

AbstractDouble-moment normalization of the drop size distribution (DSD) summarizes the DSD in a compact way, using two of its statistical moments and a “generic” double-moment normalized DSD function. Results are presented of an investigation into the invariance of the double-moment normalized DSD through horizontal and vertical displacement in space, using data from disdrometers, vertically pointing K-band Micro Rain Radars, and an X-band polarimetric weather radar. The invariance of the double-moment normalized DSD is tested over a vertical range of up to 1.8 km and a horizontal range of up to approximately 100 km. The results suggest that for practical use, with well-chosen input moments, the double-moment normalized DSD can be assumed invariant in space in stratiform rain. The choice of moments used to characterize the DSD affects the amount of DSD variability captured by the normalization. It is shown that in stratiform rain, it is possible to capture more than 85% of the variability in DSD moments zero to seven using the technique. Most DSD variability in stratiform rain can thus be explained through the variability of two of its statistical moments. The results suggest similar behavior exists in transition and convective rain, but the limited data samples available do not allow for robust conclusions for these rain types. The results have implications for practical uses of double-moment DSD normalization, including the study of DSD variability and microphysics, DSD-retrieval algorithms, and DSD models used in rainfall retrieval.

Design Rules for the Taylor‐Couette Disc Contactor

AbstractThe Taylor‐Couette disc contactor (TCDC) is a stirred liquid‐liquid phase contactor which is suitable for applications in bioseparations. For liquid‐liquid reactor design, information about the specific mass transfer area is inevitable. Therefore, the drop size distribution and holdup in the TCDC were investigated under various operating conditions and appropriate correlations for the prediction of these parameters have been determined. Experimental data of drop size distributions were correlated with lognormal, Gaussian, and Weibull drop size distribution functions.

Influence of drop size distribution function on simulated ground precipitation for different cloud droplet number concentrations

A cloud-resolving mesoscale model with a two-moment bulk microphysical scheme is used to perform cloud simulations for two different modes of the liquid water spectrum: a unified Khrgian–Mazin size distribution for the entire spectrum of drops and the monodisperse size distribution for cloud droplets with the exponential Marshall–Palmer distribution for raindrops. The cloud model calculates the mixing ratios and number concentrations of the six microphysical categories: raindrops, cloud ice, snow, graupel, frozen raindrops and hail. The cloud droplet number concentration was prescribed. The main purpose of this sensitivity study was to analyse the differences in simulated surface precipitation (rain and hail) for the two assumed approaches with different values of cloud droplet number concentration. The study showed that there are significant differences in the occurrence, amount and spatial distribution of accumulated precipitation at the surface. It can be noted that the unified Khrgian–Mazin size distribution is generally more sensitive to changes in the cloud droplet number concentration than an alternative approach. The rain showers and cloud splitting are well simulated with the unified Khrgian–Mazin size distribution, especially for smaller values of cloud droplet number concentration.

Air density of the Permian atmosphere: Constraints from lithified raindrop imprints

In contrast to the composition of Earth's ancient atmosphere, the corresponding air density is almost unknown. A unique method to estimate the palaeo-atmospheric density is to use lithified raindrop imprints. The size of imprints is amongst other factors controlled by the air density, whereas large impacts form in a less dense atmosphere and vice versa. This basic relation was used to estimate the near sea-level atmospheric density of the Permian, a period characterised by an icehouse to greenhouse transition and an atmospheric composition comparable to the modern one. This makes the Permian an important analogue to better understand the current climate and environmental changes. In this study we analysed lithified raindrop imprints preserved in Permian terrestrial sandstone from the Flechting High, northern Germany. Following former applications, we experimentally determined the relation between raindrop momentum and imprint dimension to estimate the palaeo-air density. The maximum bound of the Permian air density is ~2.3kg/m3, assuming that the maximum measured imprint area of 68±1.5mm2 was formed by the largest naturally occurring raindrop with a diameter of 6.8mm. Although we cannot rule out an air density comparable to the present day (~1.2kg/m3), more realistic estimates of maximum raindrop diameters for a rainfall rate of 100mm/h are between >3.2 and >4.3mm, yielding air density estimates of 0.3 and 0.85kg/m3, approximately one-fourth to three-quarters of the present-day value. This approach is based on a few largest observed imprints, and does take the complete raindrop imprint distribution into account. The imprint distribution primarily depends on the drop size distribution of the rainfall event that causes the imprints, which follow a defined function. In this study we present a novel method to further constrain the palaeo-air density estimation by comparing the observed distribution of lithified raindrop imprints with modelled imprint distributions for different rainfall rates and drop size distribution functions. We observed significant differences; theoretical imprint distributions are in general shifted towards smaller imprints, which we interpret as a higher probability of larger imprints to be preserved. Above a threshold imprint area of 10mm2, observed and modelled distributions match each other, suggesting that a major fraction (~50%) of raindrops did not generate recognisable imprints in the analysed substrate. Unfortunately no unique rainfall rate fits the observed imprint distribution, but instead pairs of rainfall rates and air densities yield reasonable fits. Palaeo-air density estimates therefore depend on an unknown rainfall rate. Assuming that lithified imprints formed by a rain event with a rainfall rate of <100mm/h (<50mm/h) result in an upper estimate of the Permian atmospheric air density of ~1.1kg/m3 (~0.9kg/m3). These results corroborate estimates based on the maximum observed imprint area and realistic maximum raindrop diameters, suggesting that the Permian atmospheric air density was comparable or slightly lower than the present-day density.

