Showing papers on "Knudsen number published in 1994"

Simulation of Heat and Momentum Transfer in Complex Microgeometries

Abstract: In this article we present a time-accurate computational model based on the slip-flow theory to simulate momentum and heat transport phenomena in complex microgeometries, encountered in typical components of microdevices such as microcapillaries, microvalves, microrotors, and microbearings. In the first part, we present extensions to the classical Maxwell/Smoluchowski slip conditions to include high-order Knudsen number effects as well as to take into account the coupling of momentum and heat transfer through thermal creep and viscous heating effects. The numerical method is based on the spectral element technique; validation of the method is obtained by comparison of the numerical simulation results in simple prototype flows (e.g., channel slip-flows) with analytical results. Reduction of pressure drop in microchannels, reported in similar experimental studies, is investigated using slip-flow theory and simulations. In the second part, we consider model inlet flows and a slip-flow past a microcylinder. The effect of slip-flow on skin friction reduction and associated increase in mass flow rate as well as the variation of normal stresses is investigated as a function of Knudsen number. Finally, the effect of compressibility is examined and possible extensions of the current model to take into account such effect are discussed.

Hydrodynamic limit of the stationary Boltzmann equation in a slab

Abstract: We study the stationary solution of the Boltzmann equation in a slab with a constant external force parallel to the boundary and complete accommodation condition on the walls at a specified temperature. We prove that when the force is sufficiently small there exists a solution which converges, in the hydrodynamic limit, to a local Maxwellian with parameters given by the stationary solution of the corresponding compressible Navier-Stokes equations with no-slip boundary conditions. Corrections to this Maxwellian are obtained in powers of the Knudsen number with a controlled remainder.

Electron-electron-scattering-induced size effects in a two-dimensional wire.

Abstract: The differential resistance of wires defined in the two-dimensional electron gas in an (Al,Ga)As heterostructure is observed to first increase and then decrease with increasing current. It is demonstrated that this behavior results from the interplay of an enhanced electron-electron-scattering rate (due to current heating of the electron gas), and the partly diffusive nature of boundary scattering in the wire. The data are identified as an experimental observation of the Knudsen maximum and the Poiseuille flow regime in electron transport, and confirm an analogy between electron and gas flow that has been anticipated since the 1950s.

Predicting failure of the continuum fluid equations in transitional hypersonic flows

Abstract: The manner in which the Navier–Stokes equations of fluid mechanics break down under conditions of low‐density, hypersonic flow is investigated numerically. This is performed through careful and detailed comparisons of solutions obtained with continuum and Monte Carlo simulation techniques. The objective of the study is to predict conditions under which the continuum approach may be expected to fail. Both normal shock waves and bow shocks formed by flow over a sphere are considered for argon and nitrogen. It is found that a Knudsen number based on local flow conditions and gradients is a convenient and accurate criterion for indicating breakdown of the continuum flow equations. Failure of the Navier–Stokes equations in hypersonic transitional flows occurs both in the shock front and in the region immediately adjacent to the body surface.

Method of invariant manifolds and regularization of acoustic spectra

Abstract: A new approach to the problem of reduced description for Boltzmann-type systems is developed. It involves a direct solution of two main problems: thermodynamicity and dynamic invariance of reduced description. A universal construction is introduced, which gives a thermodynamic parameterization of an almost arbitrary approximation. Newton-type procedures of successive approximations are developed which correct dynamic noninvariance. The method is applied to obtain corrections to the local Maxwell manifold using parametrics expansions instead of Taylor series into powers of Knudsen number. In particular, the high frequency acoustic spectra is obtained.

Particle Simulations of Helium Microthruster Flows

Abstract: Computational results are presented for the flow through a helium microthruster. This device is to be used for fine adjustments in attitude control for a proposed space experiment. The mass-flow rates used by the thruster are very low giving Knudsen numbers at the nozzle throat between 0.01 and 1 based on the stagnation conditions and the nozzle throat diameter. These conditions indicate that low-density effects will dominate the fluid mechanics. Therefore, the flows are computed with a particle simulation scheme [the direct simulation Monte Carlo method (DSMC)]. This study presents an application of the DSMC technique to the complete expansion process of a real thruster: from the stagnation chamber of the thruster, to the far-field expansion of the plume. The numerical approach is evaluated by comparison with existing experimental data taken in the expansion plume. The computational results are employed to assess the effect of varying the mass-flow rate on the terminal state of the gas. In addition, the effect of including the background chamber pressure measured in the experimental vacuum facility is investigated and found to be significant.

