Mean Free Path Of Molecules Research Articles

Experimental and Numerical Investigation of Slip Effect on Nanofiber Filter Performance at Low Pressures.

Nanofiber filters are widely used in air filtration applications due to their superior performance over microfiber filters. Velocity slip around nanofibers has been identified as a key factor contributing to their high figure of merit, yet its impact on filter performance, especially particle collection efficiency, remains unclear due to the difficulty in isolating the slip effect as the sole variable. This study combines experimental and simulation methods to investigate the slip effect by adjusting the air molecule mean free path, rather than varying fiber size as done in previous studies. Filter media with mean fiber sizes ranging from 16.2 to 0.084µm are utilized. An image-based regression method is developed to address the challenge of determining the solidity of thin nanofiber layers. The results show that the slip effect is enhanced as the testing pressure decreases, reducing pressure drop by less than 15% for microfiber filters and over 50% for nanofiber filters ≈100nm. The enhanced slip effect at low pressures (i.e., relatively low pressure compared to the ambient environment) significantly improves filtration efficiency, especially for particles larger than 100nm. It also proposes semi-empirical equations for predicting filter performance in slip and transition flow regimes.

A novel approach to thermal insulation modelling in soft and medium vacuum insulation systems

The accuracy of vacuum-dependent models for predicting thermal performance of Multi-Layer Insulation (MLI) and other layered insulation systems is critical for the development of novel solutions in the aerospace and energy sectors, particularly long distance superconductors and cryogenic transfer lines. This paper presents a review of the current state of the art in cryogenic vacuum insulation systems and their associated modelling techniques and test methods. Current modelling techniques, namely the Lockheed and McIntosh MLI models, are compared to cryogenic, boil-off calorimeter test data for 3 types of MLI from the current literature. Both current models provide acceptable accuracy at high vacuum pressures but deviate from the test data when gas conduction becomes the dominant heat transfer mechanism (Kn≤1). Neither of the current models follow the characteristic S-curve observed by researchers during insulation tests. This paper presents the introduction of a novel modelling approach for layered insulation systems though changes to the current state of the art, specifically at soft (pvac≤7.5×105 mTorr) and medium vacuum (pvac≤7.5×102 mTorr) pressures, by substituting the gas conduction term in both equations with alternative terms based on the system Knudsen number (Kn) and molecule mean free path (l). This results in a stronger pressure dependence across the vacuum regime. Both modified models exhibited the characteristic S-curve with significantly reduced errors over the entire range.

Study on aerosol thermophoretic force and thermophoresis deposition model in reactor building

During nuclear reactor severe accident, the input of external passive containment cooling system will result in significant aerosol thermophoresis deposition in reactor building, which plays important factor affecting the safety assessment of nuclear reactor. Ratio of aerosol particles radius and gas molecule mean free path (Kn number) in reactor building is not greater than 2, corresponding to the near-continuous medium zone and transition zone. Firstly the typical thermophoretic force equations for these zones are evaluated. Combined with key influencing factors analysis, the correction factor connected to Kn number is introduced to address the deviation in the prediction for the transition zone of the current equations. The modified thermophoretic force equation obviously improves the underestimation of the original equation and is then used to obtain the thermophoresis deposition coefficient. According to experimental validation, the obtained coefficient is more accurate than the model that is widely employed in analysis codes in predicting the thermophoresis deposition of aerosol particles. Finally, fluid continuity, momentum, energy and particle continuity equations are established with deduced thermophoresis deposition coefficient and solved numerically. The thermophoresis deposition model considering natural convection (CT model) is established by theoretical derivation and fitting with the numerical solutions. By introducing the effects of natural convection on temperature distribution and aerosol concentration distribution the CT model could accurately describe the aerosol thermophoresis deposition behavior under natural convection conditions for Kn number is not greater than 2, which can be applied to aerosol thermophoresis deposition prediction in reactor building.

