Droplet Growth Model Research Articles

Coarsening dynamics in off-critically quenched binary systems under microgravity conditions

Effects of thermo-capillary motion of spherical droplets on the coarsening dynamics in off-critically quenched binary systems are studied by means of the droplet growth model for the case where the temperature gradient is imposed under microgravity conditions. Such movement of precipitating droplets is driven by variations in interfacial tension caused by the temperature gradient (the Marangoni effect), leading to instantaneous droplet coagulation events. Large-scale computer simulation results show that (1) even in the presence of Marangoni effect the average droplet radius grows in time with the growth exponent 1 3 , and also the scaling regime for the droplet size distribution does exist, which are similar to those of the well-known Ostwald ripening process; and (2) effects of thermo–capillary motion induced coagulation result in enhanced coarsening rate and fairly broad scaling form of the size distribution function, compared with those of the Ostwald ripening.

Application of mathematical morphology in measurement of droplet size distribution in dropwise condensation

The study of the distribution of droplets formed due to underside dropwise condensation provides information about the heat transfer by the droplets. To find the count and size distribution of droplets the image processing technique is proposed so that the whole process could be automated. Moreover, more accurate results can be obtained as compared to manual tracing and theoretical droplet growth model simulation. The watershed transformation for effective segmentation of the droplet image, prior to finding the size distribution was used. The proposed method has been applied to scanning electron microscopy (SEM) images of droplets formed on condensed bismuth metal vapor on stainless steel substrate and graphite substrate.

Transitional droplet growth and diffusion coefficients

The droplet growth models of Gyarmathy and Young, valid for arbitrary Knudsen numbers, are compared with experimental growth results obtained from expansion wave tube experiments. Growth experiments of n-pentanol in helium were performed at approximately 1 bar, resulting in growth curves stretching from the transition regime ( Kn≈1) to the continuum regime ( Kn≪1). Droplet growth experiments of water in helium and water in nitrogen were performed at elevated pressures, when the mean free path is small; hence, these growth curves are situated near the continuum regime. For Kn > 0.1, the Gyarmathy model appears to describe the experimental growth curves better than the Young model. However, for Kn < 0.02, the Young model gives the best results. For the water–helium and water–nitrogen systems new experimental diffusion coefficients are obtained, which are in good agreement with literature data.

Transonic flow of moist air around a thin airfoil with non-equilibrium and homogeneous condensation

A new small-disturbance model for a steady transonic flow of moist air with non-equilibrium and homogeneous condensation around a thin airfoil is presented. The model explores the nonlinear interactions among the near-sonic speed of the flow, the small thickness ratio and angle of attack of the airfoil, and the small amount of water vapour in the air. The condensation rate is calculated according to classical nucleation and droplet growth models. The asymptotic analysis gives the similarity parameters that govern the flow problem. Also, the flow field can be described by a non-homogeneous (extended) transonic small-disturbance (TSD) equation coupled with a set of four ordinary differential equations for the calculation of the condensate (or sublimate) mass fraction. An iterative numerical scheme which combines Murman &amp; Cole's (1971) method for the solution of the TSD equation with Simpson's integration rule for the estimation of the condensate mass production is developed. The results show good agreement with available numerical simulations using the inviscid fluid flow equations. The model is used to study the effects of humidity and of energy supply from condensation on the aerodynamic performance of airfoils.

The effect of carrier gas pressure on vapor phase nucleation experiments using a thermal diffusion cloud chamber

Recent measurements of critical supersaturations for the vapor phase homogeneous nucleation of several substances using a diffusion cloud chamber technique exhibit a dependence on the pressure of the carrier gas used in the experiments. A model of droplet growth and motion in a diffusion cloud chamber, combined with the density and temperature profiles of the chamber is presented to explain the pressure dependent results. The model demonstrates that at higher carrier gas pressures the growth of the droplets is retarded and the optical scattering signal from the particles is reduced. It is concluded that the observed effect may not result from a pressure dependence of the nucleation rate, but from a pressure dependence of the droplet growth and motion.

