Particle Knudsen Number Research Articles

A general drag coefficient model for a spherical particle incorporating rarefaction and particle-to-gas temperature ratio effects

A concept and model of two critical Reynolds numbers Rep,cr1 and Rep,cr2 corresponding respectively to onsets of drag crisis and recovery are proposed. A drag model at limits of zero particle Mach and Knudsen numbers is constructed. On this basis, we develop a general drag coefficient model applicable for a spherical particle traveling in a gas by using a large number of available data derived from the previous experiments, direct numerical simulations and direct simulation Monte-Carlo methods. The scope of applicability of the proposed drag model covers all flow regimes relative to the particle characterized by particle Reynolds and Mach (or Knudsen) numbers and different particle-to-gas temperature ratios. Its comparison with two latest general drag models shows the significantly smaller relative error. Furthermore, quasi-one dimensional simulations against two supersonic nozzle gas-particle flow experiments are conducted with an in-house code to validate its accuracy in comparison with the two drag models.

Drag coefficient for micron-sized particle in high-speed flows

The drag force on the small particle in high-speed flows is influenced by the combined effects of fluid viscosity, compressibility, and rarefaction. The existing drag coefficient models are still insufficient in accuracy and efficiency for gas-particle flow simulation. This study comprehensively considers these effects and conducts high-fidelity numerical simulations. A new drag coefficient is generated using a symbolic regression method reasonably based on the particle Mach number, Reynolds number, and Knudsen number, which are related to particle diameter, gas-particle relative velocity, and other parameters. The new drag coefficient possesses clear physical significance, high predictive accuracy, low computational cost, and consistency with theory in limiting conditions. The application of the new drag coefficient to three typical gas-solid two-phase flow cases demonstrated its excellent performance.

TEMPus VoLA: The timed Epstein multi-pressure vessel at low accelerations.

The field of planetary system formation relies extensively on our understanding of the aerodynamic interaction between gas and dust in protoplanetary disks. Of particular importance are the mechanisms triggering fluid instabilities and clumping of dust particles into aggregates, and their subsequent inclusion into planetesimals. We introduce the timed Epstein multi-pressure vessel at low accelerations, which is an experimental apparatus for the study of particle dynamics and rarefied gas under micro-gravity conditions. This facility contains three experiments dedicated to studying aerodynamic processes: (i) the development of pressure gradients due to collective particle-gas interaction, (ii) the drag coefficients of dust aggregates with variable particle-gas velocity, and (iii) the effect of dust on the profile of a shear flow and resultant onset of turbulence. The approach is innovative with respect to previous experiments because we access an untouched parameter space in terms of dust particle packing fraction, and Knudsen, Stokes, and Reynolds numbers. The mechanisms investigated are also relevant for our understanding of the emission of dust from active surfaces, such as cometary nuclei, and new experimental data will help interpreting previous datasets (Rosetta) and prepare future spacecraft observations (Comet Interceptor). We report on the performance of the experiments, which has been tested over the course of multiple flight campaigns. The project is now ready to benefit from additional flight campaigns, to cover a wide parameter space. The outcome will be a comprehensive framework to test models and numerical recipes for studying collective dust particle aerodynamics under space-like conditions.

A specific slip length model for the Maxwell slip boundary conditions in the Navier–Stokes solution of flow around a microparticle in the no-slip and slip flow regimes

Abstract In the case of microscopic particles, the momentum exchange between the particle and the gas flow starts to deviate from the standard macroscopic particle case, i.e. the no-slip case, with slip flow occurring in the case of low to moderate particle Knudsen numbers. In order to derive new drag force models that are valid also in the slip flow regime for the case of non-spherical particles of arbitrary shapes using computational fluid dynamics, the no-slip conditions at the particle surface have to be modified in order to account for the velocity slip at the surface, mostly in the form of the Maxwell’s slip model. To allow a continuous transition in the boundary condition at the wall from the no-slip case to the slip cases for various Knudsen (Kn) number value flow regimes, a novel specific slip length model for the use with the Maxwell boundary conditions is proposed. The model is derived based on the data from the published experimental studies on spherical microparticle drag force correlations and Cunningham-based slip correction factors at standard conditions and uses a detailed CFD study on microparticle fluid dynamics to determine the correct values of the specific slip length at selected Kn number conditions. The obtained data on specific slip length are correlated using a polynomial function, resulting in the specific slip length model for the no-slip and slip flow regimes that can be applied to arbitrary convex particle shapes. Graphic abstract