Parameterization of the Sedimentation of Raindrops with Finite Maximum Diameter

In common cloud microphysics parameterization models, the prognostic variables are one to three moments of the drop size distribution function. They are defined as integrals of the distribution function over a drop diameter ranging from zero to infinity. Recent works (by several authors) on a one-dimensional sedimentation problem have pointed out that there are problems with those parameterization models caused by the differing average propagation speeds of the prognostic moments. In this study, the authors propose to define the moments over a finite drop diameter range of [0, Dmax], corresponding to the limitation of drop size in nature. The ratios of the average propagation speeds are thereby also reduced. In the new model, mean particle masses above a certain threshold depending on Dmax lead to mathematical problems, which are solved by a mirroring technique. An identical, one-dimensional sedimentation problem for two moments is used to analyze the sensitivity of the results to the maximum drop diameter and to compare the proposed method with recent works. It turns out that Dmax has a systematic influence on the model’s results. A small, finite maximum drop diameter leads to a better representation of the moments and the mean drop mass when compared to the detailed microphysical model.

CloudSat Precipitation Profiling Algorithm—Model Description

Abstract Identifying and quantifying the intensity of light precipitation at global scales is still a difficult problem for most of the remote sensing algorithms in use today. The variety of techniques and algorithms employed for such a task yields a rather wide spectrum of possible values for a given precipitation event, further hampering the understanding of cloud processes within the climate. The ability of CloudSat’s millimeter-wavelength Cloud Profiling Radar (CPR) to profile not only cloud particles but also light precipitation brings some hope to the above problems. Introduced as version zero, the present work uses basic concepts of detection and retrieval of light precipitation using spaceborne radars. Based on physical principles of remote sensing, the radar model relies on the description of clouds and rain particles in terms of a drop size distribution function. Use of a numerical model temperature and humidity profile ensures the coexistence of mixed phases otherwise undetected by the CPR. It also provides grounds for evaluating atmospheric attenuation, important at this frequency. Related to the total attenuation, the surface response is used as an additional constraint in the retrieval algorithm. Practical application of the profiling algorithm includes a 1-yr preliminary analysis of global rainfall incidence and intensity. These results underscore once more the role of CloudSat rainfall products for improving and enhancing current estimates of global light rainfall, mostly at higher latitudes, with the goal of understanding its role in the global energy and water cycle.

ANALYSIS OF A TWO-FLUID SPRAYER AND ITS USE TO DEVELOP THE NUMBER SIZE DISTRIBUTION MOMENTS SPRAY MODEL, PART II: COMPUTATIONAL ANALYSIS

This study presents the development of a new spray model that captures the full polydisperse nature of spray flow without using drop size classes. In this model, the size information concerning the spray is obtained by calculating three moments of the drop size distribution function from transport equations and one moment from a Gamma distribution function. The method reduces the need to presume a particular distribution function for the drop sizes and the complexity of the solution scheme. Drop size data obtained from the examination of the structures of a free spray using a laser-diffraction-based drop analyzer were used to further develop the new spray model and determine its ability to accurately predict such sprays. Test results include the axial distributions of the radial-averaged drop sizes and the local drop distribution at axial positions, and indicate that the drop size distribution is dependent on the atomizing air pressure of the injector and the position of the drop from the injector nozzle. At higher pressure cases the distributions were observed to be bimodal. These trends are captured by the computational results and compare well with those from another moments spray model.

