Homogeneous Nucleation Of Droplets Research Articles

Development of a four-fluid model for condensing steam flows

A new four-fluid model is proposed for condensing water-steam flows applicable to a wide variety of different flow conditions relevant for nuclear power plants. The model expands upon current state-of-the-art three-fluid models by adding a second droplet fluid phase to account for homogeneous nucleation of droplets from the bulk vapor field. Consistent with prior three-fluid models, liquid wall film and entrained droplet fluids share a common temperature and energy equation, resulting in a system of eleven conservation equations. An additional governing PDE is required for the nucleated droplet number density, for twelve total governing equations. A full set of closure relations for the interphase and wall interactions is given. An initial test of the model formulation in a quasi-one-dimsensional converging-diverging nozzle with condensing steam was carried out using a simple in-house solver as a proof of concept. Results showed qualitatively correct trends compared with experimental data.

Investigation of droplet nucleation in CCS relevant systems: Progress in the design and testing of the mixture preparation device

The study presents progress in the development of mixture preparation device (MPD) representing an important part of the larger experimental setup intended for investigation of homogeneous droplet nucleation in CO2-rich systems. MPD allows for accurate adjustment of flow parameters, i.e. temperature, pressure, and flow rate, of CO2 in either superheated vapor or supercritical fluid phases and of other gas components such as argon or nitrogen. Through accurate settings of flow rates of individual components, the mixture composition can continuously be defined. MPD is going to be connected to the expansion chamber, where the droplet nucleation will experimentally be observed. In this work, CO2-branch, i.e. the core part of MPD, was modified and tested. Several components, e.g., pressure transducers and safety valve, had to be calibrated and adjusted to assure well-defined and safe operation. Most attention was paid to the design and performance of throttling capillary tubes installed in thermostatic bath, which define final flow parameters of CO2 coming from the CO2 branch. The flow characteristics of two capillary tubes with lengths of 7.8 and 4.0 m and inner diameter 0.1 mm were measured and compared to the predictions of a numerical model. The 1-D model of isothermal capillary flow was found to provide quite good agreement with the measured data.

Predictions of homogeneous nucleation rates for n-alkanes accounting for the diffuse phase interface and capillary waves.

Homogeneous droplet nucleation has been studied for almost a century but has not yet been fully understood. In this work, we used the density gradient theory (DGT) and considered the influence of capillary waves (CWs) on the predicted size-dependent surface tensions and nucleation rates for selected n-alkanes. The DGT model was completed by an equation of state (EoS) based on the perturbed-chain statistical associating fluid theory and compared to the classical nucleation theory and the Peng-Robinson EoS. It was found that the critical clusters are practically free of CWs because they are so small that even the smallest wavelengths of CWs do not fit into their finite dimensions. The CWs contribute to the entropy of the system and thus decrease the surface tension. A correction for the effect of CWs on the surface tension is presented. The effect of the different EoSs is relatively small because by a fortuitous coincidence their predictions are similar in the relevant range of critical cluster sizes. The difference of the DGT predictions to the classical nucleation theory computations is important but not decisive. Of the effects investigated, the most pronounced is the suppression of CWs which causes a sizable decrease of the predicted nucleation rates. The major difference between experimental nucleation rate data and theoretical predictions remains in the temperature dependence. For normal alkanes, this discrepancy is much stronger than observed, e.g., for water. Theoretical corrections developed here have a minor influence on the temperature dependency. We provide empirical equations correcting the predicted nucleation rates to values comparable with experiments.

Surface tension of water droplets upon homogeneous droplet nucleation in water vapor

A method has been proposed for determining interfacial free energy from the data of molecular dynamics simulation. The method is based on the thermodynamic integration procedure and is distinguished by applicability to both planar interfaces and those characterized by a high curvature. The workability of the method has been demonstrated by the example of determining the surface tension for critical nuclei of water droplets upon condensation of water vapor. The calculation has been performed at temperatures of 273–373 K and a pressure of 1 atm, thus making it possible to determine the temperature dependence of the surface tension for water droplets and compare the results obtained with experimental data and the simulation results for a “planar” vapor–liquid interface.

