Showing papers on "Knudsen number published in 2000"

Gravity in a lattice Boltzmann model

Abstract: In this paper we consider the introduction of a body force, in the incompressible limit, into the lattice Boltzmann model. A number of methods are considered and their suitability to our objectives determined. When there is no density variation across the fluid, gravity can be introduced in the form of an altered pressure gradient. This method correctly satisfies the Navier-Stokes equation; however, if there is a non-negligible density variation present (produced by the body force or otherwise) this method becomes less accurate as the density variation increases and the constant density approximation becomes less valid. Three other methods are also considered for application when there is a non-negligible density variation. The equations of motion satisfied by these models are found up to second order in the Knudsen number and it is seen that only one of these methods satisfies the true Navier-Stokes equation. Numerical simulations are performed to compare the different models and to assess the range of application of each.

Drag and Torque on Clusters of N Arbitrary Spheres at Low Reynolds Number

Abstract: Hydrodynamics of particle clusters suspended in viscous fluids is a subject of considerable theoretical and practical importance. Using a multipole expansion of the flow velocity in a series of spherical harmonics, Lamb's fundamental solution of the Stokes flow outside a single sphere is generalized in this work to the case of N nonoverlapping spheres of arbitrary size with slip boundary conditions. The expansion coefficients are found by transforming the boundary conditions to the Lamb form and by transforming the spherical coordinates and solid spherical harmonics centered at different spheres. The problem is reduced to the solution of the linear system of equations for the expansion coefficients, which is carried out numerically. Based on the developed theory, the relation between the hydrodynamic and gyration radius of fractal-like aggregates with different structure is established. In another application, an asymptotic slip-regime dependence of the aggregate hydrodynamic radius on the Knudsen number and the number of particles is found by performing calculations of drag forces acting on the gas-borne fractal-like and straight chain aggregates. A good agreement is shown in comparing predictions of the described theory with available experimental and theoretical results on motion of various small sphere clusters in viscous fluid.

Solving the kinetic equation for all Knudsen numbers

Abstract: Starting from the Boltzmann kinetic equation, a system is constructed which allows one to calculate flow fields with strong deviation from equilibrium and for Knudsen numbers 0

Molecular dynamics of flows in the Knudsen regime

Abstract: Novel technological applications often involve fluid flows in the Knudsen regime in which the mean free path is comparable to the system size. We use molecular dynamics simulations to study the transition between the dilute gas and the dense fluid regimes as the fluid density is increased.

Finite Element Analysis of the Spiral Groove Gas Face Seal at the Slow Speed and the Low Pressure Conditions — Slip Flow Consideration

Abstract: A Finite Element model for the noncontacting gas face seal is developed based on the modified Reynolds Equation developed by Fukui and Kaneko (4), (5) that considers the slip flow effects. Numerical studies of a representative spiral groove seal at the slow speed (≤, 500 rpm) and the low pressure (≤ .303 MPa) conditions showed that slip flow can significantly affect the seal performances such as the lift-off speed, leakage rate, load carrying capacities. Without the consideration of the slip flow effect, the lift-off speed and the corresponding leakage rate would be greatly underestimated, especially at near ambient pressure condition. By examining the F-h characteristic curves, it was found that under the parameters presented in the present study the slip flow could be significant for Knudsen number, Kn as small as .05, and the slip flow in effect reduces the viscous pumping resulting in a loss of load carrying capacities. Presented at the 55th Annual Meeting Nashville, Tennessee May 7–11, 2000

Effects of rarefaction and compressibility of gaseous flow in microchannel using dsmc

Abstract: The direct simulation Monte Carlo (DSMC) is performed for two-dimensional gaseous flow through a microchannel in both slip and transition regimes to understand the effects of compressibility and rarefaction. Results are presented in the form of axial pressure distribution, velocity profile, local friction coefficient, and local Mach number (Ma) and are compared with the available analytical and experimental results. The effect of compressibility is examined for the inlet to outlet pressure ratios ranging from 1.38 to 4.5. Low-pressure drop simulations with Knudsen numbers (Kn) ranging from 0.03 to 0.11 are performed to identify the effect of rarefaction. It was found that compressibility makes the axial pressure variation nonlinear and enhances the local friction coefficient. On the other hand, rarefaction does not affect pressure distribution but causes the flow to slip at the wall and reduces the local friction coefficient. In addition, it was found that the locally fully developed (LFD) assumption is v...

