Kohler Equation Research Articles

Separation of an Industrial Mixture of Decalin or Naphthalene Fluorination Products: Cis-Perfluorodecalin, Trans-Perfluorodecalin and Perfluoro(butylcyclohexane): Physicochemical, Thermophysical, and Spectral Data

New physicochemical data for trans-perfluorodecalin (trans-PDF) and cis-perfluorodecalin (cis-PFD) are presented. Based on the differential scanning calorimetry, the temperature and heat of the solid−liquid phase transition are determined. The coefficients of Antoine’s equation are calculated based on the experimental temperature–pressure dependence data. This article also presents data on the rheological properties («zero» shear viscosity and apparent activation energy for the viscous flow) of the studied compounds. The dependencies of refractive index and excess volume (density) on temperature are studied. Gas chromatography–mass spectrometry data and FTIR, 13C NMR, and 19F NMR spectra are provided. The dependencies are given for the perfluoro(butylcyclohexane) (BCH)–trans-PFD, BCH–cis-PFD, and trans-PFD–cis-PFD binary systems and BCH–trans-PFD–cis-PFD ternary system: refractive index and density (liquid molar volume and excess molar volume) of composition and temperature. The dependences of the excess molar volume on the composition and temperature of the mixtures are correlated with Redlich-Kister and Kohler equations.

Thermodynamic stability of the fluorite phase in the CeO2 − CaO − ZrO2 system

The fluorite phase in CeO2 − CaO − ZrO2 is important in nuclear energy and fuel cell applications yet one lacks quantitative evaluation of its composition range and thermodynamic properties. Using experimental thermodynamic data from binary systems ZrO2 − CeO2, CaO − CeO2 and CaO − ZrO2, the present work fitted the enthalpies of mixing to regular solid solution models. The interaction parameters for binary systems are ΩCeO2-ZrO2 = 49.9 ± 2.4 kJ mol−1, ΩCaO-ZrO2 = −78.0 ± 16.7 kJ mol−1 and ΩCaO-CeO2 = 53.1 ± 21.5 kJ mol−1. Enthalpies of transformation of monoclinic zirconia to the cubic fluorite phase (stable at high temperature) and of rocksalt calcia to the hypothetical fluorite phase were constrained by experimental data. Enthalpies of mixing in the ternary CeO2 − CaO − ZrO2 system were calculated from enthalpies of drop solution measured by high temperature oxide melt solution calorimetry for a set of newly synthesized ternary fluorite samples. Comparison of measured values with those predicted from the binary systems using the Kohler equation suggests that the ternary excess term is zero within experimental error. Thus simple equations could be derived for heats of formation, mixing and phase transition in this system which can be used in the future in CALPHAD type calculations of phase equilibria that obtain the extent of fluorite solid solution at different temperatures.

Concentration Dependences of the Surface Tension and Density of Solutions of Acetone–Ethanol–Water Systems at 293 K

Concentration dependences of the surface tension and density of solutions of three-component acetone–ethanol–water systems and the bounding binary systems at 273 K are studied. The molar volume, adsorption, and composition of surface layers are calculated. Experimental data and calculations show that three-component solutions are close to ideal ones. The surface tensions of these solutions are calculated using semi-empirical and theoretical equations. Theoretical equations qualitatively convey the concentration dependence of surface tension. A semi-empirical method based on the Kohler equation allows us to predict the concentration dependence of surface tension within the experimental error.

Development of a cloud condensation nuclei (CCN) counter using a laser and charge-coupled device (CCD) camera

A continuous flow streamwise thermal gradient cloud condensation nuclei (CCN) counter with an aerosol focusing and a laser-charge-coupled device (CCD) camera detector system was developed here. The counting performance of the laser-CCD camera detector system was evaluated by comparing its measured number concentrations with those measured with a condensation particle counter (CPC) using polystyrene latex (PSL) and NaCl particles of varying sizes. The CCD camera parameters (e.g. brightness, gain, gamma, and exposure time) were optimized to detect moving particles in the sensing volume and to provide the best image to count them. The CCN counter worked well in the particle number concentration range of 0.6–8000 #·cm−3 and the minimum detectable size was found to be 0.5 μm. The supersaturation in the CCN counter with varying temperature difference was determined by using size-selected sodium chloride particles based on Kohler equation. The developed CCN counter was applied to investigate CCN activity of atmospheric ultrafine particles at 0.5% supersaturation. Data showed that CCN activity increased with increasing particle size and that the higher CCN activation for ultrafine particles occurred in the afternoon, suggesting the significant existence of hygroscopic or soluble species in photochemically-produced ultrafine particles.

