Hertz Knudsen Formula Research Articles

Физические процессы в датчике низкого вакуума типа Пирани

Auger electron spectroscopy and thermoresistive methods are used to study the physical processes leading to gas cooling of heated molybdenum filaments in a wide temperature range of 350 - 1300 K and pressures of 760 - 10-3 Torr, corresponding to the operating range of a Pirani-type vacuum sensor. Nitrogen was used as the gas. It is shown that nitrogen atoms chemisorbed on the surface do not contribute to gas cooling, which occurs only due to physisorbed N2 molecules. In the intermediate vacuum region of 10-3 – 1 Torr, the heater is cooled due to the equilibrium between the flux of incident and thermally desorbed molecules, which is well described by the Hertz-Knudsen formula and first-order desorption with an activation energy of ~ 0.55 eV. On the contrary, at high pressures close to atmospheric, this cooling occurs due to the thermal desorption of gas molecules from an almost filled monolayer, which reduces its relative efficiency by many orders of magnitude.

Sublimation of cometary ices in the presence of organic volatiles II

Evolution of the surfaces of cometary nuclei is determined by the sublimation of ice. The rate of sublimation is commonly calculated using the simple Hertz–Knudsen formula, which should be corrected by a temperature dependent sublimation coefficient αs (Kossacki et al., 1999; Gundlach et al., 2011; Kossacki and Leliwa-Kopystynski, 2014). In Kossacki et al. (2017) the temperature dependence of αs for water ice with acetone and methanol present in comets was analyzed. In this work the influence of propanal at the temperature ~195–~230 K and formamide at the temperature ~200–~235 K were investigated. Experiments were performed with slab ice and porous fine-grained ice. It is found, that the presence of propanal reduces the sublimation coefficient at T < 207 K and enhances at T > 208 K. The presence of formamide reduced the sublimation coefficient in the whole investigated temperature range. Porous fine-grained ice sublimed up to four times slower than non-porous one.

Sublimation of cometary ices in the presence of organic volatiles

Evolution of the surfaces of cometary nuclei is determined by the sublimation of ice. The rate of sublimation is commonly calculated using the simple Hertz–Knudsen formula, which should be corrected by a temperature dependent sublimation coefficient αs (Kossacki et al., 1999; Gundlach et al., 2011; Kossacki and Leliwa–Kopystynski, 2014). In this work influence of C3H6O (acetone), and CH3OH (methanol) present in cometary ice on the sublimation coefficient is investigated. We have found, that the mass fraction of admixtures ≤ 2% in water ice is sufficient for an increase of αs by a factor about 5, depending on the temperature. The sublimation coefficient is sensitive to very small concentrations of admixtures. When the admixture is acetone and the mass fraction f=0.005 is αs(215 K) ∼ 0.46 instead of ∼ 0.18 for pure water ice. When f=0.01 is αs(215 K) ∼ 0.74, and when f=0.02 is αs(215 K) ∼ 0.78. The presence of 0.01 mass fraction of methanol has similar influence on the sublimation coefficient as the presence of 0.01 mass fraction of acetone. At temperatures T > 235 K for all investigated compositions of ice αs ∼ 0.15.

The Corrected Simulation Method of Critical Heat Flux Prediction for Water-Cooled Divertor Based on Euler Homogeneous Model**supported by the National Magnetic Confinement Fusion Science Program of China (No. 2010GB104005) and National Natural Science Foundation of China (No. 51406085)

