Entire Range Of Knudsen Numbers Research Articles

Simulation study of drag force characteristics of nanoparticles in transition regime

Transport properties of nanoparticles in gases have many practical applications, such as aerosol science, combustion, and micro- and nano-scale fabrication. A nanoparticle moving in a fluid is expected to experience a drag force, which determines the transport property of the particle. According to the Einstein relationship, the diffusion coefficient of a particle is inversely proportional to the drag force coefficient. However, in the transition regime, it is usually difficult to evaluate the drag force of suspended particles. A typical method is to extend the asymptotic solution of the free molecular or continuum limit to the transition regime. According to the gas kinetic theory, Li and Wang proposed a theoretical expression for drag force on nanoparticles in the free molecular regime, which is then extended to the entire range of Knudsen number following a semi-empirical approach [Li Z G, Wang H 2003 <i>Phys. Rev. E</i> <b>68</b> 061207]. For nanoparticles, it is necessary to verify the theoretical predictions since the gas-particle non-rigid-body interactions must be taken into account. In this work, the drag force on nanoparticle in the transition regime is investigated by using molecular dynamics (MD) simulation. To evaluate the drag force, a harmonic potential is used to the nanoparticle to constrain its Brownian motion in our MD simulation. In the steady state, the drag force can be obtained by the balance between the drag force and harmonic force. It is found that the gas-particle non-rigid-body interaction has a significant influence on the drag force of nanoparticle. For weak gas-solid coupling, the MD simulation results can be in good agreement with the prediction of Li-Wang theory. However, for strong coupling, there exists significant discrepancy between the MD simulation results and the theoretical results. Due to the gas-solid intermolecular interactions, gas molecules can be adsorbed on the nanoparticle surface, and after a time period, they may be re-emitted from the surface when they gain sufficient kinetic energy. Therefore, an adsorption-desorption equilibrium and an adsorption layer can be established on the particle surface. The adsorption layer enlarges the collision cross-sectional area and enhances the momentum transfer between gas molecules and the particle, and thus the drag force increases. This can explain the inconsistencies between the theoretical results and MD simulations. In this work, we introduce an adsorption ratio to evaluate the thickness of the adsorption layer. Then, the effective particle radius can be defined by the sum of particle radius and the thickness of the adsorption layer. By using the effective particle radius, the simulation values are in very good agreement with the theoretical predictions. The results of this work provide insights into the applications of nanoparticles in aerosol science.

Effect of relative humidity on the nozzle performance in non-equilibrium condensing flows for improving the compressed air energy storage technology

Compressed air energy storage (CAES) is now becoming the focus of a lot of research as it offers broad application possibilities and enormous economic benefits in terms of environmental protection and life with low-carbon emissions. The compressor performance is closely related to energy storage efficiency in CAES. Considering the significant role of water components inside the flow of the compressor, the flow mechanism in transonic with non-equilibrium condensation should be explored in detail. First, a blend condensation model is presented with the effect of the entire Knudsen number range on droplet growth, to explore condensation flow characteristics more clearly. Moreover, the influence of relative humidity (RH) on the nucleation and droplet growth and condensation shock wave is analysed with the blend model. Finally, flow loss is calculated in the de Laval nozzle, and the direction to optimize the compressor efficiency is indicated by considering the RH conditions and wall curvature arcs. The contribution of these factors to the flow loss deserves careful consideration. The analysis conducted in the paper shows the effect of relative humidity on the nozzle, and the conclusions obtained provide meaningful reference for improving the performance of compressors used in the CAES process in the future.

Beyond size classification: The dynamic-aerosol-size electrical mobility spectrometer

This paper overviews an alternative operation mode of the Electrical Mobility Spectrometer (EMS), herein termed as the “Dynamic-aerosol-size Electrical Mobility Spectrometer (DEMS)” where the size of aerosol particles evolves during mobility classification in response to any chemical/physical processes taking place within the device's classification region. Unlike the conventional EMS designed to classify the aerosol particle size, the function of the DEMS is to probe the kinetics (in terms of corresponding growth/shrinkage rates) of aerosol related chemical/physical processes. This is achievable because the particle trajectory in the device's classification region contains full information about the particle size change history. A theoretical framework is developed for analyzing the transmission of size-changing particles in the device's classification region. Expressions for the particle mobility at the device outlet and size change rate are derived based on known initial mobility and DEMS response, using the evaporation of monodisperse and single-component aerosols as an example. The theoretical framework is applicable to the entire Knudsen number range and is compared against simulations and experiments, demonstrating excellent validity in the test conditions.