Application of the “Rock'n’Roll” (RNR) resuspension model with non-zero slip velocities on particle and substrate surfaces

Studies on the flow-induced particle resuspension have great significance for engineering applications of many areas, and several established resuspension models have been developed for the analysis of particle resuspension in an atmospheric environment. However, for flow with a non-negligible particle Knudsen number (the ratio between the mean free path of gas and the particle radius), which is commonly expected for vacuum systems and space applications, the potentially substantial influence of flow slip on solid surfaces on flow-induced particle resuspension is hardly accounted for in the existing published works. Therefore, in the present study, flow slips on the surfaces of particle and substrate are taken into consideration in the calculation of flow-induced forces, which are further integrated into the established “rock'n’roll” (RNR) resuspension model to predict the resuspension rates of micron and sub-micron particles in a viscous flow of rarefied gas, and the results are compared with available experimental measurements for validation. It is shown that flow slip on the surfaces of particle and substrate considerably affects the removal force and the corresponding resuspension rates of mono-layer particle deposition, and the predictions on particle resuspension rates given by the present slip-modified RNR (SRNR) model exhibit noticeably better agreement with the experimental measurements.

Mass, momentum, and energy transfer in supersonic aerosol deposition processes

Aerosol deposition (AD) can lead to the formation of dense coatings via deposition of particles from a gas flow; in AD the aerosol is passed through a converging-diverging nozzle, facilitating inertial particle impaction on a desired substrate at supersonic particle velocities. Unique from thermal spray methods, AD can be applied near room temperature and unique from cold spray, in AD the aerosol is typically at atmospheric pressure upstream of the nozzle. Though AD has been successfully demonstrated previously, a number of aspects related to particle motion in AD systems remain poorly understood. In this work, we simulated compressible flow field profiles and particle trajectories for typical AD working conditions for a slit type converging-diverging nozzle with a planar substrate. In examining the fluid flow profile, we show that the velocity and pressure profiles, as well as the shock structure are sensitive to the upstream and downstream operating pressures of the nozzle. These ultimately affect particle impaction speed. Importantly, in AD, the particle drag regime is dynamic; both particle Knudsen numbers and Mach numbers can vary by orders of magnitude. To aid particle trajectory simulations, we trained a neural network to predict the drag force on the particles based on existing experimental data, theoretical limits, and new direct simulation Monte Carlo (DMSC) results. The neural network based drag law, which depends upon both Mach and Knudsen numbers, shows better agreement with the DSMC simulation data than pre-existing correlations. With it, particle trajectory simulation results reveal that for a given particle density, there exists an optimal particle diameter to maximize particle impaction speed. We also find that in AD particles undergo size dependent inertial focusing, i.e. there is a particular particle diameter where the particle deposition linewidth is minimized. Particles smaller than this diameter are underfocused, and particles larger than this are overfocused, and hence have larger deposition linewidths in both cases. Using trajectory simulations, we additionally developed a framework that can be used to evaluate the position-dependent mass, momentum and kinetic energy fluxes to the deposition substrate for any aerosol size distribution function upstream of the nozzle. It is shown for typical aerosol concentrations achievable in the laboratory, the kinetic energy flux can approach a magnitude normally observed in convective heat transfer with phase change, hence translational kinetic energy to thermal energy transfer in AD is likely a key contributor to the formation of dense coatings.