Molecular dynamics of flows in the Knudsen regime

Abstract: Novel technological applications often involve fluid flows in the Knudsen regime in which the mean free path is comparable to the system size. We use molecular dynamics simulations to study the transition between the dilute gas and the dense fluid regimes as the fluid density is increased.

Bobylev's instability

Abstract: In 1982 Bobylev [A.V. Bobylev, Sov. Phys. Dokl. 27, 29 (1982)] made a linear stability analysis of the Burnett equations and showed that beyond a certain critical reduced wave number there exist normal modes that grow exponentially, concluding that the Burnett equations are linearly unstable. We have partially extended his analysis, originally made for Maxwellian molecules, for any interaction potential and argue that his results can be reinterpreted as to give a bound for the Knudsen number above which the Burnett equations are not valid.

Structural and Transport Properties of Alumina Porous Membranes from Process-Based and Statistical Reconstruction Techniques

Abstract: We study the structural and transport properties of two model porous membranes made by compaction of spherical monosize γ-alumina particles. A ballistic deposition process of spherical particles has been employed as a process-based representation method for accurately simulating the pore structure of the membranes. Comparison between the computed and experimental permeability values obtained in the Knudsen regime shows very good agreement for both membranes and indicates that sufficient representation of the original pore structure is achieved with the random sphere packs. In a further step, a medium with the same porosity and autocorrelation function as the sphere pack has been stochastically reconstructed. Comparison between the structural properties of the random sphere pack system (process-based model) and the stochastically reconstructed medium (statistical model) shows nearly identical correlation functions and pore chord length distributions but widely different mass chord length distributions. This is reflected to a significant difference in the prediction of a dynamic property like the Knudsen permeability by a factor of about 4. The results suggest that matching of the porosity and the two-point correlation function alone is not always adequate when pursuing an accurate representation of the structure of a porous material. In such cases, higher order statistical properties of the material contained in the chord length distribution of both pore and solid phase should be satisfied as well. It is also found that proper account of the formation process in the reconstruction of a porous material (process-based model) leads to representations of its structure more accurate than those of statistical reconstruction models.

A new analytic solution of the navier-stokes equations for microchannel flows

Abstract: In this article we present a new analytic solution of the Navier-Stokes equations for microchannel flows. The solution is based on the concept of the continuum approach using the Chapman-Enskog method, but built upon the proposal to introduce a hyperbolic tangent function of Kn number in the power series of the distribution function and slip boundary condition. The physics behind the mathematical modification are discussed. With the slip boundary condition accurate to O(tanh(Kn)), the solution of the Navier-Stokes equations is extended successfully to the transition flow regime. The analytic solutions are compared with results of DSMC in both slip flow and transition flow regimes. Satisfactory agreements on the velocity profiles and pressure distributions have been achieved. The extension of the upper Knudsen number limits of continuum approach is significant in molecular gas dynamics.

