Jan Hrubý

Surface Tension of Supercooled Water: No Inflection Point down to −25 °C

A dramatic increase in the surface tension of water with decreasing temperature in the supercooled liquid region has appeared as one of the many anomalies of water. This claimed anomaly characterized by the second inflection point at about +1.5 °C was observed in older surface tension data and was partially supported by some molecular simulations and theoretical considerations. In this study, two independent sets of experimental data for the surface tension of water in the temperature range between +33 and -25 °C are reported. The two data sets are mutually consistent, and they lie on a line smoothly extrapolating from the stable region. No second inflection point and no other anomalies in the course of the surface tension were observed. The new data lies very close to the extrapolated IAPWS correlation for the surface tension of ordinary water, which hence can be recommended for use, e.g., in atmospheric modeling.

Gradient theory computation of the radius-dependent surface tension and nucleation rate for n-nonane clusters

The Van der Waals-Cahn-Hilliard gradient theory (GT) is applied to determine the structure and the work of formation of clusters in supersaturated n-nonane vapor. The results are analyzed as functions of the difference of pressures of the liquid phase and vapor phase in chemical equilibrium, which is a measure for the supersaturation. The surface tension as a function of pressure difference shows first a weak maximum and then decreases monotonically. The computed Tolman length is in agreement with earlier results [L. Granasy, J. Chem. Phys. 109, 9660 (1998)] obtained with a different equation of state. A method based on the Gibbs adsorption equation is developed to check the consistency of GT results (or other simulation techniques providing the work of formation and excess number of molecules), and to enable an efficient interpolation. A cluster model is devised based on the density profile of the planar phase interface. Using this model we analyze the dependency of the surface tension on the pressure difference. We find three major contributions: (i) the effect of asymmetry of the density profile resulting into a linear increase of the surface tension, (ii) the effect of finite thickness of the phase interface resulting into a negative quadratic term, and (iii) the effect of buildup of a low-density tail of the density profile, also contributing as a negative quadratic term. Contributions (i)-(iii) fully explain the dependency of the surface tension on the pressure difference, including the range relevant to nucleation experiments. Contributions (i) and (ii) can be predicted from the planar density profile. The work of formation of noncritical clusters is derived and the nucleation rate is computed. The computed nucleation rates are closer to the experimental nucleation rate results than the classical Becker-Döring theory, and also the dependence on supersaturation is better predicted.

Measurements of the p-v-T behavior of refrigerant R134a in the liquid phase

A short description of an improved p-v-T apparatus with a constant-volume piezometer is given. p-v-T data of the environmentally safe refrigerant 1, 1, 1, 2-tetrafluoroethane (R134a), were measured for temperatures between 205 and 309 K at pressures up to 56 MPa. The uncertainty of temperature and pressure measurements was estimated to be ±0.05°C and ±0.1%, respectively; uncertainty in the specific volume was estimated to be ±0.15%. Purity of the sample used was 99.9+wt%. The data are represented analytically in order to demonstrate experimental accuracy and to facilitate calculation of thermodynamic properties.

On the effect of pressure and carrier gas on homogeneous water nucleation

Homogeneous nucleation rates of water droplets were measured at a nucleation temperature close to 240 K in a Pulse-Expansion Wave Tube (PEWT). Several measures were taken to improve the data obtained with the PEWT. For instance, the molar water vapor fraction was determined with three independent techniques. The resulting standard uncertainty of the supersaturation was within 1.8%. Results are given for water nucleation in helium at 100 kPa and at 1000 kPa and in nitrogen at 1000 kPa. Two trends were observed: (i) the values of the nucleation rate of water in helium at 1000 kPa are slightly but significantly higher (factor 3) than its values at 100 kPa and (ii) nucleation rates of water in nitrogen at 1000 kPa are clearly higher (factor 10) than in helium at the same pressure. It is argued that the explanation of the two observed trends is different. For case (i), it is the insufficient thermalization of the growing water clusters in helium at the lowest pressure that has a reducing effect on the nucleation rate, although a full quantitative agreement has not yet been reached. For case (ii), thermal effects being negligible, it is the pressure dependency of the surface tension, much stronger for nitrogen than for helium, that explains the trends observed, although also here a full quantitative agreement has not yet been achieved.