New Model for Liquid Evaporation and Vapor Transport in Nanopores Covering the Entire Knudsen Regime and Arbitrary Pore Length.

Liquid evaporation and the associated vapor transport in micro/nanopores are ubiquitous in nature and play an important role in industrial applications. Accurate modeling of the liquid evaporation process in nanopores is critical to achieving a better design of devices for enhanced evaporation. Although having high impact on evaporation rate, vapor transport resistance in micro/nanopores remains incompletely understood. In this study, we proposed a new model which, for the first time, considered vapor transport in finite-length pores under various Knudsen regimes and then coupled the transport resistance to liquid evaporation. Direct Simulation Monte Carlo and laboratory experiments were conducted to provide validation for our model. The model successfully predicts the variation of pore transmissivity with Knudsen number and nanopore size, which cannot be revealed by prior theories. The relative error of model-predicted evaporation rate was within 1% in L/r = 0 cases and within 3.5% in L/r > 0 cases. Our model is featured by its applicability under the entire range of Knudsen numbers. The evaporation of various types of liquids in arbitrarily sized pores can be modeled using a universal relation.

An extended thermal Lattice Boltzmann model for transition flow

In the last two decades, mathematical models based on the Lattice Boltzmann method have been developed in an effort to accurately capture flow characteristics in rarefied conditions, including thermal effects. In order to improve their performance to characterize fluid conditions for the entire transition flow regime, a new wall-distance function has been developed based on existing Molecular Dynamics data and implemented to a D2Q13 Lattice Boltzmann model. The ability of the improved Lattice Boltzmann model to capture thermal effects on fluid conditions due to heat conduction and convection is demonstrated by simulating rarefied Fourier flow and Couette flow within micro/nano-channels, respectively, and validated against Direct Simulation Monte Carlo data as well as data based on the linearized Boltzmann equation. The results show that the proposed Lattice Boltzmann model is capable of simulating conditions, including thermal effects, in rarefied flow for the entire Knudsen number range representing slip, transition, and the low end of molecular flow regimes.

Transient heat transfer flow through a binary gaseous mixture confined between coaxial cylinders

The unsteady transient behaviour of both single and binary mixture rarefied gas confined between two coaxial cylinders at rest caused by a sudden change of the temperature of one of the surfaces is studied numerically by using Direct Simulation Monte Carlo. For single gas flow, the results are compared to the numerical solution of the Navier–Stokes–Fourier equations. The study covers the response of the system from the initial moments, when the change in the temperature is introduced, up to the point where the system reaches the steady state for different degrees of gas rarefaction within the entire range of Knudsen number, various values of the temperature difference, from linear to strongly non-linear conditions, and several radius ratios. Numerical results are reported for two binary mixtures (Ne–Ar and He–Xe) with various mole fractions.

Determination of the Scalar Friction Factor for Nonspherical Particles and Aggregates Across the Entire Knudsen Number Range by Direct Simulation Monte Carlo (DSMC)

The friction factor of an aerosol particle depends upon the Knudsen number (Kn), as gas molecule–particle momentum transfer occurs in the transition regime. For spheres, the friction factor can be calculated using the Stokes–Millikan equation (with the slip correction factor). However, a suitable friction factor relationship remains sought-after for nonspherical particles. We use direct simulation Monte Carlo (DSMC) to evaluate an algebraic expression for the transition regime friction factor that is intended for application to arbitrarily shaped particles. The tested friction factor expression is derived from dimensional analysis and is analogous to Dahneke's adjusted sphere expression. In applying this expression to nonspherical objects, we argue for the use of two previously developed drag approximations in the continuum (Kn → 0) and free molecular (Kn → ∞) regimes: the Hubbard–Douglas approximation and the projected area (PA) approximation, respectively. These approximations lead to two calculable geometric parameters for any particle: the Smoluchowski radius, R S, and the projected area, PA. Dimensional analysis reveals that Kn should be calculated with PA/πR S as the normalizing length scale, and with Kn defined in this manner, traditional relationships for the slip correction factor should apply for arbitrarily shaped particles. Furthermore, with this expression, Kn-dependent parameters, such as the dynamic shape factor, are readily calculable for nonspherical objects. DSMC calculations of the orientationally averaged drag on spheres and test aggregates (dimers, and open and dense 20-mers) in the range Kn = 0.05–10 provide strong support for the proposed method for friction factor calculation in the transition regime. Experimental measurements of the drag on aggregates composed of 2–5 primary particles further agree well with DSMC results, with differences of less than 10% typically between theoretical predictions, numerical calculations, and experimental measurements. Copyright 2012 American Association for Aerosol Research

The Mobility of Fractal Aggregates: A Review

A review of the experimental and theoretical literature describing the mobility of fractal aggregates over the previous three decades is presented. Aggregates are those formed via both diffusion and reaction limited cluster-cluster aggregation processes, DLCA and RLCA, which form aggregates with fractal dimensions and prefactors of ca. 1.78 and 1.3 and 2.1 and 0.94, respectively. Emphasis is placed on DLCA aggregates. The entire Knudsen number range from continuum to free molecular is reviewed. The review finds a simple and general consensus description of mobility for the entire range.