Hydrodynamic interactions between aerosol particles in the transition regime

We present a method for calculating the hydrodynamic interactions between particles in the kinetic (or transition regime), characterized by non-negligible particle Knudsen numbers. Such particles are often present in aerosol systems. The method is based on our extended Kirkwood–Riseman theory (Corson et al., Phys. Rev. E, vol. 95 (1), 2017c, 013103), which accounts for interactions between spheres using the velocity field around a translating sphere as a function of Knudsen number. Results for the two-sphere problem at small Knudsen numbers are in good agreement with those obtained using Felderhof’s interaction actions for mixed slip-stick boundary conditions, which are accurate to order $r^{-7}$ (Felderhof, Physica A, vol. 89 (2), 1977, pp. 373–384). The strength of the interactions decreases with increasing Knudsen number. Results for two fractal aggregates demonstrate that one can apply a point force approach for interactions between particles in the transition regime; the interaction tensor is similar to the Oseen tensor for continuum flow. Using this point force approach, we present an analysis for the settling of an unbounded cloud of particles. Our analysis shows that for sufficiently high volume fractions and cloud radii, the cloud behaves as a gas droplet in continuum flow even when the individual particles are small relative to the mean free path of the gas. The method presented here can be applied in a Brownian dynamics simulation analogous to Stokesian dynamics to study the behaviour of a dense aerosol system.

The electrical mobilities and scalar friction factors of modest-to-high aspect ratio particles in the transition regime

In this work, we compared measured electrical mobilities of spherical and non-spherical nanoparticles to the predictions of both transition regime and free molecular theories. Mobility measurements were performed using a differential mobility analyzer (DMA) with both air and carbon dioxide as background gases, at atmospheric pressure and room temperature. For all measurements, DMA operational settings were selected in an attempt to minimize particle alignment with the electric field or flow direction, such that particles traversed the DMA rotating close to randomly. The particles included gold nanospheres (diameters of 50nm and 70nm), polystyrene latex dimers (primary particle diameter of 116nm), gold nanorods (mobility equivalent diameters of 40–70nm; aspect ratio 1.8–14) and carbon nanotubes (mobility equivalent diameters of 48.3nm –60.3nm; aspect ratios 9.5–243). Our primary objective was to test the accuracy of a transition regime theory (Dahneke’s adjusted sphere model) for non-spherical particles; this theory suggests that electrical mobilities are dependent upon a particle Knudsen number, specifically defined as Kn=λπRH/PA, where λ is the background gas mean free path, RH is the particle’s hydrodynamic radius, and PA is its projected area. For this test, electrical mobilities were calculated from the dimensions of particles collected downstream of a DMA and observed via TEM, with the DMA operated with fixed voltage and sheath flowrate. Mobility calculations for each particle were used to reconstruct electrical mobility histograms, which we show are directly comparable to the DMA transfer function for the operating conditions selected. We further compared the mean electrical mobilities of electrical mobility ranked sub-populations to the transfer function weighted mean mobilities of “windows” smaller in width than the DMA transfer function. Through analysis we find that in the 0.7<Kn<6 range, measurements are explained well by the proposed transition regime expression (the adjusted sphere model), with transfer function weighted and sub-population mean electrical mobilities typically within 10% of one another. Electrical mobilities calculated using solely free molecular models were in reasonable agreement with observations (±20%) for Kn>3, but were systematically lower (by up to 50%) than measurements for 1<Kn<3. Based upon the presented measurements and analysis, we propose that (1) an appropriate Knudsen number-dependent friction factor/electrical mobility expression needs to be used in interpreting DMA measurements not only for spheres (as is common practice), but also non-spherical nanorods and nanotubes, and (2) the expression tested here is suitable for such measurement interpretation.

Heat transfer effects on particle motion under rarefied conditions

Gas–solid flows occurring on very small spatial scales (of the order of micro and nanometres) are of great relevance in a number of industrial applications. It is currently not well established how particle motion and filtration are affected by non-isothermal conditions at such scales. Furthermore, when the particle size is comparable to the mean free path of the gas, rarefaction effects become important. In the present work we investigate the effects of heat transfer and non-isothermal conditions on the motion of small particles in rarefied flow. For that purpose, a suitable framework is developed here as a generic multiphase DNS method for rarefied flows. The resulting model is valid for low particle Reynolds number flows, irrespective of the Biot number, and for particle Knudsen numbers up to unity in unbounded flow. Using this model, we show that there is different settling behaviour of particles with an internal heat source in rarefied and continuum cases of the carrier gas respectively. It is shown that the chances for thermal levitation and/or lifting up of a particle due to buoyancy effects are significantly reduced under rarefied conditions.