Review of pumping by thermal molecular pressure

Abstract: Pumping energy is supplied by temperature changes alone. A general feature of such pumps is that the upper pressure limit is reached when the mean free path becomes small relative to the physical dimensions of the pump in the region of the temperature transition. Thus, the upper pressure limits of these pumps have been determined by the microfabrication limits of the day; they have operated at relatively low pressures, with low throughputs, and have not become main line pumps. In recent years, however, Micro-Electronic-Mechanical Systems (MEMS) has introduced a whole new level of miniaturization to devices in general, including vacuum devices, and hence has raised the upper pressure limits, and thus the throughputs of thermal molecular pumps to atmospheric levels. The purpose of this article is to isolate the various physical manifestations of thermal molecular pumps, which have been realized over the years. The general pumping phenomenon has had various names: Knudsen compressor; thermal transpiration; thermal creep; thermodynamic, thermolecular, thermal molecular, and accommodation pumping. This multiplicity of terminology can cause some confusion and it is one of the aims of the article to simplify the situation. We have chosen the title “Pumping by Thermal Molecular Pressure” following the terminology of Knudsen. Broadly speaking, it is found that these pumps divide into two classes: (a) those requiring no specially prepared surfaces, (b) those in which special surfaces are essential. The latter have no low pressure limit. A table is assembled comparing pumps which have been built and tested, rather than those calculated on paper. Scaling rules for multiple stage pumps in series, based on results obtained for single-stage pumps are presented. A Knudsen compressor in series with an accommodation pump already promises operation from atmosphere to indefinitely low pressures, whereas an accommodation pump alone may be able to cover this range in the future. A number of potential applications of the technology such as small gas chromatographs and small valves are noted. Despite this complexity, thermal molecular pressure devices all have the compelling advantage that there are no moving parts nor any fluids in the vacuum.

Some Empirical Observations on Diesel Particulate Filter Modeling and Comparison Between Simulations and Experiments

Abstract: Comparisons between 1D simulations and experiments on a mini scale SiC filter are presented. First of all, experiments with regeneration for different loading mass and soot composition enabled us to derive an improved pressure drop correlation. The assumption of constant particulate layer permeability proves unable to predict the influence of the gas temperature on the pressure drop. This discrepancy seems to be linked to the high Knudsen number of the flow in the particulate layer. A new correlation is proposed. This correlation contains four adjustable constants which have been determined on a single experimental run. Without modifying these constants, other cases have been correctly simulated. Obviously, more work is needed to substantiate this approach. In a second step, regenerations with and without additive (Cerium) for two different soot compositions have been simulated and compared with experimental results. Soluble Organic Fraction vaporization has to be taken into account to obtain the right soot mass when regeneration begins. The experimental trend is well captured by numerical simulations.

Asymptotic theory of the Boltzmann system, for a steady flow of a slightly rarefied gas with a finite Mach number: General theory

Abstract: A steady rarefied gas flow with Mach number of the order of unity around a body or bodies is considered. The general behaviour of the gas for small Knudsen numbers is studied by asymptotic analysis of the boundary-value problem of the Boltzmann equation for a general domain. The effect of gas rarefaction (or Knudsen number) is expressed as a power series of the square root of the Knudsen number of the system. A series of fluid-dynamic type equations and their associated boundary conditions that determine the component functions of the expansion of the density, flow velocity, and temperature of the gas is obtained by the analysis. The equations up to the order of the square root of the Knudsen number do not contain non-Navier–Stokes stress and heat flow, which differs from the claim by Darrozes (in Rarefied Gas Dynamics, Academic Press, New York, 1969). The contributions up to this order, except in the Knudsen layer, are included in the system of the Navier–Stokes equations and the slip boundary conditions consisting of tangential velocity slip due to the shear of flow and temperature jump due to the temperature gradient normal to the boundary.

Kinetic schemes and boundary conditions for moment equations

Abstract: A numerical scheme for moment equations of kinetic theory, due to LeTallec & Perlat, is considered for the calculation of stationary heat transfer in the Grad 13 moment system and linearized extended thermodynamics of 14 moments. It is shown that the required distance of grid points must be considerably smaller than the mean free path. Thus, the kinetic scheme is useful only in the case of large Knudsen numbers. Results of the numerical calculation for 13 and 14 moments are compared with an analytical solution for heat transfer with 13 moments. The results indicate that the boundary conditions do not guarantee conservation of energy at the walls. In order to overcome this deficiency a modification of the boundary conditions is presented and discussed.