The Bias and Error in Moment Estimators for Parameters of Drop Size Distribution Functions: Sampling from Gamma Distributions

Abstract This paper complements an earlier one that demonstrated the bias in the method-of-moments (MM) estimators frequently used to estimate parameters for drop size distribution (DSD) functions being “fitted” to observed raindrop size distributions. Here the authors consider both the bias and the errors in MM estimators applied to samples from known gamma DSDs (of which the exponential DSD treated in the earlier paper is a special case). The samples were generated using a similar Monte Carlo simulation procedure. The skewness in the sampling distributions of the DSD moments that causes this bias is less pronounced for narrower population DSDs, and therefore the bias problems (and also the errors) diminish as the gamma shape parameter increases. However, the bias still increases with the order of the moments used in the MM procedures; thus it is stronger when higher-order moments (such as the radar reflectivity) are used. The simulation results also show that the errors of the estimates of the DSD parameters are usually larger when higher-order moments are employed. As a consequence, only MM estimators using the lowest-order sample moments that are thought to be well determined should be used. The biases and the errors of most of the MM parameter estimates diminish as the sample size increases; with large samples the moment estimators may become sufficiently accurate for some purposes. Nevertheless, even with some fairly large samples, MM estimators involving high-order moments can yield parameter values that are physically implausible or are incompatible with the input observations. Correlations of the sample moments with the size of the largest drop in a sample (Dmax) are weaker than for the case of sampling from an exponential DSD, as are the correlations of the MM-estimated parameters with Dmax first noted in that case. However, correlations between the estimated parameters remain because functions of the same observations are correlated. These correlations generally strengthen as the sample size increases.

The impact of the choice of the entire drop size distribution function on Cumulonimbus characteristics

The study compares two different ways to represent the size distributions of cloud droplets and raindrops in bulk microphysical schemes in the scope of cloud-resolving mesoscale modeling of cumulonimbus clouds. A unified Khrgian-Mazin size distribution for the entire liquid water is systematically compared to a widely applied approach using a Marshall-Palmer size distribution for rain and a fitted monodisperse distribution for cloud droplets. The impact of the distribution function on cloud microphysics, precipitation characteristics, cloud appearance and dynamics of one simulated cumulonimbus case is investigated. The agreement of the model values with typical observed cloud characteristics is discussed. The results of our study reveal that there are considerable differences between the two approaches, both with respect to the microphysical production terms and with respect to the cloud appearance while the differences are less pronounced in cloud dynamics. An important result is that the Khrgian-Mazin size distribution leads to more cumulative rain compared to the one composed of a monodisperse and a Marshall-Palmer distribution. There is observational evidence that a storm splitting, hail field characteristics and cumulative total precipitations are simulated more accurately by the Khrgian-Mazin distribution function.

Drop breakage in liquid–liquid stirred dispersions: Modelling of single drop breakage

A new experimental approach has been developed for the identification of reasonable break-up mechanisms in stirred dispersions and of a physically based model for the daughter drop size distribution. In the experiments, breakage of a single organic droplet and the subsequent fragments formation are analyzed by image processing techniques. The experimental data are then fitted by means of a daughter drop size distribution function written in terms of a probability density function (pdf). The mathematical approach that matches the theoretical requirements and that provides the best fit of the experimental data is the pdf proposed by Diemer and Olson [2002. A moment methodology for coagulation and breakage problems: part 3—generalized daughter distribution functions. Chemical Engineering Science 57(19), 4187–4198]. This is a purely statistical model that does not contain any dependence on physical parameters, such as turbulence intensity and mother drop size. Based on the experimental results, an extension of the Diemer and Olson model is proposed, in which dependence on the Weber number as well as on the mother drop diameter can be inserted. Other models appeared in the literature [ Coulaloglou and Tavlarides, 1977. Description of interaction processes in agitated liquid–liquid dispersions. Chemical Engineering Science 32(11), 1289–1297; Konno et al., 1983. Scale effect on breakup process in liquid–liquid agitated tanks. Journal of Chemical Engineering of Japan 16(4), 312–319; Martínez-Bazán et al., 1999. On the breakup of an air bubble injected into a fully developed turbulent flow: part 2. Size PDF of the resulting daughter bubbles. Journal of Fluid Mechanics 401, 183–207] are also discussed and compared with experimental data. Out of them, the only model that approaches experimental data in a satisfactory way is the model by Martínez-Bazán et al. [1999. On the breakup of an air bubble injected into a fully developed turbulent flow: part 2. Size PDF of the resulting daughter bubbles. Journal of Fluid Mechanics 401, 183–207]. Eventually, the presence of the maximum break-up probability for symmetric breakage, in contrast with some theoretical models published in the literature, is explained and justified by means of energy considerations.

