Free-molecule Regime Research Articles

Direct numerical simulation of supersonic nanoparticles flow in free-molecule regime using the angular coefficient method

We employ molecular flow methods to numerically simulate the supersonic nanoparticles flow in free-molecule regime. To streamline the computational complexity, interaction forces between the gas and solid particles are disregarded. We first develop a discrete phase model (DPM) method that integrates the non-rigid body collision model, enabling an accurate simulation of nanoparticle diffusion under the influence of the drag force and Brownian motion force. The nanoparticles considered in this study have sizes below 10 nm, and the accuracy of the DPM method is verified by comparing its results with experimental data. Subsequently, we theoretically and numerically investigate the transmission probability and number density of N2 molecules flowing through two-dimensional (2D) channels and three-dimensional (3D) tubes by using the angular coefficient (AC) method and the direct simulation Monte Carlo (DSMC) method. The findings indicate that as the diameter of the nanoparticle (dp) decreases to 1 nm, the diffusion coefficient (D) and the root mean square displacement (x) of nanoparticles approach the N2 molecules. The microscopic velocity of most N2 molecules falls within the range of 62–1400 m/s, and the macroscopic velocity of N2 flow falls within the range of Ma = 1.28–1.35. In contrast to the DSMC method, the AC method exhibits enhanced accuracy even with a reduced number of grids and obviates the process for large-scale sampling. Additionally, the solution time required by the AC method is approximately 1/10 and 1/13–1/32 of the DSMC method in 3D cylindrical tubes and 2D channels, respectively. Moreover, the AC method demonstrates superior adaptability when dealing with complex geometries.

Analysis of turbulent coagulation in a jet with discretised population balance and DNS

The objective of the present study is to investigate turbulence–coagulation interaction via direct numerical simulation (DNS) coupled with the population balance equation (PBE). Coagulation is an important process in several environmental and engineering applications involving turbulent flow, including soot formation, gas-phase synthesis of nanoparticles and atmospheric processes, but its interaction with turbulence is not yet fully understood. Particle dynamics can be described by the PBE, whose Reynolds decomposition leads to unclosed terms involving correlations of number density fluctuations. In this work, we employ a discretisation (sectional) method for the solution of the PBE, which is free ofa prioriassumptions regarding the particle size distribution (PSD), and couple it with a DNS for the flow field in order to study the behaviour and significance of the unknown correlations. At present, it is not feasible to resolve the Batchelor scales that result from diffusion at high Schmidt number, hence a unity Schmidt number is employed. The investigation is conducted on a three-dimensional planar jet laden with monodisperse nanoparticles, and coagulation in the free-molecule regime is considered. The correlations due to turbulent fluctuations of the particle number density are calculated at several points in the domain and found to be positive in most cases, except close to the jet break-up. The transport equation for the moments of the PSD is also studied, and it is found that the correlations make a considerable contribution to the time-averaged coagulation source term, up to$20\,\%$on the jet centreline and$40\,\%$close to the edges.

An Approximate Analytical Solution for the Knudsen Number-Dependent Self-Preserving Size Distributions of Coagulating Fine Particles in the Transition Regime

An approximate analytical solution for the size distribution width of fine particles coagulating in the transition regime is derived. It is assumed that the particle size distribution can be represented by a time-dependent log-normal function, which passes through a series of quasi-self-preserving states. The solution extends a previous solution presented for the near-continuum regime to cover the whole transition regime by adopting Dahneke’s collision kernel based on the flux-matching theory. Good agreement was obtained when the new solution was compared to numerical calculations. The effective degree of homogeneity of collision kernel is defined and used for a comprehensive analysis of the Knudsen number-dependent behavior of the quasi-self-preserving size distributions. Although the new solution is valid for the whole particle size range from the free-molecule regime (Knudsen number larger than ~50) through the transition regime (Knudsen number between ~1 and ~50) to the continuum regime (Knudsen number smaller than ~1), it is practically applicable in the transition regime because the self-preserving behavior in the continuum regime or in the free-molecule regime had previously been investigated and understood well.