Advanced continuum modelling of gas-particle flows beyond the hydrodynamic limit

The accurate prediction of dilute gas-particle flows using Euler–Euler models is challenging because particle–particle collisions are usually not dominant in such flows. In other words, in dilute flows the particle Knudsen number is not small enough to justify a Chapman–Enskog expansion about the collision-dominated near-equilibrium limit. Moreover, due to the fluid drag and inelastic collisions, the granular temperature in gas-particle flows is often small compared to the mean particle kinetic energy, implying that the particle-phase Mach number can be very large. In analogy to rarefied gas flows, it is thus not surprising that two-fluid models fail for gas-particle flows with moderate Knudsen and Mach numbers. In this work, a third-order quadrature-based moment method, valid for arbitrary Knudsen number, coupled with a fluid solver has been applied to simulate dilute gas-particle flow in a vertical channel with particle-phase volume fractions between 0.0001 and 0.01. In order to isolate the instabilities that arise due to fluid-particle coupling, a fluid mass flow rate that ensures that turbulence would not develop in a single phase flow (Re = 1380) is employed. Results are compared with the predictions of a two-fluid model with standard kinetic theory based closures for the particle phase. The effect of the particle-phase volume fraction on flow instabilities leading to particle segregation is investigated, and differences with respect to the two-fluid model predictions are examined. The influence of the discretization on the solution of both models is investigated using three different grid resolutions. Radial profiles of phase velocities and particle concentration are shown for the case with an average particle volume fraction of 0.01, showing the flow is in the core-annular regime.

Quadrature-Based Moment Model for Moderately Dense Polydisperse Gas−Particle Flows

A quadrature-based moment model is derived for moderately dense polydisperse gas-particle flows starting from the inelastic Boltzmann-Enskog kinetic equation including terms for particle acceleration (e.g., gravity and fluid drag). The derivation is carried out for the joint number density function, f(t,x,m,u), of particle mass and velocity, and thus, the model can describe the transport of polydisperse particles with size and density differences. The transport equations for the integer moments of the velocity distribution function are derived in exact form for all values of the coefficient of restitution for particle-particle collisions. For particular limiting cases, the moment model is shown to be consistent with hydrodynamic models for gas-particle flows. However, the moment model is more general than the hydrodynamic models because its derivation does not require that the particle Knudsen number (and Mach number) be small.

Two-way coupling model for shock-induced laminar boundary-layer flows of a dusty gas

The present paper describes a numerical two-way coupling model for shock-induced laminar boundary-layer flows of a dust-laden gas and studies the transverse migration of fine particles under the action of Saffman lift force. The governing equations are formulated in the dilute two-phase continuum framework with consideration of the finiteness of the particle Reynolds and Knudsen numbers. The full Lagrangian method is explored for calculating the dispersed-phase flow fields (including the number density of particles) in the regions of intersecting particle trajectories. The computation results show a significant reaction of the particles on the two-phase boundary-layer structure when the mass loading ratio of particles takes finite values.

Dust-cloud structures behind a shock wave moving over a deposited layer of fine particles*

Abstract The present paper investigates dispersed-phase flow structures of a dust cloud induced by a normal shock wave moving a constant speed over a flat surface deposited with fine particles. In the shock-fitted coordinates, the general equations of dusty-gas boundary layer flows are formulated within the framework of a multi-fluid model and parametic numerical studies of the carrier- and dispersed-phase flow field are performed. The problem associated with crossing particle trajectories and the formation of local particle accumulation regions are solved by using the full Lagrangian method for the dispersed phase. The basic features of the near-wall two-phase flow under consideration including the role of saffman force in the particle entrainment and the development of discontinuities or singularities in the particle density profiles are discussed. The effects associated with account of the non-uniformity of particle size and the finiteness of the particle knudsen numbers are studied in detail

Comment on "Thermophoresis of a Near-Wall Particle at Great Knudsen Numbers"

An approximate expression for the thermophoretic force on a near-wall particle at large particle Knudsen number is presented. The wall has arbitrary specularity. The derivation of this expression follows that of Chen and Xu (2002) but corrects two errors in their approach. The corrected expression predicts that the near-wall thermophoretic force is only 5-10% greater than the bulk-gas (Waldmann) value and that the greatest difference occurs for a specular wall, whereas the original expression predicts that the force is increased up to 35% and that the greatest difference occurs for a diffuse wall. Independent molecular gas dynamics simulations confirm the corrected expression.