A novel approach for structure quantification of fatty acids films on rain water

A complete compositional or structural description of naturally occurring surfactants on rainwater is not currently feasible. A main limitation of previous work has been the lack of means for correlating force-area characteristics with the chemical makeup of the films. Instead of analyzing the chemical composition of rain water film-forming organics, it is postulated here to introduce the novel scaling procedures (2D virial equation of state and 2D polymer film scaling theory) applied to the surface pressure-area ( π− A) isotherms and surface pressure-temperature ( π− T) isochors, and resulting from generalized physical formalisms modified to a multicomponent surfactant film. A set of the introduced structural film state parameters could become sensitive indicators for surface-active source-specific organic matter pathways tracing, where the measurement of surfactant concentration and chemical analyses are avoided. Performed comprehensive film studies on rain, marine and snow-melted water samples exhibited significant and differentiated film structural parameters variability. The developed procedure allows one to recover the film parameters ( π, Γ, A lim, E isoth) present originally at the raindrop surface from the Langmuir trough data supplemented with the simultaneously taken rain event characteristics (rain rate and rain drop diameter distribution). It requires the partitioning effect of the surfactant molecules between the surface and bulk phases to be estimated where the entering quantities are: the partitioning coefficient K p= Γ/ c and a degree of the rain water interfacial system area development A r/ V r evaluated here using the fatty acids concentrations as model input data. The latter parameter depends on the rain rate and the form of the drop size distribution function differing significantly from the Marshall–Palmer one at low I r (<1 mm h −1). The partitioning factor K p related to the physicochemical composition of the film-composing material exhibited a seasonal and source-specific features and varies in a rather narrow range (5.3–13.7 cm) being of secondary importance in comparison to the interface development term. Original rain drop films exhibited low surface pressures (1.9–5.4 mN m −1) characteristic of a quasi-ideal 2D gas behavior found in a diluted surfactant solution (with c≈10 −3a) and are composed of slightly soluble immiscible components of mean molecular masses within 261–270 Da in agreement with chemical analyses. The concept is quantified using the Langmuir trough film and disdrometer data on a set of rainwater events observed at Gdańsk, Poland in May 2002.

On the Drop Size Distribution Functions and Drop Size Relationships in Stabilized Agitated Dispersions

Experiments on stabilized agitated dispersion of non-reacting styrene monomers were conducted at various dispersed phase hold-up fractions, impeller geometries, impeller speeds and agitation times. The experimental drop size distributions obtained were described best by the cumulative upper limit volume, upper limit number and the Nukiyama-Tanasawa ones. Plots of the maximum diameter and the mean diameter vs. the Sauter mean diameter gave straight lines with slopes 1.74 and 0.87, respectively.

The Bias in Moment Estimators for Parameters of Drop Size Distribution Functions: Sampling from Exponential Distributions

Abstract The moment estimators frequently used to estimate parameters for drop size distribution (DSD) functions being “fitted” to observed raindrop size distributions are biased. Consequently, the fitted functions often do not represent well either the raindrop samples or the underlying populations from which the samples were taken. Monte Carlo simulations of the process of sampling from a known exponential DSD, followed by the application of a variety of moment estimators, demonstrate this bias. Skewness in the sampling distributions of the DSD moments is the root cause of this bias, and this skewness increases with the order of the moment. As a result, the bias is stronger when higher-order moments are used in the procedures. Correlations of the sample moments with the size of the largest drop in a sample (Dmax) lead to correlations of the estimated parameters with Dmax, and, in turn, to spurious correlations between the parameters. These things can lead to erroneous inferences about characteristics of the raindrop populations that are being sampled. The bias, and the correlations, diminish as the sample size increases, so that with large samples the moment estimators may become sufficiently accurate for many purposes.