Thermophoresis of nanoparticles hotter/colder than the surrounding dilute gases

Aerosol particles suspended in a diluted gas with non-uniform temperature distribution are expected to experience a thermophoretic force. In theoretical treatment of thermophoresis, it is usually assumed that the particle temperature is equal to the surrounding gas temperature. However, this might not always be the case. In some particular applications, the particle temperature can significantly differ from the gas temperature. In the present paper, we theoretically investigate the effect of the particle temperature on the thermophoresis of nanoparticles in the free molecule regime. Theoretical formulas for the thermophoretic force and thermophoretic velocity are obtained based on the gas kinetic theory. As examples, a spherical Ag nanoparticle suspended in a dilute He gas is considered, and the Rudyak–Krasnolutski potential is employed to model the gas–particle interaction. It is found that the influence of the particle temperature on the thermophoresis of nanoparticles can be significant. With increasing particle size, the error due to the equal gas–particle temperature assumption can be neglected.

Thermophoretic force on nanoparticles in free molecule regime

Thermophoresis refers to the motion of small particles suspending in a fluid with non-uniform temperature distribution due to the temperature gradient around the particle. Usually, the fluid molecules coming from the hot side carry more kinetic energy than those from the cold side, which results in a net thermophoretic force in the direction opposite to the temperature gradient. Since it was discovered more than 100 years ago, thermophoresis has been of major importance in a variety of applications, where it can play either beneficial role or adverse role, including material synthesis, micro- and nano-fabrication, and environmental science. Therefore, it is necessary to accurately evaluate the thermophoretic force. In the present work, the thermophoretic force on nanoparticles is examined in the free molecule regime by using non-equilibrium molecule dynamics (MD) simulation. It has been widely accepted that the thermophoretic force conforms with the Waldmann equation for large Knudsen numbers. However, due to the effect of the nonrigid-body interactions between the particle and gas molecules, the thermophoretic force on nanoparticles might deviate greatly from the classical theory. In our MD model, a single nanoparticle with a diameter of several nanometers suspends in a diluted gas. The Lennard-Jones (L-J) potential is employed to simulate the intermolecular interactions. To avoid deforming the nanoparticle, the solid molecules within the nanoparticles are linked to their nearest neighbors through a finite extensible nonlinear elastic bonding potential. The thermophoretic force on a nanoparticle is calculated by imposing a harmonic potential on the nanoparticle, which eliminates the effect of the Brownian motion of the nanoparticle on the thermophoresis. The effective thermal conductivity of the ambient gas is employed in Waldmann equation for the thermophoretic force due to the finite volume effect. It is found that the Waldmann theory for thermophoresis is still valid for nanoparticles in the case of weak gas-particle interaction or high gas temperature. With the increase of the gas-particle interaction strength or the decrease of the gas temperature, the Waldmann theory is invalid due to the effect of gas-particle nonrigid-body collisions and the adsorption of gas molecules on the particle surface. By considering the gas-particle nonrigid-body interaction and the modified particle size, the theoretical results for thermophoretic force accord with the MD simulations quite well.

Aerodynamics of Orion in Rarefied Flow Along a Mars Entry Path

It is well known that the Orion capsule was born as a rescue shuttle for the International Space Station. NASA developed the Orion project for the transportation of instruments and astronauts to moon, Mars, and beyond. Orion is similar in shape but larger than the Apollo command module; Orion will be able to host up to six astronauts, but Apollo command module could host only three astronauts. The aerodynamics of Orion during reentry to Earth in all rarefied regimes (that is, from free molecule regime to continuum) was already studied. The forthcoming use of Orion for Mars landing stimulated the author to evaluate global aerodynamic coefficients in rarefied flow along an entry path. The study considered problems both in axisymmetric and in three-dimensional flows by means of two direct simulation Monte Carlo codes. Global aerodynamic coefficients were computed at the altitudes of 80, 90, and 100 km. Special attention was paid to longitudinal stability.