Phonon Transport Across Mesoscopic Constrictions

Phonon transport across constrictions formed by a nanowire or a nanoparticle on a substrate is studied by a numerical solution of the gray Boltzmann transport equation (BTE) resolving the effects of two length scales that govern problems of practical importance. Predictions of total thermal resistance for wire/substrate and particle/substrate combinations are made for the entire range of Knudsen number, with an emphasis on resolving transport in the mesoscopic regime where ballistic-diffusive mechanisms operate and analytical expressions are not available. The relative magnitudes of bulk and constriction resistance are established, and a correlation for overall thermal resistance spanning the range of practical Knudsen numbers is provided.

The direct simulation of acoustics on Earth, Mars, and Titan

With the recent success of the Huygens lander on Titan, a moon of Saturn, there has been renewed interest in further exploring the acoustic environments of the other planets in the solar system. The direct simulation Monte Carlo (DSMC) method is used here for modeling sound propagation in the atmospheres of Earth, Mars, and Titan at a variety of altitudes above the surface. DSMC is a particle method that describes gas dynamics through direct physical modeling of particle motions and collisions. The validity of DSMC for the entire range of Knudsen numbers (Kn), where Kn is defined as the mean free path divided by the wavelength, allows for the exploration of sound propagation in planetary environments for all values of Kn. DSMC results at a variety of altitudes on Earth, Mars, and Titan including the details of nonlinearity, absorption, dispersion, and molecular relaxation in gas mixtures are given for a wide range of Kn showing agreement with various continuum theories at low Kn and deviation from continuum theory at high Kn. Despite large computation time and memory requirements, DSMC is the method best suited to study high altitude effects or where continuum theory is not valid.

The propagation of sound on Titan using the direct simulation Monte Carlo

With the recent success of the Huygens lander on Titan, a moon of Saturn, there has been renewed interest in exploring the acoustic environment of the only moon in the solar system with a significant atmosphere. The direct simulation Monte Carlo (DSMC) method is used here for modeling sound propagation in the Titan atmosphere. DSMC is a particle method that describes gas dynamics through direct physical modeling of particle motions and collisions. DSMC is based on the kinetic theory of gas dynamics, where representative particles are followed as they move and collide with other particles. The validity of DSMC for the entire range of Knudsen numbers (Kn), where Kn is defined as the mean free path divided by the wavelength, allows for the exploration of sound propagation in the Titan atmosphere for all values of Kn. DSMC results for the absorption of sound have shown that sound absorption depends heavily on Kn and deviates significantly from the continuum classical assumption for large Kn. This talk presents the successful application of the DSMC method to acoustic waves in the Titan atmosphere for a large range of Kn. [Work supported by NASA.]

The absorption of sound on Mars using the direct simulation Monte Carlo

The physical properties that govern the absorption of sound on Mars are very similar to those on Earth: classical losses associated with the transfer of acoustic energy into heat, and relaxation losses associated with the redistribution of internal energy of molecules. The difference in molecular composition between Earth and Mars as well as the lower atmospheric pressure on Mars results in larger values for the absorption coefficient on Mars. The direct simulation Monte Carlo (DSMC) method is the simulation tool used for modeling sound propagation in the Martian atmosphere. DSMC describes gas dynamics through direct physical modeling of particle motions and collisions. DSMC is based on the kinetic theory of gas dynamics, where representative particles are followed as they move and collide with other particles. The validity of DSMC for the entire range of Knudsen numbers (Kn), where Kn is defined as the mean free path divided by the wavelength, allows for the exploration of sound propagation in the Martian atmosphere for all values of Kn. Successful application of the DSMC method to acoustic waves in the Martian atmosphere, including the details of molecular relaxation in gas mixtures, will be shown. [Work supported by NASA.]