Observation of Knudsen and Gurzhi transport regimes in a two-dimensional wire

Abstract: We have observed electronic Knudsen and Poiseuille flow in a current heating experiment on electrostatically defined wires in (Al,Ga)As heterostructures. Current heating induces an increase in the number of electron-electron collisions in the wire, leading first to an increase (Knudsen regime) and subsequently to a decrease (due to Poiseuille electron flow, and known as the Gurzhi effect) of the resistance of the wire.

Gas flows caused by evaporation and condensation on two parallel condensed phases and the negative temperature gradient: Numerical analysis by using a nonlinear kinetic equation

Abstract: A rarefied gas between its two parallel plane condensed phases with different temperatures is considered, and the steady gas flows caused by evaporation and condensation on the condensed phases are investigated numerically by using the (nonlinear) Boltzmann–Krook–Welander equation. The flow properties are clarified for a wide range of the parameters (the Knudsen number, the ratio of the temperatures of the condensed phases, and the ratio of the saturation gas pressures at these temperatures). In particular, the phenomenon of the negative temperature gradient, which has extensively been studied for weak evaporation and condensation, is confirmed to occur also for nonweak evaporation and condensation. In the course of the numerical analysis, it is demonstrated that the linearized Boltzmann equation does not give the correct description of the problem for very small Knudsen numbers (i.e., Knudsen numbers comparable to or smaller than the flow Mach number), however weak the evaporation and condensation may be. In addition, a high‐speed evaporating flow toward a perfectly absorbing wall is investigated as an extreme case of strong evaporation and condensation.

Direct simulation Monte Carlo and Navier-Stokes simulations of blunt body wake flows

Abstract: Numerical results obtained with direct simulation Monte Carlo and Navier-Stokes methods are presented for a Mach-20 nitrogen flow about a 70-deg blunted cone. The flow conditions simulated are those that can be obtained in existing low-density hypersonic wind tunnels. Three sets of flow conditions are considered with freestream Knudsen numbers ranging from 0.03 to 0.001. The focus is on the wake structure: how the wake structure changes as a function of rare faction, what the afterbody levels of heating are, and to what limits the continuum models are realistic as rarefunction in the wake is progressively increased. Calculations are made with and without an afterbody sting. Results for the afterbody sting are emphasized in anticipation of an experimental study for the current flow conditions and model configuration. The Navier-Stokes calculations were made with and without slip boundary conditions. Comparisons of the results obtained with the two simulation methodologies are made for both flowfield structure and surface quantities.

Domain Decomposition for Kinetic Problems with Nonequilibrium States

Abstract: A nonequilibrium situation governed by kinetic equations with strongly contrasted Knudsen numbers in different subdomains is discussed We consider a domain decomposition problem for Boltzmann- and Euler equations, establish the correct coupling conditions and prove the validity of the obtained coupled solution Moreover numerical examples comparing different types of coupling conditions are presented

Simulation of Rarefied Gas Transport and Profile Evolution in Nonplanar Substrate Chemical Vapor Deposition

Abstract: A simulation model is presented for nonplanar CVD over device feature scale structures. The direct simulation Monte Carlo method is used to describe the rarefied gas transport in a localized region above the feature. A new approach is outlined to simulate the evolution of the film profile which provides dynamic step‐coverage performance and microstructural detail of the growing film. The method allows simulations of nonequilibrium effects resulting from rarefaction of the gas above surface features. Results of a parametric study are presented for deposition within a long narrow trench and a cylindrical contact hole. The parameters investigated include the reactive sticking coefficient, the surface mobility of the adsorbed reactants, the Knudsen number (the ratio of the mean‐free path to the feature scale), the feature aspect ratio and feature geometry. A sample calculation is presented for deposition over a square hole structure to demonstrate the extension to realistic three‐dimensional structures.

Unique determination of solutions to the Burnett equations

Abstract: Recent success in applying the Burnett equations to the one-dimensional shock-structure problem has raised the issue of whether the full Burnett equations can be used replace the Navier-Stokes equations for solving boundary-value problems in rarefied gasdynamics. As is familiar from the classical rarefied gasdynamics literature, the Burnett equations, if not solved as a successive approximation to the Navier-Stokes equations for a small Knudsen number, would require more boundary conditions than those in the Navier-Stokes system, owing to the presence of the higher-order derivatives. In this paper, this issue is examined with concrete solution examples for the steady Couette flows, addressing specifically whether solutions to the full Burnett equations can be uniquely determined without adding more boundary conditions than those in the Navier-Stokes system. The analysis, supported by detailed numerical solutions, confirms that additional boundary conditions are needed as long as the Knudsen number is not identically zero, lest the solution to the Burnett equations is not unique.