Densities, Speeds of Sound, Excess Molar Volumes, and Excess Isentropic Compressibilities at T = (298.15 and 308.15) K for Methyl Methacrylate + 1-Alkanols (1-Butanol, 1-Pentanol, and 1-Heptanol) + Cyclohexane, + Benzene, + Toluene, + p-Xylene, and + Ethylbenzene

The densities and speeds of sound for ternary mixtures of methyl methacrylate (MMA) + 1-alkanols (1-butanol, 1-pentanol, 1-heptanol) + aromatic organic solvents (benzene, toluene, p-xylene, and ethylbenzene) or + cyclohexane have been measured at T/K = (298.15 and 308.15). Excess molar volumes and excess isentropic compressibilities were calculated across the mole fraction range, and the same were compared with the values calculated using similar data for the binary pairs through Redlich−Kister (RK), Tsao and Smith (TS), and Kohler (K) equations. As compared to others, the RK equation predicted the excess molar volumes and excess isentropic compressibilities adequately.

Excess molar volumes, excess isentropic compressibilities and relative permittivity deviations for the ternary mixtures of esters + glycols + organic solvents at different temperatures

The experimental densities, ρ at T = (298.15 to 313.15) K, speeds of sound, υ and relative permittivities, ε r at T = (298.15 and 308.15) K at atmospheric pressure are measured across the composition for ternary mixtures of methyl acetate (MA) + diethylene glycol (DG) + nitrobenzene, MA + triethylene glycol (TG) + chlorobenzene, MA + TG + bromobenzene, MA + TG + nitrobenzene, ethyl acetate (EA) + TG + chlorobenzene, EA + TG + bromobenzene, EA + TG + nitrobenzene, EA + DG + nitrobenzene, butyl acetate (BA) + TG + chlorobenzene, BA + TG + bromobenzene and BA + TG + nitrobenzene. Excess molar volumes, excess isentropic compressibilities and relative permittivity deviations are calculated from the experimental data. The excess and deviation functions of ternary mixtures are also calculated from the data of constituent binary pairs employing Redlich–Kister (RK), Tsao–Smith (TS) and Kohler (K) equations. Specific structure making interactions between the Cl- and Br- groups of organic solvents with the carboxyl oxygen of esters, dipole–dipole interactions and Cl…H–O complexes are probably predominant in these mixtures.

Thermodynamic Effects of Alloying Elements on Solid-State Interfacial Reactions in a Multicomponent System: Theoretical Treatment and Application in SiC f /Ti Composites

Alloying elements have an important effect on phase formation in solid-state chemical reactions. From the viewpoint of thermodynamics, applying the Miedema model and Kohler equation, a thermodynamic model for a solid-state multicomponent system was presented. Using this model, the activity of Ti in SiC f /Ti composites was evaluated, and the influence of different alloying elements on the interfacial chemical reactions in SiC f /Ti composites at high temperature was investigated from a thermodynamic viewpoint. The calculation results showed that most alloying elements, such as Al, Sn, Ni, Fe, Cu, Mn, Cr, Mo, and V, can reduce the activity of Ti and hinder the interfacial chemical reactions in SiC f /Ti composites, but an alloying element such as Nb can decrease the stability of the interfacial chemical reactions in SiC f /Ti composites, in most cases.

A curved multi-component aerosol hygroscopicity model framework: Part 2 – Including organic compounds