Heat conduction from a spherical nano-particle: status of modeling heat conduction in laser-induced incandescence

Laser-induced incandescence (LII) of nano-second pulsed laser heated nano-particles has been developed into a popular technique for characterizing concentration and size of particles suspended in a gas and continues to draw increased research attention. Heat conduction is in general the dominant particle cooling mechanism after the laser pulse. Accurate calculation of the particle cooling rate is essential for accurate analysis of LII experimental data. Modelling of particle conduction heat loss has often been flawed. This paper attempts to provide a comprehensive review of the heat conduction modelling practice in the LII literature and an overview of the physics of heat conduction loss from a single spherical particle in the entire range of Knudsen number with emphasis on the transition regime. Various transition regime models developed in the literature are discussed with their accuracy evaluated against direct simulation Monte Carlo results under different particle-to-gas temperature ratios. The importance of accounting for the variation of the thermal properties of the surrounding gas between the gas temperature and the particle temperature is demonstrated. Effects of using these heat conduction models on the inferred particle diameter or the thermal accommodation coefficient are also evaluated. The popular McCoy and Cha model is extensively discussed and evaluated. Based on its superior accuracy in the entire transition regime and even under large particle-to-gas temperature ratios, the Fuchs boundary-sphere model is recommended for modeling particle heat conduction cooling in LII applications.

Entropic lattice Boltzmann method for microflows

A new method for the computation of flows at the micrometer scale is presented. It is based on the recently introduced minimal entropic kinetic models. Both the thermal and isothermal families of minimal models are presented, and the simplest isothermal entropic lattice Bhatnagar–Gross–Krook (ELBGK) is studied in detail in order to quantify its relevance for microflow simulations. ELBGK is equipped with boundary conditions which are derived from molecular models (diffusive wall). A map of three-dimensional kinetic equations onto two-dimensional models is established which enables two-dimensional simulations of quasi-two-dimensional flows. The ELBGK model is studied extensively in the simulation of the two-dimensional Poiseuille channel flow. Results are compared to known analytical and numerical studies of this flow in the setting of the Bhatnagar–Gross–Krook model. The ELBGK is in quantitative agreement with analytical results in the domain of weak rarefaction (characterized by Knudsen number Kn , the ratio of mean free path to the hydrodynamic scale), up to Kn ∼ 0.01 , which is the domain of many practical microflows. Moreover, the results qualitatively agree throughout the entire Knudsen number range, demonstrating Knudsen's minimum for the mass flow rate at moderate values of Kn , as well as the logarithmic scaling at large Kn . The present results indicate that ELBM can complement or even replace computationally expensive microscopic simulation techniques such as kinetic Monte Carlo and/or molecular dynamics for low Mach and low Knudsen number hydrodynamics pertinent to microflows.

Nonlinear acoustic simulations using direct simulation Monte Carlo

In the simulation of fluid dynamics, one can either treat the fluid as a continuum or as discrete particles. Although popular for acoustics, the continuum model is limited to small Knudsen numbers (the ratio of mean free path to a length scale). Particle methods are necessary for, but not limited to, problems with Knudsen numbers greater than 0.1, which can occur in shockwaves, microdevices, high frequency sound or rarefied gases. Some well known particle methods include Monte Carlo, cellular automata, discrete velocity, lattice Boltzmann, and molecular dynamics. The direct simulation Monte Carlo (DSMC) method describes gas flows through direct physical modeling of particle motions and collisions. DSMC can model problems for the entire range of Knudsen numbers. In particular, DSMC is capable of simulating nonlinear acoustics, as well as the details of viscous dissipation, dispersion, nonequilibrium effects, and other physical properties. A DSMC method has been implemented for one-dimensional nonlinear acoustics problems on parallel computers using object-oriented C++ and the message passing interface (MPI). DSMC results will be shown and compared with continuum theory and continuum simulations.

The Self-Preserving Size Distribution Theory: I. Effects of the Knudsen Number on Aerosol Agglomerate Growth

Gas-phase synthesis of fine solid particles leads to fractal-like structures whose transport and light scattering properties differ from those of their spherical counterparts. Self-preserving size distribution theory provides a useful methodology for analyzing the asymptotic behavior of such systems. Apparent inconsistencies in previous treatments of the self-preserving size distributions in the free molecule regime are resolved. Integro-differential equations for fractal-like particles in the continuum and near continuum regimes are derived and used to calculate the self-preserving and quasi-self-preserving size distributions for agglomerates formed by Brownian coagulation. The results for the limiting case (the continuum regime) were compared with the results of other authors. For these cases the finite difference method was in good in agreement with previous calculations in the continuum regime. A new analysis of aerosol agglomeration for the entire Knudsen number range was developed and compared with a monodisperse model; Higher agglomeration rates were found for lower fractal dimensions, as expected from previous studies. Effects of fractal dimension, pressure, volume loading and temperature on agglomerate growth were investigated. The agglomeration rate can be reduced by decreasing volumetric loading or by increasing the pressure. In laminar flow, an increase in pressure can be used to control particle growth and polydispersity. For Df=2, an increase in pressure from 1 to 4 bar reduces the collision radius by about 30%. Varying the temperature has a much smaller effect on agglomerate coagulation.

