Showing papers on "Knudsen number published in 2004"

Expanded analogy between Boltzmann kinetic theory of fluids and turbulence

Abstract: We demonstrate that the effects of turbulent fluctuations have a striking resemblance to those of microscale (thermal) fluctuations in laminar flows, even to higher order in the Knudsen number. This suggests that there may be a good basis for understanding turbulence in terms of Boltzmann kinetic theory. If so, turbulence may be better described in terms of ‘mixing times’ rather than the more classical ‘mixing lengths’. Comparisons are made to Reynolds-stress turbulence models.

Validation of a Second-Order Slip Flow Model in Rectangular Microchannels

Abstract: An analytical slip-flow model based on second-order boundary conditions was proposed for gaseous flow in rectangular microchannels. An experimental setup has been designed for the measurement of gaseous micro flow rates under controlled temperature and pressure conditions. Data relative to nitrogen and helium flows through rectangular microchannels, from 4.5 to 0.5 μm in depth and with aspect ratios from 1–9%, are presented and analyzed. A method is proposed to eliminate the main source of uncertainty, which is the imprecision when measuring the dimensions of the microchannel cross-section. It is shown that in rectangular microchannels, the proposed second-order model is valid for Knudsen numbers up to about 0.25, whereas the first-order model is no longer accurate for values higher than 0.05. The best fit is found for a tangential momentum accommodation coefficient σ = 0.93, both with helium and nitrogen.

Analytical calculation of slip flow in lattice Boltzmann models with kinetic boundary conditions

Abstract: We present a mathematical formulation of kinetic boundary conditions for Lattice Boltzmann schemes in terms of reflection, slip, and accommodation coefficients. It is analytically and numerically shown that, in the presence of a non-zero slip coefficient, the Lattice Boltzmann flow develops a physical slip flow component at the wall. Moreover, it is shown that the slip coefficient can be tuned in such a way to recover quantitative agreement with analytical and experimental results up to second order in the Knudsen number.

Modelling mass transport through a porous partition: Effect of pore size distribution.

Abstract: Direct contact membrane distillation process has been studied using microporous polytetrafluoroethylene and polyvinylidene fluoride membranes. The membranes were characterized in terms of their nonwettability, pore size distribution and porosity. The mean pore sizes and pore size distributions were obtained by means of wet/ dry flow method. The mean pore size and the effective porosity of the membranes were also determined from the gas permeation test. A theoretical model that considers the pore size distribution together with the gas transport mechanisms through the membrane pores was developed for this process. The contribution of each mass transport mechanism was analyzed. It was found that both membranes have pore size distributions in the Knudsen region and in the transition between Knudsen and ordinary diffusion region. The transition region was the major contribution to mass transport. The predicted water vapor permeability of the membranes were compared with the experimental ones. The effect of considering pore size distribution instead of mean pore size to predict the water vapor permeability of the membranes was investigated.

Microchannel flow in the slip regime: gas-kinetic BGK-Burnett solutions

Abstract: In the first part of this paper presents a gas-kinetic scheme based on the Bhatnagar–Gross–Krook (BGK) model for the microflow simulations in the near continuum flow regime. The current method improves the previous gas-kinetic BGK Navier–Stokes (BGK–NS) solver by (i) implementing a general non-equilibrium state based on the Chapman–Enskog expansion of the BGK model up to the Knudsen number squared , the qualitative differences in the pressure distribution in the cross-stream direction between the Navier–Stokes and the DSMC results are resolved by the gas-kinetic BGK–Burnett scheme. It demonstrates that the BGK–Burnett method could give a more realistic description of flow motion than the Navier–Stokes method even in the slip flow regime. After that, the current method is used to simulate the microchannel flows, where the experimental data are available. In this study, the similarity in the pressure distribution along the straight microchannel is verified first. Then, the mass flow rates for different gases, such as argon, helium and nitrogen, in the long microchannel of submicron height are computed and compared with the experimental measurements.