Viscous effects in Mach reflection of shock waves and passage to the inviscid limit

The influence of viscosity on the Mach reflection of shock waves in a steady flow of a monatomic gas is studied by solving the Navier–Stokes equations numerically. Based on the nested block grid refinement technique, the flow near the shock wave intersection is simulated, and its behaviour with increasing Reynolds number is studied. The computations are performed for the interaction of both strong (free-stream Mach number$M_\infty = 4$) and weak ($M_\infty = 1.7$) shock waves. In the strong reflection of shock waves at all Reynolds numbers in the examined range, it is found that there exists a small-size zone behind the shock wave intersection where the flow parameters differ from those predicted by the Rankine–Hugoniot relations and hence deviate from the predictions of the inviscid three-shock theory. The structure of this zone is self-similar: in coordinates normalised to the mean free path of molecules in the free stream. The structure is identical at all Reynolds numbers considered in the study. As the Reynolds number increases, the size of this zone in physical coordinates decreases, but the maximum difference between the viscous and inviscid solutions in this zone remains constant, reaching approximately$10\,\%$for pressure. In the weak reflection of shock waves, the flow structure behind the shock wave intersection is not self-similar, i.e. the flow fields at different Reynolds numbers do not coincide in the normalised coordinates, but converge, as the Reynolds number increases, to the parameters predicted by the inviscid three-shock theory.

Diffusion Model of Lennard-Jones Fluids Based on the Radial Distribution Function.

In this paper, a theoretical model for diffusion in Lennard-Jones fluids over a larger density range is proposed. Instead of looking at molecular transport in real space, we treat molecular transport in radial distribution function space. Specifically, the radial distribution function, which reflects the static properties of the system, is used to calculate the mean free path of molecules, a key physical quantity for diffusion modeling. Then, on the basis of the rarefied hard sphere gas diffusion model, and with consideration of the correction of collision frequency and backscattering effect caused by the higher density, the expression of self-diffusion coefficient in Lennard-Jones fluid is obtained. Molecular dynamics was used to simulate the diffusion in Lennard-Jones fluids in a wide range of reduced densities from 0.298 to 1.19 and reduced temperatures from 0.833 to 1.67. Except for small deviations at very high densities, the simulation results agree with the model predictions when the temperature is not too low.

Application of Prandtl, von Kármán, and lattice Boltzmann methods to investigations of turbulent slip incompressible flow in a flat channel

The paper focuses on the modeling of turbulent slip incompressible flow in a flat channel. Slippage on the channel wall can be caused by two reasons. The first reason is microchannels when the mean free path of molecules exceeds a certain value, which is characterized by the Knudsen number. The second reason is hydrophobic surfaces, which are used to reduce hydraulic resistance. Two models of turbulence were used to derive analytical solutions of fully developed flow. The first model is the Prandtl model (model of mixing length). The second model is the von Kármán model (model of similarity of pulsation velocities). Analytical models were built in a two-layer approximation: a laminar sublayer and a turbulent core. Both models showed a good agreement with the lattice Boltzmann method. An increase in the Knudsen number leads to an increase in the flow rate and a decrease in shear stress on the walls, which reduces the friction factor. This is due to the weakening of the interaction between the flow and the wall, which also leads to a decrease in the shear stress on the walls. As the Reynolds number increases, this effect becomes more noticeable.

Nonisothermal Rarefied Gas Flow through a Long Cylindrical Channel under Arbitrary Pressure and Temperature Drops

The problem of rarefied gas flow through a long cylindrical channel as a function of the pressure and the temperature maintained at the channel ends is considered on the basis of the S-model of the Boltzmann kinetic equation. The pressure and temperature drops between the channel ends vary from small values at which the linear transport theory is valid to large values at which the gas molecule mean free path ceases to be constant along the channel. The solution to the model kinetic equation is found by means of the collocation method using the Chebyshev polynomials and rational functions. The mass flow and the pressure in the channel are obtained. Isobaric and isothermal flows are investigated.

Поведение куперовской пары в потенциальном поле вихря Абрикосова

At present, there is no complete theory of superconductivity, which entails the need to solve a number of important research tasks and problems. In particular, studying the dynamic behavior of a superconducting system is of key importance for understanding the processes occurring in superconductors and for their practical application. The behavior of a Cooper pair in the potential field of an Abrikosov vortex is considered, and the process physics is qualitatively described. The system characteristic length and energy are estimated on the basis of the Heisenberg uncertainty and the quasi classical method. The theoretical and experimental coherence lengths for cuprate superconductors are compared with each other, and good agreement of the estimates is achieved for many cuprate superconductors. The obtained estimate of the coherence length indicates the physics of the process, according to which the Cooper pairs in the vortex superconducting region can be considered in some approximation as a gas of particles. The obtained forms of the coherence length estimates resemble the expression for the molecule effective diameter if the London length is regarded as the molecule mean free path, and the coherence length as the effective diameter. The Schrodinger equation for the Cooper pair is analyzed, and the dependences of the wave function of a particle in the vortex superconducting and non-superconducting regions are derived. Solutions are obtained for two cases in which the particle total energy is more or less than its potential energy. In the second case, it can be seen that the wave function at the vortex center does not equal zero, but has a finite value, which points to possible quantization of the Cooper pair energy in the Abrikosov vortex inner region. The obtained results can be used for developing a complete theory of superconductivity and can serve as the starting point for developing a theory of superconducting system dynamics, which will entail, in particular, a description of the current-voltage characteristics of superconductors of the second kind.