An accurate critical heat flux (CHF) prediction method is the key factor for realizing the steady-state operation of a water-cooled divertor that works under one-sided high heating flux conditions. An improved CHF prediction method based on Euler's homogeneous model for flow boiling combined with realizable k-ϵ model for single-phase flow is adopted in this paper in which time relaxation coefficients are corrected by the Hertz-Knudsen formula in order to improve the calculation accuracy of vapor-liquid conversion efficiency under high heating flux conditions. Moreover, local large differences of liquid physical properties due to the extreme nonuniform heating flux on cooling wall along the circumference direction are revised by formula IAPWS-IF97. Therefore, this method can improve the calculation accuracy of heat and mass transfer between liquid phase and vapor phase in a CHF prediction simulation of water-cooled divertors under the one-sided high heating condition. An experimental example is simulated based on the improved and the uncorrected methods. The simulation results, such as temperature, void fraction and heat transfer coefficient, are analyzed to achieve the CHF prediction. The results show that the maximum error of CHF based on the improved method is 23.7%, while that of CHF based on uncorrected method is up to 188%, as compared with the experiment results of Ref. [12]. Finally, this method is verified by comparison with the experimental data obtained by International Thermonuclear Experimental Reactor (ITER), with a maximum error of 6% only. This method provides an efficient tool for the CHF prediction of water-cooled divertors.

Evaporation and condensation in the presence of nanoparticles in bulk vapor

The problem of evaporation/condensation on planar surfaces in the presence of nanoparticles in bulk vapor, with the particles being uniformly distributed in the space, has been solved by direct numerical solution of the Boltzmann kinetic equation. It has been suggested that droplets are formed on the particles and grow with time due to vapor condensation. The problem has been solved with the use of an original procedure of transforming the function of distribution of vapor molecules, with the transformation being caused by the interaction of the molecules with the particles. The condensation sites present in the vapor have been shown to significantly influence its macroscopic parameters. In particular, the condensation on the nanoparticles significantly reduces the mass flux transmitted from one surface to another. The results of the numerical solution confirm the possibility of calculating the growth rate of the droplets in the free-molecular regime by the Hertz–Knudsen formula.

Temperature dependence of the sublimation rate of water ice: Influence of impurities

The sublimation rate of ice is commonly calculated using the simple Hertz–Knudsen formula. This formula is derived from the kinetic theory of gases and ignores microphysical processes determining the actual sublimation rate. The microphysical processes can be accounted for by including the temperature dependent sublimation coefficient (Kossacki, K.J., et al. [1999]. Planet. Space Sci. 47, 1521–1530; Gundlach, B., Skorov, Y.V., Blum, J. [2011]. Icarus 213, 710–719). Kossacki and Markiewicz (Kossacki, K.J., Markiewicz, W.J. [2013]. Icarus 224, 172–177) discussed to what extent inaccuracy of the simple Hertz–Knudsen equation affects the calculated temperature of Comet 67P/Churyumov-Gerasimenko. Numerical simulations presented in Kossacki and Markiewicz (Kossacki, K.J., Markiewicz, W.J. [2013]. Icarus 224, 172–177) indicate, that derivation of the temperature below the dust mantle from the measured water production rate ignoring temperature dependence of the sublimation coefficient can lead to an underestimate of the temperature by more than 10K. Thus, it is important to know the dependence on the sublimation coefficient of the composition of the real cometary ice, which can be far from purity.We intended to check whether a small amount of dissolved minerals can affect the temperature dependence of the sublimation coefficient of ice. According to our experiments the answer is positive.

Comet 67P/CG: Influence of the sublimation coefficient on the temperature and outgassing

The sublimation rate of ice is commonly calculated using simple Hertz–Knudsen formula. This formula is derived from the kinetic theory of gases and ignores microphysical processes determining the actual sublimation rate. The microphysical processes can be accounted for by including in the Herz–Knudsen equation a temperature dependent sublimation coefficient (Kossacki, K.J., Markiewicz, W.J., Skorov, Y., Koemle, N.I. [1999]. Planet. Space Sci. 47, 1521–1530; Gundlach, B., Skorov, Y.V., Blum, J. [2011]. Icarus, 213, 710–719). Here we address the question to what extent inaccuracy of the simple Hertz–Knudsen equation affects the calculated temperature of a cometary nucleus and the emission rate of water vapor to space. We performed numerical simulations dealing with evolution of a model comet of the orbit the same as Comet 67P/Churyumov-Gerasimenko, target comet of the Rosetta mission (Glassmeier, K.H., Boehnardt, H., Koshny, D., Kuhrt, E., Richter, I. [2007]. Space Sci. Rev. 128, 1–21). We have found, that the temperature below dust mantle is most sensitive to the value of the sublimation coefficient when the mantle is coarse grained, while the sublimation rate is most affected when the mantle is fine grained. We also conclude that derivation of the temperature below the mantle from the measured water production rate ignoring temperature dependence of the sublimation coefficient leads to an underestimate of the temperature by more than 10K when the nucleus is fine grained.