Space-Time Discretization of Series Expansion Methods for the Boltzmann Transport Equation

Abstract: The approximate solution of the Boltzmann transport equation via Galerkin-type series expansion methods leads to a system of first order differential equations in space and time for the expansion coefficients. This system is extremely stiff close to the fluid dynamical regime (for small Knudsen numbers), and exhibits a mildly dispersive behavior, due to the acceleration of waves by the external force (the electric field). In this paper a class of difference methods is presented and analyzed which represent a generalization of the well-known Scharfetter--Gummel exponential fitting approach for the drift-diffusion equations. It is shown that, by using appropriate operator splitting methods for the time discretization, one obtains stability properties which are only mildly dependent on the Knudsen number and essentially independent of the size of the electric field.

Slip flow in an annulus with corrugated walls

Abstract: The effects of deterministic surface wavy roughness on the fluid flow inside annuli with microfabricated solid walls was investigated. The Navier-Stokes equations are solved using perturbation methods with incorporated microscopic slip conditions at the wavy wall. The volume flow rate thus obtained, considering fully-developed parallel flow cases, contains terms which are related to the Knudsen number (Kn ) of the fluid, roughness ratio ( ), phase shift ( ) and the wave number (k ) of the small wavy-roughness elements. The results show that if the phase shift is zero once Kn is ~0.1 then the increase of k will reduce the second-order volume flow rate significantly. Meanwhile, there is a critical k for the Kn = 0.1 case, in which increases and then this critical k increases.

Saturated Vapor Pressure of Iridium(III) Acetylacetonate

Abstract: The temperature dependency of the saturated vapor pressure of Ir(acac)3 has been measured by the method of calibrated volume (MCV), the Knudsen method, the flow transpiration method, and the membrane method. The thermodynamic parameters of phase transition of a crystal to gas were calculated using each of these methods, and the following values of ΔH T 0 (kJ mol−1) and ΔS T 0 (J mol−1K−1), respectively, were obtained: MCV: 101.59, 156.70; Knudsen: 130.54, 224.40; Flow transpiration: 129.34, 212.23; Membrane: 95.45, 149.44 Coprocessing of obtaining data (MCV, flow transportation method and Knudsen method) at temperature ranges 110−200°C as also conducted:ΔH T 0 =127.9±2.1 (kJ mol−1 ); ΔS T 0 =215.2±5.0 (J mol−1 K−1 ).

Monte Carlo analysis of macroscopic fluctuations in a rarefied hypersonic flow around a cylinder

Abstract: From consideration of the length scales characteristic of molecular and turbulent phenomena, it is proposed that flow instabilities and structural motions should be generated under certain rarefied, hypersonic flow conditions. This proposal is investigated using the direct simulation Monte Carlo (DSMC) method for hypersonic, rarefied flow over a cylinder. The overall mean flow field contains a number of regions including an undisturbed free stream, a bow shock, a recompression in the wake, and a recirculation zone behind the cylinder. Analysis of the fluctuations in velocity and number density predicted by the DSMC technique for this flow is conducted. An important aspect of this analysis is the clarification of the nature of the observed fluctuations. It is found that different types of fluctuations are found in the bow shock and in the wake. These are attributed to different types of physical phenomena. In the bow shock, large fluctuations are caused by the macroscopic field gradients in the shock front. In the wake, one can also observe an area with enhanced scatter of the mean flow characteristics caused by a macroscopic structural motion. The calculated instantaneous flow fields show small oscillations of the vortices immediately downstream of the cylinder for a Mach number of 26 and a Knudsen number of 0.0025. When the Knudsen number is reduced to 0.001, an irregular secondary flow motion containing vortex structures is exhibited in the far wake.