Super and subcritical nonlinear nonlocal analysis of NSGT nanotubes conveying nanofluid

An advanced nonlinear continuum model is presented to analyse the super and subcritical nonlinear behaviour of nanotubes. The nanoscale system is used to convey fluid flow at nanoscale levels. Due to the restrictions of one-parameter size-dependent models, a more comprehensive nonlinear coupled model containing two different size parameters is introduced using the nonlocal strain gradient theory (NSGT). Both axial and transverse inertial terms are taken into consideration, leading to more accurate results for nanotubes conveying fluid. In addition, since the mean free path of molecules is not negligible compared to the diameter of the tube at nanoscales, the Beskok–Karniadakis approach is implemented. The NSGT, Galerkin’s technique and continuation method are finally employed to derive, discretise and solve the coupled nonlinear equations, respectively. The frequency–amplitude response, modal interactions and the possibility of energy transfer between modes are examined in both supercritical and subcritical flow regimes.

Numerical Simulation of the Chemical Combination and Dissociation Reactions of Neutral Particles in a Rarefied Plasma Arc Jet

The expansion of neutral particles in a plasma arc jet is crucial for the distribution of the ions and electrons, especially in an unsteady rarefied plasma arc jet with chemical reactions. A 3-D unsteady investigation of neutral particles in a rarefied flow with chemical combination and dissociation reactions is numerically simulated based on an in-house direct simulation Monte Carlo (DSMC) code. The evolution of the neutral particles flow in vacuum cylinders is presented, and the influence of the chemical reactions has been investigated for the neutral particles. The predicted results imply that the dissociation reaction plays a key role in the expansion of the neutral particles process. In order to study the expansion of the neutral particles in an electric field, an electrostatic particle-in-cell (PIC) and DSMC are combined to simulate the axisymmetric rarefied plasma flows with chemical reactions. Two sets of grids are employed for the DSMC/PIC method by considering the different requirements of both the methods based on the molecule mean free path and the Debye length. The properties of both the flow and electric fields are analyzed in detail. It is found that the electric potential increases if the initial velocity of the ions from the inlet is sufficiently large, and accordingly, the number density of the ions in the flow field increases further.

The use of the finite element method for calculating the thermophoresis velocity of large aerosol particles

The finite-element method has been employed to calculate the photophoresis velocity of solid aerosol particles, the sizes of which are much larger than the mean free path of molecules in a gas. The thermal electromagnetic radiation from the particle surface and the temperature dependences of the density, viscosity, and thermal conductivity of the gaseous medium and particle material have been taken into account. The photophoresis velocity has been numerically calculated for a number of axially symmetric particles moving along their rotation axes. Cylindrical particles, particles having a shape resulting from rhomb rotation around one of its diagonals, and spheroidal particles have been considered.

Analysis of Gas Molecule Mean Free Path and Gaseous Thermal Conductivity in Confined Nanoporous Structures

This study comprehensively analyzes the mean free path of gas molecules and gaseous thermal conductivity in confined nanoporous structures through a wide range of temperatures and pressures. A simplified unit cell cubic array structure of nanospheres is used to correlate microstructure features with specific surface area and density of nanoporous materials. Zeng’s model is used to describe the mean free path of the gas molecules and the gaseous thermal conductivity in confined nanoporous structures, and experimental gaseous thermal conductivity data from the literature is used to validate the model. The results show that a material’s nanoporous structure features are directly related to specific surface area and density. The mean free path of gas molecules in a confined nanoporous structure decreases with increasing specific surface area and density. Thus, nanoporous materials with a relatively high specific surface area and a higher density are more favorable for confining gaseous thermal conductivity in nanopores. This work shows that \(p=10^{4}\hbox { Pa}\) and \(10^{6}\hbox { Pa}\) are two characteristic pressures at ambient temperatures for the investigated silica aerogel materials. When \(p 10^{6}\hbox { Pa}\), the limiting effect of the nanoporous structure on the movement of gas molecules can be ignored, and so the mean free path of gas molecules in the nanoporous material approaches the mean free path of gas molecules in free space, while the gaseous thermal conductivity approaches the gaseous thermal conductivity in free space. As temperature increases, there exists a maximum value for gaseous thermal conductivity in confined nanoporous materials, but this maximum increases as pressure increases. The maximum gaseous thermal conductivity for the material is also determined in the paper.