Early stage of critical clusters growth in phenomenological and molecular dynamics simulation models

The growth of critical clusters is discussed in the paper according to the classical and molecular dynamics (MD) approaches. A new formula for molecule numbers in critical clusters has been derived within the framework of the classical approach. A set of equations controlling the early stage of growth in a neighborhood of a critical size is presented. As far as molecular dynamics simulation is concerned, a computational technique based on the DL_POLY code is described in brief. Computation results are presented concerning cluster formation of H 2O vapor, distribution of clusters versus time, cluster growth and radial density distribution of isolated clusters. A comparison with the classical results is made for a case of dense vapor, where the mechanism of strong condensation is predominant. The Hertz–Knudsen formula seems to be verified by the molecular dynamics results.

On the behavior of micro-spheres in a hydrogen pellet target

A pellet target produces micro-spheres of different materials, which are used as an internal target for nuclear and particle physics studies. We will describe the pellet hydrogen behavior by means of fluid dynamics and thermodynamics. In particular, one aim is to theoretically understand the cooling effect in order to find an effective method to optimize the working conditions of a pellet target. During the droplet formation the evaporative cooling is best described by a multi-droplet diffusion-controlled model, while in vacuum, the evaporation follows the (revised) Hertz–Knudsen formula. Experimental observations compared with calculations clearly indicated the presence of supercooling, the effect of which is discussed as well.

Cathode erosion in high-current high-pressure arc

Cathode erosion rate was experimentally investigated for two types of arcs: one with tungsten cathode in nitrogen atmosphere and one with hafnium cathode in oxygen atmosphere. Conditions were typical for plasma arc cutting systems: gas pressure from 2 to 5 atm, arc current from 200 to 400 A, gas flow rate from 50 to 130 litre min−1.It was found that the actual cathode evaporation rate G is much lower than G0, the evaporation rate that follows from the Hertz–Knudsen formula: G = νG0. The difference is because some of the evaporated particles return back to the cathode. For conditions of our experiments, the factor ν could be as low as 0.01. It was shown experimentally that ν depends strongly on the gas flow pattern close to the cathode. In particular, swirling the gas increases ν many times. To explain the influence of gas swirling, model calculations of gas flows were performed. These calculations revealed difference between swirling and non-swirling flows: swirling the gas enhances the gas flow from the cathode. This enhanced flow sweeps the evaporated particles away from the cathode and thus increases the net evaporation rate.Electrode erosion associated with arc ignition (start erosion) was observed for both types of cathodes. Models of start erosion were suggested. In case of hafnium cathode the start erosion is due to cracking out the hafnium oxide layer, which was created during previous arc operation. In the case of the tungsten cathode the start erosion is produced by a cold cathode arc (vacuum arc), which exists at the cathode during first moments after arc ignition.

Investigation of the Dynamics of Evaporation of Condensed Bodies on the Basis of the Law of Spectral‐Radiation Intensity of Particles

The fundamentals of radiation theory and the mechanism of evaporation of condensed bodies are presented. The distribution functions of particles of a body by energies and by the intensity of their transition from one energy level to another in the process of evaporation have been obtained based on the law of spectral‐radiation intensity of the body particles. The temperature dependence of the resulting vapor flow on the outer surface of a massive condensed body and a thin layer in equilibrium and nonequilibrium states, which, in the limit, transforms to the known Hertz–Knudsen formula, has been found.