Stable transport equations for rarefied gases at high orders in the Knudsen number

Abstract: An approach is presented to derive transport equations for rarefied gases from the Boltzmann equation within higher orders of the Knudsen number. The method focuses on the order of magnitude of the moments of the phase density, and the order of accuracy of the transport equations, both measured in powers of the Knudsen number. The method is developed up to the third order, and it is shown that it yields the Euler equations at zeroth order, the Navier–Stokes–Fourier equations at first order, Grad’s 13 moment equations (with omission of a nonlinear term) at second order, and a regularization of these at third order. The method is discussed in detail, and compared with the classical methods of kinetic theory, i.e., Chapman–Enskog expansion and Grad moment method. The advantages of this method above the classical approaches are discussed conclusively. An important feature of the method presented is that the equations of any order are stable, other than in the Chapman–Enskog method, where the second and third ...

Experimental Investigation of Gas Flow in Microchannels

Abstract: We present an experimental investigation of laminar gas flow through microchannels. The independent variables: relative surface roughness, Knudsen number and Mach number were systematically varied to determine their influence on the friction factor The microchannels were etched into silicon wafers, capped with glass, and have hydraulic diameters between 5 and 96 μm. The pressure was measured at seven locations along the channel length to determine local values of Knudsen number, Mach number and friction factor. All measurements were made in the laminar flow regime with Reynolds numbers ranging from 0.1 to 1000

A lattice Boltzmann BGK model for simulation of micro flows

Abstract: We propose a lattice Boltzmann BGK model for simulation of micro flows. This model is based on the kinetic theory and the entropic lattice Boltzmann method (S. Ansumali and I. V. Karlin, J. Stat. Phys. 107 (2002) 291) but the relaxation time is re-defined in terms of the Knudsen number, and a diffuse-scattering boundary condition (DSBC) is adopted to consider the velocity slip at the wall. Simple theoretical analysis and numerical validation show that the proposed model gives good predictions of the micro fluidic behaviors.

Numerical simulation of rarefied gas flow through a thin orifice

Abstract: Rarefied gas flow through a thin orifice is studied on the basis of the direct simulation Monte Carlo method. The mass flow rate and the flow field are calculated over the whole range of the Knudsen number for various values of the pressure ratio. It is found that at all values of the pressure ratio a significant variation of the flow rate occurs in the transition regime between the free-molecular and hydrodynamic regimes. In the hydrodynamic regime the flow rate tends to a constant value. In the case of finite pressure ratio the flow field qualitatively differs from that for outflow into vacuum, namely vortices appear in the downflow container on approaching the hydrodynamic regime. Then, in the hydrodynamic regime the gas flow forms a strong jet. A comparison of the numerical results with experimental data available in the open literature has been performed.

Tractable molecular theory of transport of Lennard-Jones fluids in nanopores

Abstract: We present here a tractable theory of transport of simple fluids in cylindrical nanopores, which is applicable over a wide range of densities and pore sizes. In the Henry law low-density region the theory considers the trajectories of molecules oscillating between diffuse wall collisions, while at higher densities beyond this region the contribution from viscous flow becomes significant and is included through our recent approach utilizing a local average density model. The model is validated by means of equilibrium as well nonequilibrium molecular dynamics simulations of supercritical methane transport in cylindrical silica pores over a wide range of temperature, density, and pore size. The model for the Henry law region is exact and found to yield an excellent match with simulations at all conditions, including the single-file region of very small pore size where it is shown to provide the density-independent collective transport coefficient. It is also shown that in the absence of dispersive interactions the model reduces to the classical Knudsen result, but in the presence of such interactions the latter model drastically overpredicts the transport coefficient. For larger micropores beyond the single-file region the transport coefficient is reduced at high density because of intermolecular interactions and hindrance to particle crossings leading to a large decrease in surface slip that is not well represented by the model. However, for mesopores the transport coefficient increases monotonically with density, over the range studied, and is very well predicted by the theory, though at very high density the contribution from surface slip is slightly overpredicted. It is also seen that the concept of activated diffusion, commonly associated with diffusion in small pores, is fundamentally invalid for smooth pores, and the apparent activation energy is not simply related to the minimum pore potential or the adsorption energy as generally assumed.