Rarefied Gas Flows in Long Circular and Square Microchannels

Comprehensive reviews of rarefied gas flows in circular and square microchannels over wide ranges of rarefaction show large differences in Poiseuille numbers and reduced flow rates when identical length scales are used. The continuum Poiseuille numbers differ by 12.5%, and the free molecule reduced flow rates differ by 11%. This work demonstrates that the product of reduced flow rate and continuum Poiseuille number differs by a maximum of 1.6% over wide ranges of Knudsen numbers, and maximum differences occur at and near the Knudsen minimum, which occurs at the same rarefaction when identical length scales are used. A novel definition of dimensionless mass flow rate shows that it is related to the product of the reduced flow rate and continuum Poiseuille number, and accurate correlations are proposed for two ranges of the rarefaction from the free molecule regime to the Knudsen minimum and from the Knudsen minimum to the hydraulic regime. Models are proposed for Poiseuille–Knudsen number relations for the slip flow regime and the transition from the Knudsen minimum to the hydraulic regime. This work demonstrates that rarefied gas flows in circle and square microchannels are equivalent over wide ranges of rarefaction. The circle–square models can be used to model rarefied gas flows in regular polygonal microchannels and selected trapezoid microchannels having flow characteristics similar to flow in square microchannels.

Drag forces on nanoparticles in the free-molecule regime: Effect of the particle temperature.

In the present paper, we theoretically study the drag force on nanoparticles in the free-molecule regime. It has been widely assumed that the particle temperature is equal to the gas media temperature in the open literature. However, this assumption can be invalid in some real applications. Based on the kinetic theory, we obtain the generalized formulas for the drag force on nanoparticles in the free-molecule regime. It is found that there exists a significant error induced by the assumption of equal temperature between the particle and the surrounding gas. Therefore, it is necessary to consider the effect of the particle temperature in the analysis of the particle transport properties.

Development of a nanoscale hot-wire probe for supersonic flow applications

A new nanoscale thermal anemometry probe (NSTAP) was designed and fabricated to measure mass flux in supersonic flows. This sensor was evaluated in the Trisonic Wind Tunnel Munich (TWM) at both subsonic and supersonic speeds. Subsonic compressible flow tests were performed to confirm the new sensor’s repeatability and to compare its behaviour to measurements from a conventional cylindrical hot-wire, while supersonic tests were performed to investigate the nature of the convective heat transfer from the nanoscale sensor at those conditions. For the range of mass fluxes tested in the supersonic regime, a linear relationship between the Nusselt number and the Reynolds number fit the data well. A linear relationship has previously been noticed at length scales close to the molecular mean free path of the flow and has been attributed to the free-molecule flow regime, where the Knudsen number is on the order of unity.

The role of molecule-surface interaction in the photophoresis of an aerosol particle

The role of the interaction of gas molecules with a surface on the photophoresis of a spherical aerosol particle is studied under the conditions of a free molecule regime. An expression for the photophoretic force is obtained for an arbitrary scattering kernel. Calculations are carried out using various scattering-kernel models.

The role of molecule-surface interaction in thermophoresis of an aerosol particle

The role of the interaction of gas molecules with a surface on the drag force, thermophoretic force and velocity of a spherical aerosol particle, as well as its thermal polarization, are studied under the conditions of a free molecule regime. Expressions for these physical quantities are obtained for an arbitrary scattering kernel. Calculations are carried out using various scattering-kernel models.