Abstract. This paper describes the inclusion of organic particulate material within the Aerosol Diameter Dependent Equilibrium Model (ADDEM) framework described in the companion paper applied to inorganic aerosol components. The performance of ADDEM is analysed in terms of its capability to reproduce the behaviour of various organic and mixed inorganic/organic systems using recently published bulk data. Within the modelling architecture already described two separate thermodynamic models are coupled in an additive approach and combined with a method for solving the Kohler equation in order to develop a tool for predicting the water content associated with an aerosol of known inorganic/organic composition and dry size. For development of the organic module, the widely used group contribution method UNIFAC is employed to explicitly deal with the non-ideality in solution. The UNIFAC predictions for components of atmospheric importance were improved considerably by using revised interaction parameters derived from electro-dynamic balance studies. Using such parameters, the model was found to adequately describe mixed systems including 5–6 dicarboxylic acids, down to low relative humidity conditions. By comparison with electrodynamic balance data, it was also found that the model was capable of capturing the behaviour of aqueous aerosols containing Suwannee River Fulvic acid, a structure previously used to represent the functionality of complex oxidised macromolecules often found in atmospheric aerosols. The additive approach for modelling mixed inorganic/organic systems worked well for a variety of mixtures. As expected, deviations between model predictions and measurements increase with increasing concentration. Available surface tension models, used in evaluating the Kelvin term, were found to reproduce measured data with varying success. Deviations from experimental data increased with increased organic compound complexity. For components only slightly soluble in water, significant deviations from measured surface tension depression behaviour were predicted with both model formalisms tested. A Sensitivity analysis showed that such variation is likely to lead to predicted growth factors within the measurement uncertainty for growth factor taken in the sub-saturated regime. Greater sensitivity was found for the value of dry density used in the assumed form of the dried out aerosol. Comparison with a coupled thermodynamic approach showed that assumed values for interactions parameters may lead to erroneous results where a simple additive approach may provide more accurate results. However, where available, the use of coupled thermodynamics can better reproduce measured behaviour. Further work (and laboratory data) is required to assess whether this difference lies within the experimental uncertainty of observed hygroscopic behaviour for a variety of systems.

A curved multi-component aerosol hygroscopicity model framework: Part 1 – Inorganic compounds

Abstract. A thermodynamic modelling framework to predict the equilibrium behaviour of mixed inorganic salt aerosols is developed, and then coupled with a technique for finding a solution to the Kohler equation in order to create a diameter dependent hygroscopic aerosol model (Aerosol Diameter Dependent Equilibrium Model – ADDEM). The model described here provides a robust and accurate inorganic basis using a mole fraction based activity coefficient model and adjusted energies of formation for treating solid precipitation. The model framework can accommodate organic components, though this added complexity is considered in a companion paper, this paper describes the development of the modelling architecture to be used and predictions of an inorganic model alone. The modelling framework has been developed to flexibly use a combination of mixing rules and other potentially more accurate techniques where available to calculate the water content. Comparisons with other state-of-the-art general equilibrium models and experimental data are presented and show excellent agreement. The Kelvin effect can be considered in this scheme using a variety of surface tension models. Comparison of predicted diameter dependent phenomena, such as the increased relative humidity for onset of deliquescence with decreasing diameter, with another diameter dependent model is very good despite the different approach used. The model is subject to various sensitivities. For the inorganic systems studied here, the model is sensitive to choice of surface tension scheme used, which decreases for larger aerosol. Large sensitivities are found for the value of dry density used. It is thus likely that the history of the aerosol studied in a hygroscopic tandem differential mobility analyser (HTDMA), specifically the nature of the drying process that will influence the final crystalline form, will create systematic uncertainties upon comparisons with theoretical predictions. However, the magnitudes of all of the above sensitivities are potentially less than those introduced when using a semi ideal growth factor analogue for certain conditions.

Densities, Excess Molar Volumes at T = (298.15 to 313.15) K, Speeds of Sound, Excess Isentropic Compressibilities, Relative Permittivities, and Deviations in Molar Polarizations at T = (298.15 and 308.15) K for Methyl Methacrylate + 2-Butoxyethanol or Dibutyl Ether + Benzene, Toluene, or p-Xylene

The experimental densities at T = (298.15 to 318.15) K, speeds of sound, and relative permittivities at T = (298.15 and 308.15) K are measured for the six ternary mixtures: methyl methacrylate (1) + 2-butoxyethanol (2) or dibutyl ether (2) + benzene (3), toluene (3), and p-xylene (3). Excess molar volumes, excess isentropic compressibilities, and deviations in molar polarizations are calculated. The above functions of ternary systems were also calculated using the data of constituent binary pairs and employing Redlich−Kister, Tsao and Smith, and Kohler equations. Compared to others, the Redlich−Kister equation has been found to reproduce the experimental excess molar volumes adequately, while excess isentropic compressibilities and deviations in molar polarizations of the above ternary mixtures could not be adequately calculated using the respective binary contributions.