Microphysics of Aerodynamic Contrail Formation Processes

Abstract Aerodynamic condensation is a result of intense adiabatic cooling in the airflow over aircraft wings and behind propeller blades. Out of cloud, condensation appears as a burstlike fog (jet aircraft during takeoff and landing, propellers) or as an iridescent trail visible from the ground behind the trailing edge of the wing (jet aircraft in subsonic cruise flight) consisting of a monodisperse population of ice particles that grow to sizes comparable to the wavelength of light in ambient humidities above ice saturation. In this paper, the authors focus on aerodynamic contrail ice particle formation processes over jet aircraft wings. A 2D compressible flow model is used to evaluate two likely processes considered for the initial ice particle formation: homogeneous droplet nucleation (HDN) followed by homogeneous ice nucleation (HIN) and condensational growth of ambient condensation nuclei followed by their homogenous freezing. The model shows that the more numerous HDN particles outcompete frozen solution droplets for water vapor in a 0.5–1-m layer directly above the wing surface and are the only ice particles that become visible. Experimentally verified temperature and relative humidity–dependent parameterizations of rates of homogeneous droplet nucleation, growth, and freezing indicate that visible aerodynamic contrails form between T = −20° and −50°C and RH ≥ 80%. By contrast, combustion contrails require temperatures below −38°C and ice-saturated conditions to persist. Therefore, aerodynamic and combustion contrails can be observed simultaneously.

Homogeneous droplet nucleation modeled using the gradient theory combined with the PC-SAFT equation of state

In this work, we used the density gradient theory (DGT) combined with the cubic equation of state (EoS) by Peng and Robinson (PR) and the perturbed chain (PC) modification of the SAFT EoS developed by Gross and Sadowski [1]. The PR EoS is based on very simplified physical foundations, it has significant limitations in the accuracy of the predicted thermodynamic properties. On the other hand, the PC-SAFT EoS combines different intermolecular forces, e.g., hydrogen bonding, covalent bonding, Coulombic forces which makes it more accurate in predicting of the physical variables. We continued in our previous works [2,3] by solving the boundary value problem which arose by mathematical solution of the DGT formulation and including the boundary conditions. Achieving the numerical solution was rather tricky; this study describes some of the crucial developments that helped us to overcome the partial problems. The most troublesome were computations for low temperatures where we achieved great improvements compared to [3]. We applied the GT for the n-alkanes: nheptane, n-octane, n-nonane, and n-decane because of the availability of the experimental data. Comparing them with our numerical results, we observed great differences between the theories; the best results gave the combination of the GT and the PC-SAFT. However, a certain temperature drift was observed that is not satisfactorily explained by the present theories.

Surface Tension of Binary Mixtures Including Polar Components Modeled by the Density Gradient Theory Combined with the PC-SAFT Equation of State

In this study, the Cahn–Hilliard density gradient theory (GT) is used for predicting the surface tension of various binary mixtures at relatively wide temperature ranges and for testing the application of the GT for predictions of homogeneous nucleation. The GT was combined with two physically based equations of state (EoS), namely the perturbed-chain (PC) statistical associating fluid theory (SAFT) and its modification for polar substances the perturbed-chain polar (PCP) SAFT. The GT applied to the planar phase interface was employed to predict the interfacial tension for various quadrupolar (CO2 and benzene) and dipolar (difluoromethane, i.e., R32; pentafluoroethane, i.e., R125; and 1,1,1,2-tetrafluoroethane, i.e., R134a) substances and for five binary mixtures including polar components (n-decane + CO2, benzene + CO2, R32 + R125, R32 + R134a, R134a + R125). The PCP-SAFT EoS combined with the GT provides more accurate results for both the quadrupolar and dipolar substances than the original PC-SAFT EoS. Besides the planar phase interface, the GT was also applied to the spherical phase interface simulating a critical cluster occurring in homogeneous nucleation of droplets. Carbon dioxide was considered, because it has a relatively high quadrupole moment and because of its relevance to natural gas processing. Application of the PCP-SAFT EoS provides a significant improvement compared to the PC-SAFT EoS, and it is clearly superior to the classical cubic Peng–Robinson EoS, which is still used for modeling droplet nucleation.