An experimental study of pressure distribution nonlinearity in a circular micro-tube

This paper presents an optical absorption method for determining the magnitude of pressure distribution nonlinearity in gas flowing through a circular micro-tube, based on the technique of tunable diode laser absorption spectroscopy (TDLAS), and its application to quantify the dependence on pressure ratio and rarefaction of the nonlinearity of such pressure distributions. This technique has the advantages of straightforward deployment and non-intrusive measurement, avoiding the complex fabrication of embedded mechanical sensors inside microfluidic devices used in previous studies. A dimensionless nonlinearity quantity is defined to examine the deviation of the pressure profile from a linear profile. The nonlinearity is measured for different pressure ratios between inlet and outlet. We also operate the system in a regime where the influence of pressure ratio on the nonlinearity is constant, in order to unambiguously quantify the nonlinearity as a function of mean Knudsen number (Kn). It is found that increased rarefaction changes the pressure profile from convex to linear. Our measurements were conducted over a range of mean Kn of gas flow from 0.02 to 6.6, covering the slip flow regime (0.01 < Kn < 0.1) and the bulk of the transition flow regime (0.1 < Kn < 10), extending the lower limit of pressure compared with previous investigations of pressure profiles in micron-scale channels. Furthermore, it is shown that existing theoretical models provide differing predictions of the spatial pressure profile, and this disagreement becomes significant in the transition and free molecule regimes. In particular, except for the continuum regime and the early slip regime, the existing models have not been experimentally verified, due to the lack of available experimental data. In this paper, the existing models are examined based on our experiments from the slip regime (Kn = 0.02) to the near free molecule regime (Kn = 6.6), with detailed discussions provided. Moreover, an empirical correlation is proposed that characterizes the nonlinearity as a function of both Knudsen number and pressure ratio. With this correlation, one can predict the nonlinearity of the pressure profile at a given inlet/outlet pressure condition.

Thermophoretic force on nonspherical particles in the free-molecule regime.

The present paper is devoted to studying the thermophoresis of a nonspherical convex particle suspended in a gas with nonuniform temperature distribution in the free-molecule regime. Based on the gas kinetic theory and the assumption of a rigid-body collision for the gas-particle interaction, analytical expressions for the thermophoretic forces are obtained for several typical nonspherical bodies, including cylinders, spheroids, needles, disks, and cuboids. The orientation dependences of the thermophoretic forces and thermophoretic velocities are evaluated based on these expressions. It is found that the influence of the pitching effect of the nonspheres can be significant. The expressions for the orientation-averaged thermophoretic forces are also obtained under the assumption of a uniform particle orientation distribution.

Thermal characterization of microscale heat convection in rare-gas environment by a steady-state “hot wire” method

In this work, a steady-state “hot wire” method is developed to study thermal transport around a microwire in a rare-gas environment. The convection heat transfer coefficient (h) around a 25 µm platinum wire is determined to be from 14 W m−2 K−1 at 7 Pa to 629 W m−2 K−1 at atmospheric pressure. In the free-molecule regime, the Nusselt number is found to be inversely proportional to the Knudsen number, the slope of which is used to determine the slip length as 7.03 × 10−4 m. In the transition regime, the slip length is found to be 20 times the wire diameter. Under a certain pressure, h decreases with increasing wire temperature, and this trend is more sensitive at high pressures.

Shear lift forces on nanocylinders in the free molecule regime

In the present paper, analytical formulae for the shear lift forces on nanocylinders moving in linear shear flows in the free molecule regime are derived on the basis of the gas kinetic theory. The model takes into account the intermolecular interactions between the nanocylinders and gas molecules, i.e., the non-rigid-body effect. It is shown that the resulting formulae are consistent with the previous theory in the limit of rigid-body collisions. The lift forces acting on carbon nanotubes and long-chain$n$-alkanes are evaluated as examples. It is found that the non-rigid-body effect is of great importance for small nanocylinders at low temperatures.