An analytical formulation of thermodynamic properties of dry and metastable steam suitable for computational fluid dynamics modelling of steam turbine flows

Application of computational fluid dynamics to real steam flows including non-equilibrium condensing flows requires an accurate and, at the same time, computationally inexpensive formulation of thermodynamic properties of steam. The present formulation enables fast computation of all thermodynamic properties, using mass density and specific internal energy as independent variables. This choice of independent variables follows the needs of time-marching computational fluid dynamics computations of dry and metastable steam flows in steam turbines. The formulation comprises an ideal-gas part and a residual part. The residual part is expressed in a form analogous to the virial equation of state but the coefficients are functions of internal energy, rather than temperature. Here we present a variant retaining only one coefficient of the density series, corresponding to the second virial coefficient. The uncertainties of properties computed from the present formulation only slightly exceed those of the fundamental formulation IAPWS-95. The formulation is valid from 253.15 K to 1073.15 K. Up to pressures given by the isentrope 7.5 kJ/(kg K), the computed properties are within the experimental uncertainties. Beyond this limit, the accuracy continuously decreases as it can be determined from presented deviation plots.

Prediction of the homogeneous droplet nucleation by the density gradient theory and PC-SAFT equation of state

We combined the density gradient theory (DGT) with the PC-SAFT and Peng-Robinson equations of state to model the homogeneous droplet nucleation and compared it to the classical nucleation theory (CNT) and experimental data. We also consider the effect of capillary waves on the surface tension. DGT predicts nucleation rates smaller than the CNT and slightly improves the temperature-dependent deviation of the predicted and experimental nucleation rates.

Nucleation rates of droplets in supersaturated steam and water vapour–carrier gas mixtures between 200 and 450 K

We compare experimental nucleation rates for water vapour in various carrier gases, estimated nucleation rates for steam, and nucleation rates obtained from molecular simulations. The data for steam are deduced from empirical adjustments of the classical nucleation theory developed by various authors to reproduce pressure and optical data for condensing steam flows in converging-diverging nozzles and turbine stages. By combining the data for nucleation in carrier gases and the data for steam nucleation, an unprecedented temperature range of 250 K is available to study the temperature dependence of nucleation rate. Original results of molecular dynamic simulations for TIP4P/2005 force field in the NVE (system constrained by number of particles, volume and energy) conditions are provided. Correction of classical nucleation theory for non-isothermal nucleation conditions is applied to experimental and simulated data. The nucleation rate data for steam follow a similar temperature trend as the nucleation rate data for water vapour in carrier gases at lower temperatures. The ratio of observed nucleation rates to classical nucleation theory predictions decreases more steeply with temperature than the empirical correlation by Wölk et al. (J Chem Phys 2002; 117: 4954–4960). On the contrary to experimental data, the ratios of nucleation rates computed from molecular simulations to classical nucleation theory predictions do not show a significant temperature trend.

Water Nucleation Measurements in a Pulse-Expansion Wave Tube

Wave experiments have been carried out in a pulse-expansion wave tube [1] to study water condensation, which plays an important role in a variety of industrial processes and in cloud formation models. We focused on homogeneous condensation, in which foreign bodies are absent and stable clusters (aggregates of molecules) are formed due to thermal fluctuations. Homogeneous condensation is a stochastic process of molecular collisions with a probability that colliding molecules stick together to form a condensation nucleus. Some of these nuclei will grow by catching other molecules to reach a so-called critical cluster size (nucleation) beyond which they grow to macroscopic sizes (droplet growth); other nuclei will simply evaporate into individual molecules again. The two underlying physical processes of condensation that can be distinguished are nucleation and droplet growth. Our pulse-expansion wave tube is specifically designed to produce a nucleation pulse [2] allowing to separate the processes of nucleation and droplet growth in time. We can therefore accurately measure the nucleation rate, which describes the number of water droplets that is formed per unit of time and space. Droplet growth is studied by combining two different optical techniques based on light scattering of a laser beam by the droplet cloud [3]. Once the droplet growth curve (droplet radius vs. time) is determined, the droplet number density follows from the measured extinction. The nucleation rate is finally obtained as the ratio of the droplet number density and the nucleation pulse duration. The homogeneous water nucleation rate is strongly dependent on the water vapor fraction of the gas–vapor mixture [4]. In previous work, the water vapor fraction was determined by the apparatus [5] that prepares these mixtures. In this work, we implemented two additional techniques to investigate the produced water vapor fractions. We shall present new experimental homogeneous nucleation rates of droplets of supercooled water (liquid water at temperatures lower than the equilibrium freezing temperature) in nitrogen at an elevated pressure of 1.0 MPa and a temperature of 240 K.

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.

Corrections to the classical work of formation of critical clusters

We show that some corrections to the work of formation predicted by the Classical Nucleation Theory (CNT) can be derived from a simple cluster model based on the density profile for the planar phase interface. These corrections are related to the Tolman length and to the thickness of the phase interface. We analyze the temperature dependencies of these corrections based on critical scaling and compare them with results of the density gradient theory (DGT).