Activated instability of homogeneous droplet nucleation and growth

For the pure-component supercooled Lennard-Jones vapor, the free energy of forming a droplet with a given particle number and volume is calculated using density-functional theory. In contrast to what was noted in previous studies, the free energy surface beyond the pseudosaddle point no longer exhibits a valley but rather channels the nuclei toward a locus of instabilities, initiating an unstable growth phase. Similar to a previous study of bubble formation in superheated liquids [M. J. Uline and D. S. Corti, Phys. Rev. Lett. 99, 076102 (2007)], a new picture of homogeneous droplet nucleation and growth emerges.

Homogeneous nucleation of droplets from supersaturated vapor in a closed system.

Kinetic equations describing homogeneous nucleation kinetics within standard model are solved numerically under the condition of a constant number of molecules in the considered system. It has consequences to decrease the supersaturation of the supersaturated vapor during the process of the formation of small droplets of a new phase. The decrease of supersaturation occurs in a short time and reaches some value which remains unchanged for a relatively long time (quasistationary regime), especially at lower initial supersaturations. This time interval decreases with increasing value of the initial supersaturation. In the quasistationary regime the nucleation rate reaches its stationary value. At higher initial supersaturation, the rate of formation of nuclei goes to some maximum value corresponding to the stationary nucleation rate and then decreases with time due to the decrease of supersaturation.

Statistical Characteristics of the Process of Homogeneous Nucleation in the Vapor–Gas Medium during Free Molecular Regime of the Growth of Forming Droplets

Statistical approach to the study of the process of homogeneous nucleation of droplets in the vapor–gas medium in the presence of originally generated growing droplet at free molecular regime of droplet growth after the instantaneous creation of initial vapor supersaturation is proposed. The probability density of the creation of a new droplet in the vicinity of originally generated droplet is found. The mean distance between two neighboring droplets and the relative scatter of this distance are determined. The mean expectation time for the appearance of neighboring droplet estimating the duration of the droplet nucleation stage is found. The average number of droplets in a unit volume of the vapor–gas medium by the end of the droplet nucleation stage is estimated. The results obtained are compared with the predictions of the theory based on the assumption of the homogeneity of metastable phase.

Homogeneous nucleation of droplets from a supersaturated vapor phase

An approximate semianalytic theory is developed to describe the homogeneous nucleation of droplets from a supersaturated vapor, beginning with a partition function and including rigorously the translation and surface tension contributions. The liquid and vapor phases are treated as uniform (step density profile) and may be described by any accurate equation of state. It is shown that the classical approximation for the free energy of droplet formation may be derived from the present theory by making additional approximations (ideal gas, incompressible liquid), and the two are compared for the case where the vapor phase forms a reservoir (constant supersaturation). In the case of a finite-sized vapor phase, where the supersaturation decreases as the droplet grows, a free energy minimum exists beyond the critical radius, and this stable droplet equilibrium is examined in detail. Comparison with computer simulations proves the quantitative accuracy of the present theory for a Lennard-Jones system. Also derived is a new, formally exact expression for the surface tension that is useful for computer simulations.

