Kinetics of binary condensation on the droplet growing in diffusion regime

A relaxation equation determining the regular tendency of the concentration of binary solution in the growing droplet to the stationary value, at which there is a self-similar solution to the problem of the condensation in a binary mixture, is derived. An analytical solution of the relaxation equation is obtained and it is demonstrated that the stationary value of concentration is achieved via the power law. The time interval that elapses from the emergence of the droplet until the diffusion regime of droplet growth and derived relaxation equation become effective is revealed. The stationary value of concentration is found for the model of ideal solution.     

Self-similar theory of nonisothermal vapor condensation on a droplet growing in a vapor-gas medium

Rigorous self-similar solutions to the joint problems of vapor diffusion toward a droplet growing in a vapor-gas medium and the removal of heat released during vapor condensation are found. An equation for the temperature of a droplet ensuring the existence of a self-similar solution is derived. This equation sets the constancy of the temperature of a droplet throughout the time of its growth and unambiguously determines this temperature. In the case of the strong heat effects, when the rate of droplet growth decreases substantially, the analytical solution to this equation is obtained. This temperature coincides precisely with the temperature, which is established in the droplet at the diffusion regime of its growth. At the found droplet temperature, interconnected fields of vapor concentration and temperature of vapor-gas medium around the droplet are expressed through the initial (prior to the droplet nucleation) parameters of a vapor-gas medium. These parameters are used to express the dependence of the radius of a droplet on the time at the diffusion regime of its growth and the time required to establish the diffusion regime of droplet growth. The case of weak heat effects is also studied.     

Stationary concentration establishment in the growing or evaporating droplet of ideal binary solution

The establishment of stationary solution concentration in a growing or evaporating droplet of an ideal binary solution (binary droplet) placed in a vapor mixture of constituting substances and passive gas is described analytically. Relations defining time dependences of solution concentration in a droplet, the number of molecules of each constituting component, and droplet radius are derived at known parameters of the vapor-gas mixture and the initial composition of a binary droplet. The results of calculations of time dependences of aforementioned values are reported for several variants of the initial composition of a droplet and a vapor mixture.     

Regularities of binary condensation of vapors when one of them is undersaturated

Simple analytical expressions are derived for the stationary concentration of a binary solution in a markedly supercritical droplet growing exothermically in diffusion or free-molecular regimes in mixed vapors when one of condensing vapors is supersaturated and present in a small amount and another vapor is slightly undersaturated and present in a large amount. The condensation of sulfuric acid and water vapors on a droplet under the conditions of Earth atmosphere is considered as an example of practical importance. Under isothermic conditions, analytical expressions are obtained for the time of establishing a power law for variations in a droplet radius with time under the diffusion and free-molecular regimes of the droplet growth. The power laws are derived in an explicit form under these regimes, which describe the rapid establishment of a stationary concentration of a solution in a growing droplet.     

Thermal effects accompanying stationary binary condensation of vapors into overcritical droplet

A set of equations is derived to calculate the stationary temperature and concentration of a solution in a overcritical droplet with regard to the heat release accompanying the condensation of a binary mixture of vapors in a diffusion or free-molecular regime. In the approximation of an ideal solution, relations are found for the stationary temperature of droplet growing under the conditions of strong and weak thermal effects. For the general case and the cases of strong and weak thermal effects, the temperature and concentration of the droplet and the coefficient of the thermal deceleration of the droplet growth are calculated as functions of the density of a passive gas. The influence of the condensation heat values of the first and second components of the mixture on the stationary temperature and concentration of the solution in the growing droplet is investigated separately.     

Dynamics of variations in size and composition of supercritical droplet at binary condensation

A general analytical solution is found in quadratures for the radius and concentration of a solution droplet, which isothermally grows or evaporates in a diffusion or free-molecular regime in a binary mixture of vapors. The obtained solution describes the dynamics of variations in the size and composition of a super-critical droplet during the binary condensation in mixed vapors at an arbitrary initial droplet composition. It is shown that, at small (linear) deviations of the growth regime and droplet composition from the stationarity, these quadratures lead to the results that were recently obtained for the composition relaxation in a growing droplet. Moreover, it is demonstrated that, in terms of the nonlinear theory, when the deviation of solution concentration in a droplet from its stationary value is not small, it is invalid to use the law of stationary variations in the size of a droplet with time to describe the relaxation process for its chemical composition.     