COMPETITIVE GROWTH OF CLOUD DROPLET DUE TO CONDENSATION OF ATMOSPHERIC WATER VAPOR AND ITS EQUILIBRIUM RADIUS

Cloud droplets grow on cloud condensation nuclei (CCN) with condensing the ambient water vapor in the atmospheric air parcel. The size of the cloud droplet in the equilibrium state is estimated usually by Kohler Equation. However, Kohler Equation is based on the assumption that the cloud droplet grows in an infinitely large reservoir of water vapor at constant pressure, i. e., at constant saturation ratio. This assumption causes some erroneous results in the estimation of real size in the air parcel, although the assumption makes the calculation easy. There are two apparent deficiencies in Kohler Equation. One is that the equilibrium size cannot be decided in case of the larger saturation ratio than the critical ratio. The other is that the variation of the size with the number density of CCN, i. e., the competitive growth, cannot be considered in the estimation of the size. In order to improve these deficiencies, the conventional Kohler Equation has been modified using the mass conservation of the water contained in the air parcel.

Effect of Number Density of CCN on Equilibrium Radius and Temperature of Cloud Droplet during Competitive Growth

Usually, for the estimation of the cloud droplet size in the equilibrium state, Kohler equation is utilized, assuming constant vapor content and constant temperature. However, the condensational growth of the cloud droplet consumes the ambient water vapor and releases the latent heat. Therefore, the ambient water vapor pressure and the temperature are changed. Taking account of the variations of the water vapor content and the temperature, a new model for the estimation of the equilibrium cloud droplet size and temperature has been developed by modifying Kohler equation. The modification of Kohler model is based on the mass and the heat conservation laws for the water (vapor and liquid) in the air parcel occupied by one cloud droplet. Variations of size and temperature with number density of CCN are simulated numerically.

Property Changes of Mixing for the 1-Butanol + Methanol + 2-Methoxy-2-methylbutane System at 298.15 K and Atmospheric Pressure

Densities, refractive indices, and speeds of sound at 298.15 K and atmospheric pressure for the 1-butanol + methanol + 2-methoxy-2-methylbutane ternary system have been measured. Excess molar volumes, molar refraction, and isentropic compressibility changes of mixing were determined from experimental measurements. The results were fitted to the Redlich−Kister equation to obtain the coefficients. These property changes were also predicted using several empirical equations from the values of their constituent binary subsystems, and the best results were obtained with the Kohler equation.

Effect of Number Density of CCN on Competitive Growth of Cloud Droplet by Condensation of Atmospheric Water Vapor

Cloud droplets grow on cloud condensation nuclei (CCN) with condensing the ambient water vapor in the atmospheric air parcel. The size of the cloud droplet in the equilibrium state is estimated usually Kohler Equation. However, Kohler Equation is based on the assumption that the cloud droplet grows in an infinitely large reservoir of water vapor at constant pressure, i. e., at constant saturation ratio. This assumption causes some erroneous results in the estimation of real size in the air parcel, although the assumption makes the calculation easy. Thereare two apparent deficiencies in Kohler Equation. One is that the equilibrium size cannot be decided in case of the larger saturation ratio than the critical ratio. The other is that the variation of the size with the number density of CCN, in other words the competitive growth, cannot be considered in the estimation ofthe size. In order to improve these deficiencies, the conventional Kohler Equation has been modified using the mass conservation of the water contained in the air parcel.

Predicting Particle Critical Supersaturation from Hygroscopic Growth Measurements in the Humidified TDMA. Part II: Laboratory and Ambient Studies

Abstract Laboratory studies are used to test the method proposed in Part I for estimating the critical supersaturation of quasi-monodisperse, dry particles from measurements of hygroscopic growth at relative humidities below 100%. An advantage of the proposed technique is that it directly links dry particle size to cloud condensation nuclei (CCN) activity and simultaneously provides some information on particle chemical composition. Studies have been conducted on particles composed of NaCl, (NH4)2SO4, NH4HSO4, internally and externally mixed NaCl–(NH4)2SO4, and on ambient particles of unknown chemical composition. A modified form of the Kohler equation is fit to measurements from a humidified tandem differential mobility analyzer to derive two chemical composition–dependent parameters and the critical supersaturation for a given dry particle size. A cloud condensation nucleus counter is used to simultaneously observe the critical supersaturation of the same dry particles. Results show that for particles c...