Rarefied gas flow in a channel with specular-diffuse boundary conditions for arbitrary Knudsen numbers. Onsager relations

The problem of a rarefied gas flow in a channel for arbitrary Knudsen numbers has been solved analytically for the first time in the case where the scattering of gas molecules on the channel walls can be described by specular‐diffuse boundary conditions. The mean free path of gas molecules is assumed to be constant, i.e., the collision frequency is proportional to molecular velocity. The gas moves under the action of a streamwise temperature gradient. Exact relations for heat and mass fluxes and for mean‐mass velocity are obtained. It is shown that the Onsager relations are valid within the entire range of Knudsen numbers in the problem of heat and mass transfer in a channel. The dependence of heat and mass fluxes on the Knudsen number (channel thickness) is analyzed. A comparison with available results is performed.

Diffusive mobility of fractal aggregates over the entire Knudsen number range.

We determine the effective mobility radius for fractal aggregate particles. Our method is to use static light scattering to measure the radius of gyration R(g) of the aggregates, and dynamic light scattering to measure the diffusion coefficient hence the mobility radius R(m). The range of our results can be specified by the Knudsen number Kn, which is the mean free path of the medium molecules divided by the radius of the aggregate. Our results apply to the entire range of Kn from the continuum limit (Kn=0) to the free molecular limit (Kn>>1). In the continuum regime we find R(m)/R(g)=0.97+/-0.05 when the aggregate fractal dimension is D(f) approximately 2.15, and 0.70+/-0.05 when D(f) approximately 1.75. The latter result is independent of Kn for Kn < or approximately 1.3. The free molecular mobility goes as R(m)=aN(0.44+/-0.03), where a is the monomer radius and N is the number of monomers per aggregate. Since R(g) approximately aN(1/D(f)), R(m)/R(g) is not a constant when Kn is large. We find for all Kn that the functionality of R(m)/R(g) must always begin with the correct N-->1 limit, and this affects experimental observation.

Simulation of vapor diffusion in anisotropic particulate deposits

Abstract In this study we determine the effective diffusivity tensor of computer-generated, particulate, anisotropic deposits. Deposits of polydispersed, spherical particles were generated by several algorithmic models, which simulate different deposition conditions. Depending on the nature of the arriving particle trajectories (diffusive/deterministic), the angle of incidence and the subsequent motion after they establish contact with an already deposited particle, both the deposit anisotropy and final solid fraction can be varied. Monte Carlo simulations of vapor molecules diffusing through cubic samples extracted from the generated deposits provided accurate estimates of their effective diffusivity tensor and associated tortuosities. The effective diffusivities were directly determined over the entire Knudsen number range, assuming self-diffusion in a constant pressure pure gas that consists of hard-sphere elastic molecules with exponentially distributed free-paths. In the continuum limit the effective diffusivities of the porous solids considered here could be closely correlated with porosity alone. Archie's power law [eq. (17)] with an exponent m = 1.45 fitted our computational results accurately. In the Knudsen limit on the other hand the details of the local microstructure became important and so an accurate prediction/correlation of the effective diffusivity required higher order information. An interpolation based on the experimentally measured effective diffusivities in the two Knudsen limits (Scott and Dullien, 1962) was used to correlate the diffusivities in the transition regime. For the deposits analyzed here the diffusivity properties became anisotropic only in the high Knudsen regime, that is, when diffusing molecule/solid interactions dominate. In the Knudsen limit we have also found significant “rectification” effects. Specifically, the transmission probability as determined by the test-molecule method could be up to 25% higher when the molecules diffused away from the target than when they diffused towards the target. Such “rectification” effects are attributed to the tree-like microstructures of the deposits.

A unified approach to plasma-particle heat transfer under non-continuum and non-equilibrium conditions

A model for the heat transfer from a stationary, infinite plasma to a particle is developed so as to include rarefaction effects over the entire range of Knudsen numbers, and thermal and chemical nonequilibrium in the vicinity of the particle surface. The overall heat transfer rate is obtained by separately determining the contributions due to heavy species conduction and to energy transport by ion flow to the particle surface. Knudsen effects are accounted for by means of Sherman's interpolation formula for the conduction problem and its equivalent in electrostatic probe theory for the ion transport to the surface. Results are presented for argon plasmas in the pressure range 0.1–1 atm, and for particle sizes in the range 10–100 μm.