A semi-analytical solution to Brownian coagulation of fractal aggregates valid for the entire particle size range

ABSTRACTBrownian coagulation is an important particle growth mechanism in a number of aerosol processes. Analytical solution to Brownian coagulation has been an interesting subject in aerosol population dynamics field because it can provide deep insight into the nature of the coagulation process. In the case of spherical particles, analytical solutions have been derived for all different particle size regimes. For fractal aggregates that can represent the morphology of many important industrial aerosol products and combustion soot particles, however, analytical solutions have been reported only for the free-molecule and continuum regimes. In this work, a semi-analytical solution to the Brownian coagulation of fractal aggregates valid for the entire particle size range is presented using the method of moments based on the assumption of log-normal particle size distribution. The new solution is validated by limiting-case analyses as well as by comparison with a more accurate numerical solution. The characteristics of the Brownian coagulation of fractal aggregates in the transition regime are discussed using the new solution.

On the growth rate of particle surface area for Brownian coagulation

The surface area of fine particles corresponds to the Gibbs free energy, and the particle number density corresponds to the entropy of particle system. In this study, the growth rate of particle surface area is given quantitatively for Brownian coagulation both in the free molecule and continuum regime based on the moment method. The results show that the relative growth rate for particle surface area is one-thirds of that for particle number density. The relationships together with the second law of thermodynamics form a kind of physical constraints for Brownian coagulation, which will be helpful to understand the population balance equation from point of view of thermodynamics.

Calculating the rotational friction coefficient of fractal aerosol particles in the transition regime using extended Kirkwood-Riseman theory.

We apply our extended Kirkwood-Riseman theory to compute the translation, rotation, and coupling friction tensors and the scalar rotational friction coefficient for an aerosol fractal aggregate in the transition flow regime. The method can be used for particles consisting of spheres in contact. Our approach considers only the linear velocity of the primary spheres in a rotating aggregate and ignores rotational and coupling interactions between spheres. We show that this simplified approach is within approximately 40% of the true value for any particle for Knudsen numbers between 0.01 and 100. The method is especially accurate (i.e., within about 5%) near the free-molecule regime, where there is little interaction between the particle and the flow field, and for particles with low fractal dimension (≲2) consisting of many spheres, where the average distance between spheres is large and translational interaction effects dominate. Our results suggest that there is a universal relationship between the rotational friction coefficient and an aggregate Knudsen number, defined as the ratio of continuum to free-molecule rotational friction coefficients.

Analytical expression for the friction coefficient of DLCA aggregates based on extended Kirkwood–Riseman theory

ABSTRACTWe use a self-consistent field method, which we have previously validated, to calculate the translational friction coefficient of fractal aerosol particles formed by diffusion-limited cluster aggregation (DLCA). Our method involves solving the Bhatnagar–Gross–Krook model for the velocity around a sphere in the transition flow regime. The velocity and drag results are then used in an extension of Kirkwood–Riseman theory to obtain the drag on the aggregate. Our results span a range of primary sphere Knudsen numbers from 0.01 to 100 for clusters with up to N = 2000 primary spheres. Calculated friction coefficients are in good agreement with experimental data and approach the correct continuum and free molecule limits for small and large Knudsen numbers, respectively. Results show that particles exhibit more continuum-like behavior as the number of primary spheres increase, even when the primary particle is in the free molecule regime; as an illustrative example, the friction coefficient for aggregate...

Thermophoretic force on nanocylinders in the free molecule regime.

We theoretically investigate the thermophoresis of nanocylinders in the free molecule regime, wherein the nonrigid-body effect is taken into account. The analytical formulas of the thermophoretic forces on cylinders suspended in both simple gases and binary gas mixtures are derived for the two limiting collision models, i.e., specular and diffuse scatterings, based on the gas kinetic theory. In the limit of the rigid-body collision, the expressions can be reduced to the theoretical results of Garcia-Ybarra and Rosner [AIChE J. 35, 139 (1989)10.1002/aic.690350115]. The thermophoretic velocities of the nanocylinders (including carbon nanotubes and long-chain n-alkanes) have been evaluated and it is found that the influence of the nonrigid-body effect on thermophoresis is significant for the case of small cylinder radius and low temperatures.