Sublimation coefficient of water ice under simulated cometary-like conditions

In papers dealing with evolution of cometary nuclei it is commonly assumed that the coefficients of sublimation α s and condensation α c of vapour are both equal to one. The experimental investigation of ice samples under simulated cometary-like conditions (Kossacki, K.J., Kömle, N.I., Leliwa-Kopystyński, J., Kargl, G., 1997. Thermal and structural evolution of cometary subsurface layer: selfconsistent model and experimental verification. Icarus 128, 127–144) suggests, however, that the sublimation flux calculated with the Hertz–Knudsen formula and the above assumption is nearly an order of magnitude too high. This may imply that actual values of α s for the ice/dust sample used in these experiments are of the order of 0.1. A similar conclusion can be drawn for α c from the results of various experiments concerning growth of ice crystals from the vapour phase and their sublimation (Lamb, D., Scott, W.D., 1972. Linear growth rates of ice crystals grown from the vapor phase. Journal of Crystal Growth 12, 21–31; Beckmann, W., Lacmann, R., 1982. Interface kinetics of growth and evaporation of ice II. Journal of Crystal Growth 58, 433–442; Sei, T., Gonda, T., 1989. The growth mechanism and the habit change of ice crystals growing from the vapour phase. Journal of Crystal Growth 94, 697–707). The exact values of both of these coefficients depend on various parameters such as temperature, concentration of surface impurities and deviation of the vapour pressure from that of the phase equilibrium. In this work the temperature dependence of the sublimation and condensation coefficients is discussed and an appropriate formula is proposed to fit the experimental results. This new formulation is then used to analyse the implications for the thermal conductivity of a porous cometary-like ice and the rate of vapour flux from a cometary nucleus.

A comparison of nucleation theories by the asymptotic solution of condensing nozzle flows

The classical and Dillmann-Meier nucleation rates are compared in transonic nozzle flows of moist air under atmospheric supply conditions using the asymptotic solution of Delale, Schnerr and Zierep. The comparison is made in slender nozzles using two distinct expressions for the poorly known surface tension, one fitted to the experiments of Peters and Paikert by the classical theory and the other extrapolated from room temperatures to the range of temperatures investigated. The droplet growth law is fixed by the Hertz-Knudsen formula. It is shown that the Dillmann-Meier theory predicts higher nucleation rates than the classical theory together with a delay of the onset of condensation when either of the surface tension expressions is employed.

Sublimation of Cleaved Surface of Zinc Single Crystals. II. Interpretation and Discussion

Cleaved surfaces of zinc single crystals were sublimed. The fact that the sublimed layer thickness of 29 µm evaluated from the observed deposited thickness 0.3 µm of the film is almost equal to the depth of 22 µm of the recess engraved on the cleaved surface indicates the possibility of particle ejection. The sublimed mass 1.9 g evaluated from the Hertz-Knudsen formula with the evaporation coefficient α=1 is larger than the initial specimen mass of 0.57 g. The thickness, 34 µm, of the deposited film, estimated from the Hertz-Knudsen formula with α=1, is larger than the observed thickness of 0.3 µm. Scanning electron micrographs and evaluation of the sublimed mass and the film thickness indicate the inapplicability of the atomic evaporation theory.

Theory of evaporation and condensation

The theory of evaporation and condensation is considered from a kinetic theory approach with a particular interest in the continuum limit. The moment method of Lees is used to solve the problem of the steady flow of vapor between a hot liquid surface and a cold liquid surface. By incorporating the singular nature of the problem, the forms of the continuum flow profiles found by Plesset are recovered. The expression for mass flux has the form of the Hertz–Knudsen formula but is larger by a factor of 1.665. A result of the theory is that the temperature profile in the vapor for the continuum problem is inverted from what would seem physically reasonable. This paradox is significant in that it casts a shadow of doubt on the fundamental theory.

Role of the Knudsen layer in the problem of droplet evaporation

A study is made of the part played by the Knudsen layer in the problem of weak unsteady evaporation of a spherical droplet in its own vapor. It is shown that use of the classical Hertz—Knudsen formula may lead to appreciable errors, in particular, in the determination of the time required by the droplet temperature to relax to the state corresponding to steady evaporation.