Diffusion of Vapor in the Presence of a Growing Droplet

As is shown, the solution to the diffusion equation for the concentration of vapor in the presence of a droplet growing in it, derived for the usual initial condition and equilibrium boundary conditions at the droplet surface, fails to ensure an equality between the numbers of molecules that have left the vapor due to diffusion by the current moment and those that have been included in the growing droplet. The difference between the total numbers of vapor molecules at the initial moment (when the vapor had a given uniform concentration) and at the current moment (when the size of the growing droplet is much larger than its initial size) differs from the total number of molecules in the droplet by a factor of 3/2. By substituting the usual boundary condition at the droplet surface by a time-dependent boundary condition at the surface of a constant-radius sphere with the center in the center of the growing droplet, a solution to the diffusion problem for the vapor concentration is derived. This solution describes the evolution of the vapor concentration field, which agrees with the rate of the vapor absorption by the growing droplet and with the law of the conservation of matter.     

Self-similar solution to the problem of vapor diffusion toward the droplet nucleated and growing in a vapor-gas medium

The problem of vapor diffusion toward a droplet nucleated and growing in the diffusion regime is exactly solved using the similarity theory. The surface motion of droplets is taken into account in the solution. The constructed nonstationary concentration field of vapor satisfies the diffusion equation, the boundary condition of equilibrium on the surface of growing droplet, and the initial homogeneous condition. According to the found solution, the radius of a droplet is proportional to the square root of the time of its growth. Far from the critical point, at a low ratio between the densities of excess vapor and a liquid droplet, the proportionality coefficient coincides with that resulting from an approximate solution. The balance between the numbers of molecules removed from vapor and those composing a growing droplet exactly corresponds to the obtained solution.     

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.     

Dynamics of variations in size and composition of a binary droplet in a mixture of two condensing vapors and a passive gas under arbitrary initial conditions

The dynamics of variations in the size and composition of a droplet in a mixture of two vapors and a noncondensable carrier gas has been studied at an arbitrary initial droplet size and two limiting initial concentrations of the solution in the droplet corresponding to a pure first or pure second component. The conditions for nonmonotonic variations in the droplet radius with time have been analyzed. The physical situation has been investigated for the course of binary condensation, in which, at an initial stage, before the droplet begins to grow and the stationary concentration of the solution in it is established, the droplet size markedly decreases. The opposite situation is also considered, in which the droplet grows at the initial stage and then passes to the regime of monotonic evaporation.     

Nonstationary evolution of the size,composition, and temperature of microdroplets of nonideal two- and three-component aqueous solutions

Results of numerical solution have been presented for a set of equations describing the nonstationary and nonisothermal growth or evaporation of microdroplets consisting of ethanol and water, sulfuric acid and water, and sulfuric and nitric acids and water. Time dependences of droplet size, temperature, and composition have been determined at low concentrations of a condensable vapor, as compared with the concentration of a carrier gas in an ambient vapor–gas mixture. The calculations have been performed using different initial conditions and approximations for the dependences of saturation vapor pressures, activity coefficients, and partial heats of condensation of the components, as well as average volumes per molecule on droplet composition and temperature. By the examples of ethanol–water and sulfuric acid–water droplets, it has been shown that nonmonotonic variations in the droplet radius are possible. Regimes of nonmonotonic variations in the temperature of a droplet that precede the onset of its stationary growth or evaporation have been revealed for all systems under consideration.     

Evaporation of two solution droplets and a droplet located near a flat liquid surface

The mutual influence of two moderate-sized droplets of a dilute nonvolatile substance solution on the processes of their evaporation or condensation is theoretically analyzed under the assumption of a uniform concentration distribution inside the droplets. The conditions for the applicability of this approach are revealed. The evaporation or condensation of a droplet near a flat liquid surface is considered as a limiting case. The fluxes of water molecules to and from the surface of aqueous glycerol solution droplets occurring in air are numerically estimated depending on the droplet radii, distances between their surfaces, and air humidity. Analogous estimates are obtained for an aqueous glycerol solution droplet growing near a flat water surface.     

Supersaturated vapor recondensation: Analytical theories and numerical calculation

The time dependences of droplet size within the period from the end of nucleation until the onset of asymptotic regime of recondensation are determined by the numerical integration of equations describing droplet growth in a supersaturated vapor. The obtained dependences are compared with those derived in terms of the analytical theories of condensation. The character of variations in the final self-similar size distribution of droplets is studied as depending on variations in their initial distribution function and in the system parameters.     