Comparisons of cloud microphysics with cloud condensation nuclei spectra over the summertime Southern Ocean

In spite of the many factors that determine cloud droplet concentrations, relationships were found between cloud condensation nuclei (CCN) and cloud droplet concentrations for the First Aerosol Characterization Experiment (ACE 1). Measurements were made in summertime stratocumulus clouds over the Southern Ocean far from anthropogenic sources. The closest relationship of CCN and droplet concentrations was found for CCN measured just below near‐adiabatic cloud parcels. Moreover, an adiabatic droplet growth model successfully predicted these droplet concentrations from the CCN spectra and updraft velocities. Correlations of the averages of CCN and cloud droplet concentrations over the entirety of each of the flights were also good. The model also predicted, with reasonable accuracy, these flight‐wide average droplet concentrations from flight‐wide average CCN spectra and flight‐wide average cloud updraft velocities. Drizzle was a major reason for the larger variations in droplet concentrations than in CCN concentrations. Vertical differences in CCN concentrations may have also affected cloud droplet concentrations. Despite variabilities in droplet concentrations, there was a clear relationship between CCN and droplet concentrations. Many of the inferred supersaturations (0.5 to 1.4%) in the “adiabatic” clouds ranged higher than some previous estimates (e.g., 0.2%) for stratus clouds. This may have been due to the cleaner air of the Southern Ocean but is more likely attributable to the more intensive analysis used in this study. Nonetheless, even where the inferred cloud supersaturations were lower (i.e., 0.2%), due to reductions of the droplet concentrations, there was still a relationship between CCN and droplet concentrations.

Modeling nucleation and droplet growth for ion-induced nucleation experiments in diffusion cloud chambers

A model of ion-induced nucleation, droplet growth and motion processes in a diffusion cloud chamber is presented. The model predicts the arrival time of the droplets with an accuracy greater than 95%. The model also explains the increase in the time corresponding to the maximum of the nucleation pulse with increasing carrier gas pressure. Using this model it is possible to correlate the arrival time of the droplets with the height (temperature and supersaturation) where the nucleation events take place.

Asymptotic Solution of Cloud Formation in Expansion Chambers

Asymptotic solution of homogeneous condensation in expansion cloud chambers in different droplet growth regimes is presented. In particular an exactly solvable droplet growth model with growth rates ranging between the Hertz-Knudsen and continuum droplet growth laws is introduced. The distinct condensation zones in each droplet growth regime are identified by the asymptotic solution of the condensation rate equation and the results are compared with those of direct numerical simulations using the classical nucleation theory. Excellent qualitative agreement is reached despite some minor quantitative differences in some of the condensation zones arising from the nature of the asymptotic solution in these zones.

Asymptotic solution and numerical simulation of homogeneous condensation in expansion cloud chambers

Asymptotic solution of homogeneous condensation in expansion cloud chambers in different droplet growth regimes is presented. In particular an exactly solvable droplet growth model ranging between the Hertz–Knudsen and continuum droplet growth laws is introduced. The distinct condensation zones in each droplet growth regime are identified by the asymptotic solution of the condensation rate equation and the results are compared with those of direct numerical simulations using the classical nucleation theory. Excellent qualitative agreement is reached despite some minor quantitative differences in some of the condensation zones arising from the nature of the asymptotic solution in these zones.

Encounter effects on the dynamics of fluctuations in phase separations of off-critically quenched binary systems

The encounter effect on the dynamics of fluctuations in phase separations of off-critically quenched binary systems is studied by performing three-dimensional computer simulations of the droplet growth model proposed recently by the authors. The dynamical behavior of the scattering structure function, including the dynamical scaling, the temporal power laws, and the crossovers have been investigated. At higher volume fractions, the encounter effect is shown to strongly influence not only the dynamics of the single-droplet-size distribution function, leading to flattening and broadening of the distribution function, but also the dynamics of fluctuations in phase- separating systems, leading to sharpening and narrowing the structure function.

In‐situ observations of an Antarctic polar stratospheric cloud: Similarities with Arctic observations

Measured particle volumes in a type I polar stratospheric cloud near the Antarctic polar vortex during ASHOE/MAESA show that the onset of the cloud occurred near 193 K, 3 degrees colder than nitric acid trihydrate (NAT) saturation. The onset temperature, the smooth increase of volume with decreasing temperature, the inverse correlation of particle volume and enhanced NOy (HNO3 in the particles) with temperature, and comparisons of observations with an equilibrium model of ternary droplet growth all support the notion that much of this type I PSC was ternary solution droplets. This provides confirmation of previous findings in the northern hemisphere. However, the ternary model does not fit the observations in all regions. This may be due to the presence of some solid phase growth in agreement with impactor observations.