A simple analytical model of aerosol properties with account for hygroscopic growth: 2. Scattering and absorption coefficients

A new microphysical model of aerosol size spectra accounting for hygroscopic growth is used to calculate the aerosol scattering and absorption coefficients of solar and longwave radiation. Dependence of the extinction coefficient is explored theoretically, starting from the Junge [1963] power spectra of the dry aerosol and the Kohler equation [Khvorostyanov and Curry, this issue]. It is shown that the general dependence of the aerosol scattering and absorption coefficients on λ and H can be presented in the form σ∼λ−γ(1 − H)R. This model explains both the empirical angstrom wavelength law and Kasten [1969] Hanel [1976] humidity law and identifies a functional relationship between them. The empirical coefficients of these laws are related to the aerosol microstructure and to each other by a very simple relationship. An explanation is offered for the spectral behavior of the scattering coefficient and its variation with relative humidity for various types of atmospheric hazes. It is shown that at very high humidities a dense haze with visibility less than 1 km may form under subsaturated conditions and without activated droplets. Application of this model to the longwave spectrum allows us to explain quantitatively the negative temperature dependence of atmospheric absorption in the atmospheric window without depending on the presence of water vapor dimers. Recommendations for aerosol optical and microphysical measurements are given. This simple model can be used as a tool in remote sensing for evaluation of aerosol scattering and absorption optical thicknesses and for parameterization of aerosol optical properties in general circulation and cloud models.

Erroneous Use of the Modified Kohler Equation in Cloud and Aerosol Physics Applications

Abstract The phase equilibrium equation between water vapor and a liquid droplet is an important tool of cloud physics. Although several different forms of the modified Kohler equation were derived in the past, the authors show that a thermodynamically consistent treatment leads to the original Kohler equation. Recent experimental measurements of the condensational growth of a single levitated ammonium sulfate solution droplet are compared with theoretical calculations using 1) the Kohler equation with a constant van’t Hoff factor, 2) using the Kohler equation with the van’t Hoff factor as a function of ammonium sulfate concentration, and 3) using the recently derived equilibrium equation containing an additional term proportional to the first derivative of the van’t Hoff factor with respect to solute concentration. It was found that the Kohler equation with van’t Hoff factor as a function of solute concentration provides the best agreement with experimental data.

Molar Volumes, Molar Refractions, and Isentropic Compressibilities of (Ethanol + Methanol + 2-Methoxy-2-methylpropane) and (Ethanol + Methanol + 2-Methoxy-2-methylbutane) at 298.15 K

Excess molar volumes (VE) and deviations in molar refractions (ΔR) and the isentropic compressibility (Δκs) were determined for the systems ethanol + methanol + 2-methoxy-2-methylpropane and ethanol + methanol + 2-methoxy-2-methylbutane at 298.15 K and atmospheric pressure. For both systems, these results were satisfactorily correlated by the Redlich−Kister polynomial. Several empirical equations allowing prediction of the results for the ternary systems from the VE, ΔR, and Δκs of their constituent binary subsystems were also examined. The Kohler equation gives the best predictions of VE, while the equations of Radojkovic et al. and of Jacob and Fitzner give the best predictions of ΔR and Δκs.

A Reexamination of the Derivation of the Equilibrium Supersaturation Curve for Soluble Particles

Abstract Following the methods known from the heteromolecular homogeneous nucleation theory, a reexamination of the Kohler equation is presented. In contrast to the original Kohler's approach and the later reexamination of Young and Warren, activity coefficients dependent on the droplet composition and the Gibbs-Duhem relation are taken into account. The results conform with the heteromolecular homogeneous nucleation theory and show that the additional terms derived by Young and Warren completely compensate each other. The nonideal corrections for the classical Kohler equation are investigated for an H2SO4-H2O system. An increase of critical supersaturation by up to about 25% must be expected if nonideality is taken into account.

Critical evaluation and optimization of the thermodynamic properties of liquid tin solutions

Thermodynamic and phase equilibrium data for the following 18 elements in molten Sn were collected and critically evaluated: Al, Ca, Ce, Co, Cr, Cu, Fe, H, Mg, Mo, Na, Ni, O, P, S, Se, Si, and Ti. Binary and ternary data were optimized to give polynomial expressions for the excess Gibbs energies as functions of temperature and composition. For some solutes, the optimized expressions are valid over the entire composition range 0 ≤ XSn ≤ 1. In other cases, the expressions apply to Snrich solutions. Solute-solute interaction terms were estimated where data were not available. The optimized Gibbs energy expressions are also presented in the form of interaction parameters, and the equivalence between the polynomial and interaction parameter formalisms is discussed. Through the Kohler equation, or the modified interaction parameter formalism, the thermodynamic properties of the multicomponent solution of 18 elements in Sn can be calculated. The database is suitable for computer storage and manipulation.