Equations for the evolution of a growing or evaporating free microdroplet under nonstationary conditions of diffusion and heat transfer in a multicomponent vapor–gas medium

A set of equations has been derived for the nonstationary composition, size, and temperature of a growing or evaporating multicomponent microdroplet of a nonideal solution under arbitrary initial conditions. Equations for local nonstationary diffusion molecular and heat fluxes in a mixture of a multicomponent vapor with a noncondensable carrier gas have been obtained within the framework of nonequilibrium thermodynamics with allowance for hydrodynamic flow of the medium. The derived closed set of equations takes into account the nonstationarity of the diffusion and heat transfer, effect of thermodiffusion and other cross effects in the multicomponent vapor–gas medium, the Stefan flow, and droplet boundary motion, as well as the nonideality of the solution in the droplet. The general approach has been illustrated by the consideration of the multicomponent medium at low concentrations of vapors taking into account its thermal expansion due to the Stefan flow in the case of a nonstationary diffusion regime of the nonisothermal condensation growth of a one-component droplet.     

Non-Steady-State Temperature Field of the Vapor-Gas Mixture in the Vicinity of a Growing Droplet: Balance of the Heat of Phase Transition

The non-steady-state temperature field of the vapor-gas medium in the vicinity of a droplet growing in supersaturated vapor is constructed. In the conduction problem, a time-dependent boundary condition is used which ensures the fulfillment of the balance condition of the heat of phase transition. The resultant temperature field is compared with the one obtained in the heat conduction problem with the equilibrium boundary condition on the surface of a droplet of a fixed radius. Although the solution with the equilibrium boundary condition does not ensure the balance between the heat released on the growing droplet and the heat distributed due to heat conduction in the vapor-gas medium, the difference between the two solutions is not very large. This difference is important for describing the homogeneous nucleation of supersaturated vapor in the vicinity of a growing droplet, as is indicated by comparison of the vapor supersaturation fields constructed with and without allowance for thermal effects, as well as with the use of solutions to the diffusion and heat conduction problems with various boundary conditions.__________Translated from Kolloidnyi Zhurnal, Vol. 67, No. 3, 2005, pp. 333–341.Original Russian Text Copyright © 2005 by Grinin, Zhuvikina, Gor.     

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.     

New approach to the kinetics of heterogeneous unary nucleation on liquid aerosols of a binary solution

The formation of a droplet on a hygroscopic center may occur either in a barrierless way via Kohler activation or via nucleation by overcoming a free energy barrier. Unlike the former, the latter mechanism of this process has been studied very little and only in the framework of the classical nucleation theory based on the capillarity approximation whereby a nucleating droplet behaves like a bulk liquid. In this paper the authors apply another approach to the kinetics of heterogeneous nucleation on liquid binary aerosols, based on a first passage time analysis which avoids the concept of surface tension for tiny droplets involved in nucleation. Liquid aerosols of a binary solution containing a nonvolatile solute are considered. In addition to modeling aerosols formed through the deliquescence of solid soluble particles, the considered aerosols constitute a rough model of "processed" marine aerosols. The theoretical results are illustrated by numerical calculations for the condensation of water vapor on binary aqueous aerosols with nonvolatile nondissociating solute molecules using Lennard-Jones potentials for the molecular interactions.     

Analysis of the Results of Numerical Simulation of the Supersaturated Vapor Condensation Relaxation

The process of bulk vapor condensation from the vapor-gas mixture after the fast creation of the supersaturated state as a result of vapor expansion is discussed. Scaling relations are derived for the time of condensation relaxation and droplet number density. The principal possibility of the experimental determination of nucleation rate is demonstrated based on the results obtained. The effect of initial phase, frequency, and pulsation amplitude on the condensation relaxation is analyzed for the case, when thermodynamic parameters of a mixture in the process of condensation are subjected to perturbations in the form of low-amplitude harmonic pulsations. The domain of applicability of results obtained on the scale of pulsation frequencies is determined.     

On the evaporation and motion of a volatile droplet near a volatile liquid surface

A problem concerning the free evaporation or condensation growth of a droplet near an infinite planar surface of the same liquid is solved. The behavior of the droplet is considered at vapor temperature and concentration gradients preset at an infinite distance from it. The boundary conditions take into account effects that are linear with respect to the Knudsen number. Equations are derived for the rate of variations in the radius of the droplet and the velocity of its steady motion induced by nonuniform temperature and concentration of the vapor. Dependences of the rate of variations in the radius and the velocity of the steady motion of the droplet on the distance from the planar surface are presented for a droplet 1 ??m in radius suspended in air.