The fundamental equations of gas-droplet multiphase flow

This paper describes the derivation of an equation set for the multiphase flow of small polydispersed liquid droplets in a continuous gas-phase consisting of either a pure vapour (of the same chemical species as the liquid droplets) or a mixture of pure vapour and an inert gas. Some difficulties of previous formulations are resolved by more judicious definitions of the interphase transfer terms. The analysis includes a consistent model to represent the surface energy and entropy of the liquid droplets. Surface effects are normally neglected but must be included if consistency is to be maintained with droplet growth models in which the droplet temperature depends on its radius due to the effects of capillarity. A derivation of the equation for the rate of entropy creation due to departures from equilibrium is also presented. Entropy production in non-nucleating flows can be represented by precisely four terms, three of which are associated individually with the interphase transfer of mass, momentum and energy. The fourth term represents the entropy change due to the homogeneous nucleation of liquid droplets from the vapour and is in exact agreement with the results of classical nucleation theory. The form of the entropy creation equation allows an interpretation using the methods of linear irreversible thermodynamics and indicates that some mathematical models of droplet growth in common use, derived on an informal basis, may not be physically realistic in certain circumstances.

The influence of CCN spectra and updraft velocities on the validity of the b2 parameterization of the condensation process

A detailed parcel model of condensational droplet growth, including surface tension and dissolved salts effects, is used for studying the validity of a parameterization based on the simplified droplet growth equation, the b 2 scheme. In the b 2 scheme, spectra are derived from the droplet distribution at the activation level, and the liquid water mixing ratio using a translation of the initial spectrum in the r 2 scale. Spectra derived from the detailed model are thus compared to translated spectra. Four simplified CCN distributions are tested for updraft velocities in the range of 0.5 m/s to 5.0 m/s. It is shown that the b 2 scheme is suitable for use in cloud models, except when the proportion of large nuclei in the CCN distribution is extremely high.

Coarsening of three-dimensional droplets by two-dimensional diffusion: Part I. Experiment

An experimental study of diffusional coarsening, or Ostwald ripening, in a liquid-liquid two-phase system is described. An experiment performed at its isopycnic point, 42°C, allowed observations for the long times required to investigate coarsening. A holographic technique was instrumental in this work. Holograms takenin situ permit investigation of details regarding both the influence of local environmental conditions on individual droplet size histories and measurement of global averages. This study utilized a 100 μm pathlength test cell. The discrete phase was nucleated on one wall of the cell. This configuration resembles island formation in thin film growth. Observation of Ostwald ripening over a period of 1×107 s (∼4 mo.) reveals that droplet number decays as t−0.733 and the average radius increases as t−0.247, in the asymptotic limit. This shows good agreement with theoretical predictions for diffusional growth of spherical caps on a two-dimensional substrate which is a valid approximation for the geometry of this experiment. Part I of this paper describes the experimental results. Part II discusses a numerical model for droplet growth in a comparison with the experimental results.

On Mist Formation in Natural Gas

Mist formation that occurs during production, handling, and transport of natural gas is shown to be the result of non equilibrium thermodynamic processes. A simple analysis is presented of these nucleation and droplet growth phenomena in natural gas, and an estimate is given of the rate at which droplets are formed for typical process conditions. A model of droplet growth due to condensation is presented. The nucleation and condensation behaviour of two samples of a natural gas is investigated in a Wilson expansion cloud chamber for the initial pressure range of 5 to 50 bar and initial temperatures of 294-297 K (21-24°C). For a cooling rate of about 1 K/ms, a typical undercooling at the onset of condensation is found of 32 ý 3K and 60 ý 5K respectively for the two samples, which emphasizes the importance of non-equilibrium effects.