Solving the Boltzmann equation in the case of passing to the hydrodynamic flow regime

Abstract: As the hydrodynamic regime is approached, the gas flow is usually accompanied by the formation of narrow highly nonequilibrium zones (Knudsen layers) with the characteristic size on the order of the mean free path λ for molecules. The structure of these zones is determined by fast kinetic processes. In unsteady flows, an initial layer with the time scale on the order of the mean free time τ0 = λ/vT (here, vT is the molecular thermal velocity) occurs as well. In the macroscopic scale l0 @ λ, the flow parameters vary smoothly beyond these zones [1]. From a computational standpoint, solving the Boltzmann equation with steps hx < λ and τ < τ0 is inefficient everywhere over the calculation domain. Moreover, it can result in the early cessation of the iterative process as soon as the error of the numerical method becomes equal to the small difference of two successive approximations. When passing to the macroscopic steps λ ! hx < l0, τ0 ! τ < t0 , where t0 = l0/vT , the problem of a large factor standing ahead the collision integral arises [2]. In terms of the dimensionless variables t = t*/t0, x = x*/l0, ξ = ξ*/vT (here, the asterisk denotes dimensional variables), the Boltzmann equation takes the form

Dissipative and dispersive behaviors of lattice-based models for hydrodynamics

Abstract: Both dissipation and dispersion are present in many complex systems; their interactions through nonlinearity can lead to interesting features. We investigate in this paper the dissipation-dispersion interactions that exist in lattice-based kinetic models for hydrodynamics. The classical Chapman-Enskog expansion is used to derive the dispersion coefficients at third order of Knudsen number. Unlike the dissipation coefficient (viscosity) that is always positive, the dispersion coefficient can be either positive or negative. It would be interesting to know if there is any other physics in these models as compared with the traditional dispersionless Navier-Stokes dynamics. Traveling wave solutions in one dimension are studied and two different solutions have been found: (1) monotonic shock solutions and (2) oscillatory shock solutions, according to different conditions. In two- and three-dimensional systems, whether or not these oscillatory behaviors caused by the interactions between nonlinearity, dissipation, and dispersion have anything to do with vortex cascades (direct or inverse) would be an interesting question and we leave it for future studies. (c) 2000 The American Physical Society.

Thermodynamic study of the gaseous molecules Al2N, AlN, and Al2N2 by Knudsen cell mass spectrometry

Abstract: The Knudsen effusion mass spectrometric method has been employed to measure the equilibrium partial pressures of the Al2N molecule over the AlN–Au–graphite system. Theoretical computations were carried out to determine the structure, molecular parameters, and thermodynamic properties of Al2N. The partial pressures have been combined with the calculated thermal functions to determine the atomization enthalpy, ΔaH0o, and enthalpy of formation, ΔfH298.15o, in kJ mol−1, of 783.2±15 and 342.7±15 for Al2N, respectively. Upper values for the dissociation energy of AlN, D0o(AlN,g)⩽368±15 kJ mol−1, and for the atomization enthalpy of Al2N2, ΔaH0o(Al2N2,g)⩽1402 kJ mol−1 have been obtained. These results are discussed and compared with recent theoretical literature values.

Gas Flow Characteristics in Microtubes.

Abstract: Microchannel is one of the essential components that construct various micro systems. However, it has been reported that the flow and heat transfer behavior in microchannel deviates from predictions based on the conventional assumptions generally accepted in macro scale. In this study, frictional characteristics of nitrogen (N2), argon (Ar) and helium (He) flowing through microtubes whose diameter ranges from 5 to 100 μm have been investigated experimentally. Inlet / outlet pressure difference and volumetric flow rate were measured. In the range of Reynolds number (Re=0.03∼29.7) tested in this study, the measured friction constant was observed to take the values around 50, which is about 20% less than 64, the value regarded to be correct for macro scale tube predicted by the incompressible flow assumption. Transition from incompressible to compressible flow regimes was observed experimentally. The onset of compressibility effect was dependent on the inlet / outlet pressure difference (or the pressure ratio) as well as on the Mach number. The frictional resistance of nitrogen flow showed a Knudsen number dependence, which is in rough agreement with the first-order slip model.