Parameterization of drop effective radius for drizzling marine stratocumulus

We describe a parameterization of cloud drop effective radius Re developed for drizzling marine stratocumulus clouds. The parameterization was derived using data from the CIMMS LES model with explicit formulation of drop size resolving microphysics. We showed that in the case of moderate drizzle an accurate parameterization of Re in the 8 ÷ 20 μm radius range can be obtained on the basis of two microphysical variables: liquid water content and drop concentration. For heavy drizzle, only a three‐variable parameterization proves to be sufficiently accurate. We developed two versions of the parameterization. The first one is based on traditional cloud microphysical prognostic variables representing partial moments of the drop distribution function: cloud water content, drizzle water content, and cloud drop concentration. The second one is based on the full moments of the drop size distribution function: drop concentration, total liquid water content, and radar reflectivity. The first version of the parameterization is rather sensitive to the value of the threshold radius that divides cloud and drizzle water. As a result, the errors are larger than in the second full moment version. It is concluded that a three‐variable full moment parameterization provides the most accurate representation of effective radius in precipitating stratocumulus cloud layers.

Effect of Polydispersity of Droplets in the Prediction of Flux Distribution in a Venturi Scrubber

The effect of polydispersity of droplets on the liquid flux distribution in a Venturi scrubber is investigated using a 2-D model along the length of a Venturi scrubber. Two average drop-sizes (the Sauter mean diameter and the mass median diameter) and two drop-size distribution functions (the upper-limit distribution function and the universal root-normal distribution function) are used in the model to predict liquid flux distribution. It is found that the flux distribution is a strong function of the polydispersity of droplets. It is also found that the throat gas velocity has a greater effect on the polydispersity of droplets than L/G ratio. At low throat gas velocity, flux distributions, predicted by the model using drop-size distributions, match with experimental data for a wide range of L/G ratios. At high throat gas velocities, flux distributions, predicted using average drop-sizes, follow experimental results closely. An analysis of overall collection efficiency reveals that it is nearly independent of polydispersity of droplets at low liquid-to-gas ratios and high gas velocities.

On the sensitivity of cloud microphysics under influence of cloud drop size distribution

The unique Khrgian–Mazin size distribution is suggested as an alternative approximation of the entire drop spectra besides the conventional approximation in which the monodisperse and the Marshall–Palmer size distributions for cloud droplets and raindrops are used, respectively. This approach is employed in bulk microphysics. It is shown that the effects of the changed drop size distribution are particularly pronounced in the case of the microphysical production terms with rain. This is a consequence of the fact that the Khrgian–Mazin size distribution produces more small raindrops and less large ones compared to the Marshall–Palmer one, where both have the same rain water mixing ratio. The new production terms with cloud water do not differ from those with the monodisperse size distribution used. In order to show the sensitivity of cloud microphysics with respect to the change of drop size distribution function, the experiments are perform with the two versions of a forced 1-D time-dependent model. The first one involves the microphysical production terms with the Khrgian–Mazin size distribution, while the other one uses the conventional approach. Experiments clearly show that cloud microphysics essentially depends on cloud drop size distribution. Some microphysical aspects of a model cloud with the new drop size distribution are in fair agreement with observations.

Self-Preservation of the Drop Size Distribution Function and Variation in the Stability Ratio for Rapid Coalescence of a Polydisperse Emulsion in a Simple Shear Field

Coalescence of oil-in-water emulsion droplets in a simple shear flow produced by a Couette device is considered. A phase Doppler anemometer was used to measure the droplet size distribution as a function of time for shear rates ranging from 55 to 213 s−1and for sodium chloride salt concentrations from 0.095 to 0.6 M. The initial droplet size distribution was log-normal. During the coalescence process, the size distribution was self-preserving in accordance with D. L. Swift and S. K. Friedlander's analysis [J. Colloid Sci.19,621 (1964)]. In the limiting case of negligible repulsive force due to the electric double layer, the calculated stability ratios, corrected for droplet polydispersity, agree well with the theoretical analyses of G. R. Zeichner and W. R. Schowalter [AIChE J.23,243 (1977)] and D. L. Feke and W. R. Schowalter [J. Fluid Mech.133,17 (1983)] for the case of solid particle aggregation. The good agreement between the stability ratios for the case of coalescence of droplets in the present study and those for aggregation of solid particles indicates that resistance to film deformation and thinning present in the case of coalescence is not important compared with the collision process.