Thermophoretic deposition of soot aerosol particles under experimental conditions relevant for modern diesel engine exhaust gas systems

A plate-to-plate thermal precipitation system has been used to investigate the deposition of agglomerate submicrometer soot aerosol particles under flow and temperature conditions relevant for modern heavy-duty diesel engine exhaust gas systems. Over a flow channel length of 45 cm , thermophoretic particle deposition efficiencies of 4–95% have been observed as a function of plate distance (0.45– 1.45 mm ), temperature gradient (10 4– 10 5 K m −1) , and flow velocity (2– 11 m s −1 ). Throughout the investigated particle size range of 34– 300 nm (electrical mobility diameter), the observed deposition efficiencies are nearly independent of particle size and Knudsen number. The experimental results are in good agreement with theoretical calculations assuming a constant thermophoretic coefficient K th=0.55 which had been derived by Waldmann and Schmitt (in C. N. Davies (Ed.), Aerosol Science, London, 1966) for the free molecular regime ( Kn⪢1). Our results demonstrate that K th≈0.55 is also applicable for agglomerate soot particles in the transition regime ( Kn≈1), which is supported by a recent theoretical study by Rosner and Khalil (J. Aerosol Sci. 31 (2000) 273) on the effect of reduced thermal conductivity on the thermophoretic coefficient of agglomerate particles. To our knowledge, the measurement data presented in this paper provide the first direct experimental validation for the theory of Rosner and Khalil under conditions relevant for diesel engine exhaust gas treatment. Moreover, they confirm the applicability of a simple formula of approximation for the thermophoretic deposition efficiency derived by Tsai and Lu (Aerosol Sci. Technol. 22 (1995) 172).

Estimation of Thermophoresis Acting on a Particle in Rarefied Gas by Use of DSMC Method. 2nd Report. Effect of the Position and Thermal Conductivity of the Particle in a Finite Space.

Thermophoretic force acting on a particle in a finite space is numerically estimated from the point of view of the position and thermal conductivity of the particle. Simulations are performed by Direct Simulation Monte Carlo Method in which the force is calculated from momentum transports of molecules on the particle surface provided that particle Knudsen number is in the free-molecule region. It is confirmed that the thermophoretic force becomes higher toward lower temperature region in the intermediate region of space Knudsen number 0.01<KnH<10. The profiles of the force are nonlinear in the slip region 0.01<KnHl0.1but are linear in the transition region 0.1<KnHl10. The force gradient becomes small as KnH increases and the force is almost constant in the free-molecule region. The effect of the particle conductivity on the force becomes larger as KnHincreases but the effect is very small compared with a radiation effect.

Hydrodynamic interactions between two equal spheres in a highly rarefied gas

This paper describes hydrodynamic interactions between two spherical particles having equal radii, a, and translating with velocities U1 and U2 in a highly rarefied gas. The center-to-center distance between the two spheres is aχ. The gas is at rest far from the two particles. The spheres move with speeds that are much smaller than the mean thermal speed of the gas molecules so that the Mach number, M≡max(U1,U2)/c̄, characterizing the deviation from equilibrium is much less than one. Here c̄ is the mean thermal speed of the gas molecules. Gas molecules are assumed to be diffusively reflected from the particle surfaces. Our analysis is confined to the case where the particle Knudsen number is very large, i.e., Kno≡λo/a→∞, λo being the mean free path of the gas far from the two particles. We first study the free-molecular drag on the two sphere configuration for arbitrary translations of the spheres. For small Mach number, the general time-dependent, nonlinear problem may be approximated by a quasisteady, linear problem in which the spheres are held fixed and molecules reflected from each sphere have a modified Maxwell–Boltzmann distribution of velocities. A standard integral equation formulation based on flux balances at the particle surfaces is then employed to calculate the drag force acting on the spheres. The results obtained can be used as leading estimates for the forces acting on the spheres when Kno≫1 and 2⩽χ≪Kno. We then consider the case where the flow in the vicinity of each sphere is nearly free-molecular, but the flow in the O(aχ) space between the spheres is nearly continuum in nature. In this limit, the flow in the gap between the spheres is studied using the method of reflections. This approach can be used for arbitrary Kno provided Kno≪χ≪Kno M−1. The leading correction to the drag force due to the hydrodynamic interactions between the spheres when Kno≫1 is obtained. In all cases studied, the temperature of the two spheres is assumed to be the same as that of the surrounding gas.