Low-Knudsen-number transport and deposition

Abstract: We present our model for transport and deposition in features for situations in which intermolecular collisions dominate the species transport. Species transport in this ‘‘low’’‐Knudsen‐number regime is modeled using continuum diffusion. Our model allows both homogeneous reactions and heterogeneous (surface) reactions. We use the Galerkin finite element method to estimate reactant species concentration profiles for infinite trenches with arbitrary cross section, for which two‐dimensional profile evolution is appropriate. We simulate deposition processes using a solution domain which includes the trench as well as a specified region above the surface of the wafer. As an example application, we present results for the deposition of tungsten using the hydrogen reduction of tungsten hexafluoride in trenches of rectangular cross section with initial aspect ratio 4. The model predicts that step coverage increases with decreasing temperature and increasing tungsten hexafluoride partial pressure, while keeping hydrogen partial pressure and the height of the solution domain constant. These trends are in agreement with experimental observations for tungsten deposition processes. Trends in film conformality with deposition conditions are explained in terms of the ‘‘step coverage modulus’’ and reactant concentration ratios for given initial trench dimensions. The step coverage modulus is the ratio of a characteristic deposition rate to a characteristic transport rate.

Numerical analysis of a rarefied gas flow past a volatile particle using the Boltzmann equation for hard‐sphere molecules

Abstract: A spherical condensed phase with a uniform surface temperature that is placed in a slow uniform flow of its vapor gas is considered. The steady behavior of the gas accompanied by evaporation and condensation on the sphere is investigated mainly numerically on the basis of the Boltzmann equation for hard‐sphere molecules. The numerical method is a combination of the hybrid‐difference‐scheme method, capable of describing the discontinuity of the velocity distribution function in the gas, and the numerical kernel method [Phys. Fluids A 5, 716 (1993)]. The velocity distribution function of the gas molecules, the macroscopic variables such as the density, velocity, and temperature of the gas, and the force (drag) acting on the sphere are obtained precisely for the whole range of the Knudsen number (the mean free path of the uniform flow divided by the radius of the condensed phase). In particular, the behavior of the discontinuity of the velocity distribution function in the gas is described accurately.

Molecular dynamics simulations of droplet evaporation

Abstract: The complete evaporation of a three-dimensional submicron droplet under subcritical conditions has been modeled using molecular dynamics. The two-phase system consisted of 2048 argon atoms modeled using a Lennard-Jones 12-6 potential distributed between a single droplet and its surrounding vapor. The system was first allowed to relax to equilibrium, then the droplet was evaporated by increasing the temperature of the vapor phase atoms at the boundaries of the system until only the vapor phase remained. The computed evaporation rate agrees with that predicted by the Knudsen aerosol theory.

A quartz crystal microbalance to determine enthalpies of sublimation at intermediate temperatures by the Knudsen effusion method

Abstract: The principle of operation, design, construction and testing of a quartz crystal microbalance for indirect determination of the enthalpy of sublimation is described. The instrument allows fast and reliable measurement of the mass flow from a Knudsen effusion cell at different temperatures. The enthalpy of sublimation can be derived using the Knudsen equation for the vapour pressure combined with the Clapeyron equation. The test substance used in this study was thiourea and experiments were made in the range 378.45-395.89 K. The derived value of Delta subHm(387.17 K)=(103.95+or-0.32) kJ mol-1 is in good agreement with the reported calorimetric value Delta Hsub(389.17 K)=(103.54+or-0.43) kJ mol-1.

Near-wake structure for a generic configuration of aeroassisted space transfer vehicles

Abstract: Results of a numerical study are presented for hypersonic low-density nitrogen gas flow about a 70-deg blunt cone using the direct simulation Monte Carlo method.The flow conditions simulated are attainable in existing lowdensity hypersonic wind tunnels; encompassing freestream Knudsen numbers of 0.03 to 0.001. Particular emphasis is given to the near-wake flow and its sensitivity to rarefaction and other parametric variations. A stable vortex forms in the near wake at and below a freestream Knudsen number of 0.01, and the size of the vortex increases with decreasing freestream Knudsen number. The base region of the flow remains in thermal nonequilibrium for all cases. There is no formation of a lip separation shock or a distinct wake shock at these rarefied conditions.

Molecular Dynamics Computer Simulations of Diffusion in Porous Silicates

Abstract: In this work a newly developed dual control volume grand canonical molecular dynamics technique simulates the diffusion of gas in a cylindrical pore. This allows spatial variation of chemical potential and hence an accurate simulation of steady state pressure driven diffusion. The molecular sieving nature of imicroporous imogolite models and the Knudsen effect are discussed and compared with experimental data.