Condensation Coefficient: Definitions, Estimations, Modern Experimental and Calculation Data

A brief analysis of different approaches to the calculation and measurement of the condensation coefficient of a vapor is presented. It is shown that, on frequent occasions, calculations give values of this coefficient that are at variance with the corresponding experimental data and that the condensation coefficient is determined most exactly on the basis of the molecular-kinetic theory. It was established that the spread in the literature data on the measured values of this coefficient is explained mainly by the fact that these values were obtained not in the immediate vicinity from the boundary between the gas and liquid phases but at a large distance (as compared to the mean free path of molecules) from it. Results of calculations of the condensation coefficient of argon by the method of moleculardynamic simulation are presented.

A Fokker–Planck based kinetic model for diatomic rarefied gas flows

A Fokker–Planck based kinetic model is presented here, which also accounts for internal energy modes characteristic for diatomic gas molecules. The model is based on a Fokker–Planck approximation of the Boltzmann equation for monatomic molecules, whereas phenomenological principles were employed for the derivation. It is shown that the model honors the equipartition theorem in equilibrium and fulfills the Landau–Teller relaxation equations for internal degrees of freedom. The objective behind this approximate kinetic model is accuracy at reasonably low computational cost. This can be achieved due to the fact that the resulting stochastic differential equations are continuous in time; therefore, no collisions between the simulated particles have to be calculated. Besides, because of the devised energy conserving time integration scheme, it is not required to resolve the collisional scales, i.e., the mean collision time and the mean free path of molecules. This, of course, gives rise to much more efficient simulations with respect to other particle methods, especially the conventional direct simulation Monte Carlo (DSMC), for small and moderate Knudsen numbers. To examine the new approach, first the computational cost of the model was compared with respect to DSMC, where significant speed up could be obtained for small Knudsen numbers. Second, the structure of a high Mach shock (in nitrogen) was studied, and the good performance of the model for such out of equilibrium conditions could be demonstrated. At last, a hypersonic flow of nitrogen over a wedge was studied, where good agreement with respect to DSMC (with level to level transition model) for vibrational and translational temperatures is shown.

THERMAL ESCAPE IN THE HYDRODYNAMIC REGIME: RECONSIDERATION OF PARKER's ISENTROPIC THEORY BASED ON RESULTS OF KINETIC SIMULATIONS

The one-dimensional steady-state problem of thermal escape from a single-component atmosphere of mon- and diatomic gases is studied in the hydrodynamic (blow-off) regime using the direct simulation Monte Carlo method for an evaporative-type condition at the lower boundary. The simulations are performed for various depths into an atmosphere, indicated by a Knudsen number, Kn 0, equal to the ratio of the mean free path of molecules to the radial position of the source surface, ranging from 10 to 10–5, and for the range of the source Jeans parameter, λ0, equal to the ratio of gravitational and thermal energies, specific to blow-off. The results of kinetic simulations are compared with the isentropic model (IM) and the Navier-Stokes model. It is shown that the IM can be simplified if formulated in terms of the local Mach number and Jeans parameter. The simulations predict that at Kn 0 < ~ 10–3 the flow includes a near-surface non-equilibrium Knudsen layer, a zone where the flow can be well approximated by the IM, and a rarefied far field. The corresponding IM solutions, however, only approach Parker's critical solution as λ0 approaches the upper limit for blow-off. The IM alone is not capable for predicting the flow and requires boundary conditions at the top of the Knudsen layer. For small Kn 0, the scaled escape rate and energy loss rate are found to be independent of λ0. The simulation results can be scaled to any single-component atmosphere exhibiting blow-off if the external heating above the lower boundary is negligible, in particular, to sublimation-driven atmospheres of Kuiper belt objects.