Steady state behavior in a model for droplet growth, sliding and coalescence: the final stage of dropwise condensation

Abstract The final (steady state) stage of dropwise condensation has been explored using a simple model for droplet deposition and coalescence with the rapid sliding of droplets that exceed a critical size S ∗ . In this steady state regime the mean droplet size and the total mass density both decrease algebraically with increasing distance from the upper edge of the inclined substrate (apart from pronounced oscillations at very shot distances). The droplet number density on the other hand, varies at most logarithmically with this distance. The steady state droplet size distribution can be represented quite well by a stretched exponential form.

The random choice method applied to non-linear wave propagation in gas-vapour-droplets mixtures

The random choice method (RCM) has been used to calculate non-linear wave propagation in gas-vapour-droplets mixtures. The theoretical model of heterogeneous nucleation and droplet growth is discussed. The equations for wave propagation are formulated in such a way that only one source term due to condensation or evaporation remains. The specific source term properties are investigated and the implementation of the source term into the RCM by operator splitting is explained. Numerical calculations for an unsteady centred rarefaction wave and a shock wave are compared with experimental data. The formation of a condensation discontinuity is demonstrated.

A kinetic model for droplet growth in the transition regime

A variant of the moment expansion method, used in an earlier paper to describe the flow of a gas toward an absorbing sphere, is applied to a more realistic model of a droplet condensing from a supersaturated vapor. In the simplest version a spherical droplet absorbs all incoming vapor molecules, but spontaneously emits molecules with a Maxwellian distribution at the droplet temperature and with the corresponding saturated vapor density. From a solution of the stationary linearized Boltzmann equation with these boundary conditions we obtain expressions for the heat and mass currents toward the sphere as a function of the supersaturation and the temperature difference between the droplet and the vapor at infinity. For small droplet radii the known free flow limit is obtained in a natural way. From the calculated expressions for the heat and mass current we derive evolution equations for the radius and temperature of the droplet. The temperature evolves more rapidly and can thus be eliminated adiabatically; the resulting growth curve for the radius shows a sharp transition from a kinetically controlled regime for small radii to a regime dominated by heat conduction for large radii. The effect of incomplete absorption at the surface is also studied. The actual calculations are carried out for Maxwell molecules, with parameters corresponding to argon at 0.65T c and 100% supersaturation.

Kinetics of droplet growth processes: Simulations, theory, and experiments.

The formation of a distribution of various size droplets is a characteristic feature of many systems from thin films and breath figures to fog and clouds. In this paper we present the results of our investigations of the kinetics of droplet growth and coalescence. In general, droplet formation occurs either by spontaneous nucleation or by growth from heterogeneously distributed nucleation centers, such as impurities. We have introduced two models to describe these two types of processes. In the homogeneous nucleation model droplets can form and grow anywhere in the system. The results of the simulations of the model are presented and it is shown that the droplet size distribution has a bimodal structure consisting of a monodispersed distribution of large droplets superimposed on a polydispersed distribution of smaller droplets. A scaling description for the evolution of the time-dependent droplet size distribution and its moments is presented and it is found that the scaling predictions are in excellent agreement with the simulations. A rate-equation similar to the Smoluchowski equation is also introduced for describing the kinetics of homogeneous droplet growth. The results of the simulations of the homogeneous nucleation model are also compared with the experiments on droplet growth in thin films obtained by vapor deposition of tin on sapphire substrate. It appears that this model captures the essential features of the distribution of droplets in the vapor deposition experiments. We also introduce a heterogeneous nucleation model for studying processes in which droplets only form and grow at certain nucleation centers which are initially chosen at random. Simulations, scaling theory, and a kinetic equation approach for describing the heterogeneously nucleated droplet growth model are also presented. The theoretical predictions are found to be in excellent agreement with the simulations.

A robust ‘time-marching’ solver for one-dimensional nucleating steam flows

A mixed Lagrangian/Eulerian ‘time-marching’ solver capable of predicting one-dimensional nucleating steam flows is described. Simple nucleation and droplet growth models are employed which avoid the use of variable empirical factors and which have been validated using existing experimental data from nozzle experiments performed in the steam tunnel of the Central Electricity Research Laboratories. Theoretical predictions are compared against experimental results encompassing all flow regimes likely to be encountered in a one-dimensional analysis of flow in a low pressure steam turbine. These include supercritical heat addition cases which display both steady and unsteady shock wave formation.