Experimental investigation of rarefied gas flow through rectangular slits and nozzles

Abstract: The flow of a rarefied gas through rectangular configuratons of different geometries has been experimentally studied to determine their discharge coefficient characteristics. The configurations used are a set of sharp-edged slit orifices, a smooth converging nozzle and a tube. The range of the Reynolds number based on the throat conditions varied from 0.01 to 100. The equivalent Knudsen number range based on the upstream conditions and inlet diameter varied from 0.0521 to 2.521. The results for the smooth nozzle are compared with calculations using a numerical method with one-dimensional stream tube approximation based on integrated boundary layer equations. The slit and the tube results are compared with the experimental results of Sreekanth and Davis [1988].

Pressure‐dependent gas heat transport in a spherical pore

Abstract: A mean free path gas kinetic theory is used to model the conductive heat transport of a gas within a void volume enclosed in a Fourier solid. A variational upper bound principle is derived for a void of arbitrary shape and applied to obtain a rigorous upper bound equation for the void gas conductivity in a spherical void. The variational void gas conductivity equation is exact in both the large and small Knudsen number (Kn) limits and provides a means to determine the accuracy of the reciprocal additivity interpolation formula as applied to thermal conductivity rather than diffusive mass transfer (maximum error 6% and Kn = 0.5 and [alpha] = 1). Temperature jump will occur even at atmospheric pressures and higher for sufficiently small thermal accommodation coefficients ([alpha]<0.1). Experimental void gas heat conductivities vs. pressure data for H[sub 2], He, Ne, N[sub 2], CO[sub 2], and F12 in a polyurethane foam are compared with theoretical mean free path void gas conductivity vs. inverse Knudsen number curves drawn for various [alpha]. Estimates of the thermal accommodation coefficients for the gas- polyurethane surface exhibit a maximum with increasing molecular mass of the gas molecules, which qualitatively agrees with the predictions of Baule'smore » classical theory. Results also point to a rather sharp shift of the S curve to higher pressures with decreasing thermal accommodation.« less

Gas Permeation Characteristics of Zirconia-Silica Composite Membranes at High Temperature

Abstract: Permeabilities for pure gases of H2, He, CH4, NH3, H2O, N2, N2, O2 and CO2 were measured at trans-membrane pressures up to 0.2 MPa and 303, 423, 523 and 773 K for two types (with small pores and large pores) of composite zirconia-silica membranes coated on the surface of porous ceramic tubes (0.5μm). The composition of the metal-alkoxides solution used successfully in the coating process was Zr (OC3H7) 4 : 3.09, Si (OC2H5) 4 : 7.20, Y (CH3COO) 3.4H2O : 0.21, i-PrOH : 89.50 in molar %.At low temperatures, with large-and small-pore membranes, the permeation mechanism for H2O, NH3 and CO2 is surface diffusion, and for other gases it is Knudsen flow.At higher temperature, with a large-pore membrane, the permeation mechanism for H2O, NH3, CO, and CH4 is Knudsen flow, and for other gases it is a combination of Knudsen flow and activated diffusion. With a small-pore membrane, for H2O it is Knudsen flow, and for all other gases it is activated diffusion.

An analysis of the isothermal isobaric chemical vapor infiltration of TiB2

Abstract: A model has been developed for the isothermal isobaric chemical vapor infiltration processing of ceramic matrix composites and subsequently applied to the deposition of TiB2 on the walls of a narrow pore. The model was next utilized to explore the effect of the Knudsen diffusivity on uniformity of the materials deposition during the infiltration of narrow pores found in the ceramic preform. It was found that the collisions of the gas molecules with the pore walls (the Knudsen diffusivity) may dominate the mass transport during the chemical vapor infiltration process at higher temperatures and at lower pressures.

Gas Flow in Micr-Channels

Abstract: An experimental and theoretical investigation of low Reynolds number, high subsonic Mach number, compressible gas flow in channels is presented. Nitrogen, helium, and argon gases were used. The channels were microfabricated on silicon wafers and were typically 100 μm wide, 104 μm long, and ranged in depth from 0.5 to 20 μm. The Knudsen number ranged from 10-3 to 0.4. The measured friction factor was in good agreement with theoretical predictions assuming isothermal, locally fully developed, first-order, slip flow. Disciplines Engineering | Mechanical Engineering Comments Suggested Citation: Harley, John C., Yufend Huang, Haim H. Bau and Jay N. Zemel. (1994) Gas flow in micro-channels. Journal of Fluid Mechanics. Vol 284. p. 257-274. Copyright 1994 Cambridge University Press. http://dx.doi.org/10.1017/S0022112095000358 This journal article is available at ScholarlyCommons: http://repository.upenn.edu/meam_papers/196 J. Fluid Mech. (1995), vol. 284, pp. 251-274 Copyright