Anisotropic mean free path in simulations of fluids traveling down a density gradient

Hard sphere molecular dynamics simulations are used to study the mean free path of molecules traveling down a density gradient at fluid densities ranging between 0.05sigma(-3) and 0.7sigma(-3). Gradients are developed using semipermeable boundaries in the x-direction and, as a result, a net flow develops in the positive x-direction. Over the course of the simulation, the free paths of colliding molecules are calculated and it was determined that the mean free path in the positive x-direction is greater than the mean free path in the negative x-direction at each density studied. These results are compared to the mean free paths in the positive and negative y- and z-directions (in which there is no net flow) and the distribution of free paths for molecules traveling in the positive and negative x-directions gives insight into the physics of the system. In addition, the dependency of the mean free path on speed is studied and compared to kinetic theory predictions. The results have application in the modification of the classical model of diffusion for low density systems undergoing flow in which the mean free path is finite, large, and can be anisotropic.

Ostwald Ripening in Rarefied Systems

Mass exchange between spherical molecular clusters is analyzed in the case when the mean free path of molecules in the intercluster space is much larger than the cluster sizes but much smaller than average intercluster separation. It is shown that there is a steady-state regime when big clusters grow at the expense of the smaller ones (Ostwald ripening), which is characterized by a one-parametric family of self-similar cluster size distribution functions and by an exponentially growing average cluster size. The obtained self-similar cluster size distributions are entirely different from those given by the classical theory of Ostwald ripening. Implications of the obtained results are discussed.

A history and state-of-the-art of accommodation coefficients

This paper reviews theory and measurements of transport processes between small particles and the surrounding gas. Evaporation and condensation coefficients and gas uptake coefficients are of particular interest. There has long been a great difference in coefficients reported by different experimentalists, and much of this disagreement is considered in this overview. A brief review of the kinetic theory of gases is provided to describe molecular transport to or from a surface when the mean free path of molecules, ℓ, is large compared with the particle dimensions, that is, when the Knudsen number is large. For a sphere of radius a the Knudsen number, Kn = ℓ / a. The condition Kn >> 1 is called the free molecule regime, and for Kn << 1 continuum theory applies. It is shown that accommodation coefficients cannot be determined by experiments operating in the continuum regime. At intermediate Knudsen numbers (the Knudsen regime) transport theory is more difficult, but results based on solution of the Boltzmann equation describing the evolution of the molecular velocity distribution are reviewed. With the advent of high-speed computers transport theory has been supplemented by molecular dynamics calculations, and these calculations are often at odds with experimental measurements of accommodation coefficients. Examples are provided. Experimental methods surveyed include Knudsen cell methods, jet tensimetry, electrodynamic levitation experiments, expansion cloud chamber measurements, and vibrating orifice aerosol generator (VOAG) techniques. VOAG measurements are particularly useful for studying processes over small times, and theory and results related to VOAG experiments are presented. More recent experimental measurements of condensation coefficients involving the use of molecular beams are reported. There is a growing body of evidence that accommodation coefficients are of order unity in many cases, but coefficients smaller than 0.01 are still reported, particularly for uptake coefficients.

Hydrodynamics of a superheated steam vacuum fluidized bed

Hydrodynamics of a superheated steam vacuum fluidized bed was experimentally studied. In these experiments, eight different types of large particles (1970–7430 μm) were used. In all cases, a behavior similar to that found in an air fluidized bed was observed. The minimum fluidization velocity was found to be increasing with decreasing operating pressure. In the case of employing superheated steam, the minimum fluidization conditions are established at a lower velocity than using air as the fluidizing medium. These tendencies are attributed to the variation of the mean free path of molecules. On the other hand, the experiments showed that the bed voidage in the minimum fluidization conditions is almost insensitive to the variation of the operating pressure. Several equations were developed to predict the minimum fluidization velocity. The values provided by these equations were compared with the experimental data as well as with the predictions of the correlations presented in the technical literature.

Were planetesimals formed by dust accretion in the solar nebula?

The growth of meter-sized bodies in the solar nebula by dust accretion is examined. The meter-sized bodies have velocity about 50 m/s relative to the gas and small dust aggregates. When a small dust aggregate hits a meter-sized body, the aggregate breaks into dust monomers. These monomers accrete onto the body after several bouncing as proposed by Wurm et al., Icarus (2001), if the mean free path of the gas molecules is larger than the radius of the body. On the other hand, the monomers never hit the surface of the body again, if the body is much larger than the mean free path of the molecules. The sizes of bodies would be limited to the order of 10 times the mean free path. Kilometer-sized planetesimals were hardly formed by dust accretion in the region within 5 AU from the sun where the mean free path is less than 1 m. The planetesimals were probably formed by the gravitational instabilities in this region.