Measurement of the thermophoretic force by electrodynamic levitation: Microspheres in air

The thermophoretic force on single microspheres has been measured over a wide range of Knudsen numbers and particle thermal conductivities for solid and liquid spheres in air. The microspheres were dioctyl phthalate (DOP) droplets, metallic nickel, polystyrene latex (PSL), and glass. This covers a thermal conductivity ratio (gas to particle) range of 2 × 10 −4−0.17. In a companion paper, results are reported for monatomic (helium) and polyatomic gases (carbon dioxide and air) to examine the effects of gas properties. The measurements were accomplished by levitating the particle between heated and cooled plates mounted in a vacuum chamber. Light-scattering phase functions were used to determine the size of all materials except the nickel spheres, and for the highly absorbing metallic spheres the size was obtained by measuring the marginal stability limit of the particle. The apparatus developed for the research and the experimental techniques are discussed, and the experimental results in the Knudsen number range 0.05–20 are compared with previously proposed theories. For particle Knudsen numbers greater than about 10, the effects of the temperature jumps (thermal slip) at the heated and cooled plates become significant. The data in the transition regime overlap Loyalka's (1992, J. Aerosol Sci. 23, 291–300) solution of the linearized Boltzmann equation for hard-sphere molecules, and the data in the slip regime agree with Brock's (1962, J. Colloid Sci. 17, 768–780) theory. The effects of particle thermal conductivity are shown to be nearly negligible in the Knudsen regime, for the data for particles having greatly different thermal conductivities overlap.

Slip Correction Measurements for Solid Spherical Particles by Modulated Dynamic Light Scattering

The theoretical expression for the drag force on a spherical particle moving with low Reynolds number in a gas is customarily written by multiplying the Stokes' law expression by a slip correction factor of the form C x = 1 +Kn[γ1 + γ2 exp(-γ3/Kn)], where Kn is the particle Knudsen number and γ1, γ2, and γ3 are empirical constants. We have measured the drag forces on spherical polystyrene latex particles suspended in dry air using the modulated dynamic light scattering technique which is fundamentally different from the Millikan cell approach. Data are time autocorrelation functions of the intensity of light scattered by single test particles from the intersection volume of two coherent laser beams. The data provide detailed information about test particle Brownian motion including the value of the particle diffusion coefficient. Each test particle was held in air suspension by an electrostatic trap to permit measurements to be made on the same particle at air pressures ranging from 760 to 0.2 torr. Data ...

Correction for sampling errors due to coagulation and wall loss in laminar and turbulent tube flow: direct solution of canonical ‘inverse’ problem for log-normal size distributions

Aerosol sampling from industrial environments (e.g. combustion engines) or natural environments (e.g. the troposphere) frequently involves conveying the sample to a downstream (‘sheltered’) instrument via an upstream tube or duct. While the instrument may be capable of characterizing, say, the particle size distribution (PSD) of the aerosol actually presented to it, the investigator is, of course, usually more interested in the PSD of the aerosol entering the upstream sampling tube. Invariably, this differs from that measured because of several systematic phenomena—perhaps the two most obvious of which are particle size-dependent losses to the tube walls (i.e. incomplete ‘penetration’) and PSD distortion due to suspended particle-particle coagulation when the particle concentrations are sufficiently high. We show here how recent research on the use of ‘moment methods’ to predict the effects of size-dependent walls loss and/or Brownian coagulation in flow systems can now be brought to bear to conveniently solve this ‘inverse’ problem by numerically integrating the quasi-one-dimensional coupled moment equations in the upstream direction, using downstream (measured) aerosol properties in the definitions of all dimensionless dependent variables and parameters. Illustrative ‘universal’ graphs are presented here for the sampling of log-normally distributed ‘inertialess’ (Brownian) aerosols in long straight adiabatic ducts for both commonly encountered extremes of particle Knudsen number Kn p ⪢ 1(free molecule) or Kn p ⪡ 1 (continuum), as well as convenient rational approximations derived from the leading terms of a Taylor series expansion of the above-mentioned dimensionless moment equations.