Energy fluctuations in homogeneous nucleation theory for aerosols

For the analysis of the release of fission product aerosols from reactor cores in a severe accident, it is necessary to understand the process of homogeneous nucleation of droplets from a supersaturated vapour, including the dependence on system pressure, which can vary greatly depending on the type of accident. A model of the effect of carrier gas on the nucleation process is presented, based on the changes to the energy distribution of nucleating vapour molecule clusters induced by collisions with carrier gas molecules. The rate of nucleation is altered since the cluster decay rate is strongly energy dependent. The approach is compared with previous treatments of the problem, illustrating the importance of using an equilibrium cluster energy distribution which goes beyond a Gaussian approximation, and clarifying previous confusion between cluster temperature and cluster energy in the literature. Calculations of nucleation rates are performed for n-nonane in argon and water vapour in air.

On the Dillmann-Meier theory of nucleation

A phenomenological model of the homogeneous nucleation of droplets proposed by Dillmann and Meier has been quite successful in accounting for the condensation behaviour of a number of different substances. However, a close inspection of the development reveals an inconsistency leading to two possible forms for the free energy. This can be resolved by a more careful treatment of the chemical potential. The necessary changes made to Dillmann and Meier's calculated nucleation rates are large, and to some extent spoil the agreement with experiment. Some suggestions are made which can help to restore agreement.

Homogeneous nucleation in metal tetrachloride vapors: Tin and titanium tetrachlorides

The upward thermal diffusion cloud chamber was used to measure the critical supersaturations for homogeneous nucleation of droplets from the supersaturated vapors of tin and titanium tetrachlorides. The measured supersaturations are in good agreement with the predictions of the classical theory of nucleation. Trends in free energy barrier are correctly predicted by the classical theory for carbon, silicon, tin, and titanium tetrachlorides. However, the classical barrier does not reflect the binding energies in the critical clusters. According to the classical theory, the size of the nucleus of SiCl4 ranges from 60 to 120 molecules while for TiCl4 it varies from 35 to 65 molecules in the temperature range of 240–310 K. The classical theory predicts a linear log J vs 1/T relationship. Corresponding states correlations are used to locate general trends and correlate them with molecular properties. Titanium tetrachloride deviates from the ‘‘simple fluid’’ behavior observed for CCl4, SiCl4, and even SnCl4. This deviation must reflect a modified intermolecular potential for TiCl4 as compared to a simple fluid potential.

Selected Values of Critical Supersaturation for Nucleation of Liquids from the Vapor

Selected values of critical supersaturations for homogeneous nucleation of droplets from the vapor and for heterogeneous nucleation of droplets on the natural stationary concentration of gaseous ions are tabulated and plotted, and a rationale is given for selection of these data.

Homogeneous nucleation of droplets and interfacial tension in the liquid system methanol-water-tribromomethane

Unstable homogeneous solutions within the miscibility gap of the ternary liquid system methanol-water-tribromomethane were made by rapid mixing of stable solutions. Characteristic times (induction periods) for the appearance of turbidity were measured at 25 °C. These induction periods are of the order 0.1 to 10 ms, depending on the composition of the unstable solution. For induction periods shorter than about 1 ms the measured data can be explained by assuming homogeneous nucleation of droplets followed by diffusion controlled growth. The ration between the interfacial tension calculated from the nucleation data and the bulk interfacial tension determined by the classical drop weight method is less than one, and approaches unity when the composition of the unstable system approaches the critical composition, i.e., when the interfacial tension approaches zero, and the critical nuclei become infinitely large.

Free energy of formation of droplets from vapor and dependence of surface tension on radius

It is demonstrated that a heretofore neglected contribution to the chemical potential of a small droplet arises from any variation of surface tension with radius. This contribution is evaluated by integration of the Gibbs-Duhem equation for several assumed relationships between surface tension and radius. It is concluded that the dependence of the free energy of formation of critical nuclei on surface tension is definitely not cubic, and exact relationships are given for the several assumed relationships. In the case of homogeneous nucleation of droplets from water vapor, the increase in surface tension required to bring about agreement between theory and cloud chamber observations is more nearly 50% rather than 17% as heretofore believed.