Rarefaction effects on shear driven oscillatory gas flows: A direct simulation Monte Carlo study in the entire Knudsen regime

Abstract: A complete mathematical description of oscillatory Couette flows within the framework of kinetic theory is not available in the literature. Motivated by this and their vast engineering applications, we present a parametric study of time-periodic oscillatory Couette flows using the unsteady direct simulation Monte Carlo (DSMC) method. Computations are performed as a function of the Knudsen (Kn) and Stokes (β) numbers, in the entire Knudsen regime (Kn⩽100) and a wide range of Stokes numbers (β⩽7.5). The DSMC results are validated using a recently developed semianalytical/empirical model that is applicable for quasisteady flows (β⩽0.25) in the entire Knudsen regime, and for any Stokes number flow in the slip flow regime (Kn⩽0.1). In addition, we derived an analytical solution of the linearized collisionless Boltzmann equation for oscillatory Couette flows, and utilized this to validate the DSMC results in the free-molecular flow regime. Dynamic response of the flow, including the velocity profiles, phase ang...

Gaseous slip models based on the Langmuir adsorption isotherm

Abstract: On the basis of Langmuir’s theory of adsorption of gases on solids, a robust gaseous slip model is presented. The concept of accommodation coefficient and the difference of gas particles are explained within the new framework. It turned out that the Langmuir model recovers the Maxwell model in the first-order approximation in the case of the microchannel gas flow. In order to validate the new approach, the model is applied to problems of technical interests: pressure-driven microchannel gas flow and low Reynolds number gas flow past a sphere. With the help of previous theoretical and experimental results it is shown that with an adjustable parameter the model in low-speed creeping regime with moderate Knudsen numbers yields a prediction in qualitative agreement with the data.

Examination of the lbm in simulation of microchannel flow in transitional regime

Abstract: The gas flows in micro-electro-mechanical systems possess relatively large Knudsen number and usually belong to the slip flow and transitional flow regimes. Recently the lattice Boltzmann method (LBM) was proposed by Nie et al. in Journal of Statistical Physics, vol. 107, pp. 279–289, in 2002 to simulate the microchannel and microcavity flows in the transitional flow regime. The present article intends to test the feasibility of doing so. The results of using the lattice Boltzmann method and the direct simulation Monte Carlo method show good agreement between them for small Kn (Kn = 0.0194), poor agreement for Kn = 0.194, and large deviation for Kn = 0.388 in simulating microchannel flows. This suggests that the present version of the lattice Boltzmann method is not feasible to simulate the transitional channel flow.

Entropy Generation Due to Laminar Incompressible Forced Convection Flow Through Parallel-Plates Microchannel

Abstract: The entropy generation due to steady laminar forced convection fluid flow through parallel plates microchannel is investigated numerically. The effect of Knudsen, Reynolds, Prandtl, Eckert numbers and the nondimensional temperature difference on entropy generation within the microchannel is discussed. The fraction of the entropy generation due to heat transfer to the total entropy generation within the microchannel is studied in terms of Bejan number. The entropy generation within the microchannel is found to decrease as Knudsen number increases, and it is found to increase as Reynolds, Prandtl, Eckert numbers and the nondimensional temperature difference increase. The contribution of the viscous dissipation in the total entropy generation increases as Knudsen number increases over wide ranges of the flow controlling parameters.

Lattice boltzmann method for simulating gas flow in microchannels

Abstract: Isothermal gas flows in microchannels is studied using the lattice Boltzmann method. A novel equation relating Knudsen number with relaxation time is derived. The slip-velocity on the solid boundaries is reasonably realized by combining the bounce-back reflection with specular reflection in a certain proportion. Predicted characteristics in a two-dimensional microchannel flow, including slip-velocity, nonlinear pressure drop, friction factors, velocity distribution along the streamwise direction and mass flow rate, are compared with available analytical and experimental results and good agreement is achieved.

Pressure distributions of gaseous slip flow in straight and uniform rectangular microchannels

Abstract: This paper presents analytical derivations of the pressure distribution in straight and uniform rectangular microchannels in the slip flow regime and new experimental data in those channels. The flow is to be steady state, two-dimensional, isothermal, and to have negligible transverse velocities with a first order slip boundary condition. The measured pressure distributions of airflows are compared with newly derived analytical results. There is close agreement between the measurements and calculation by the slip flow formula. The dimensionless location of the maximum deviation from the linear pressure distribution is found analytically and compared with the measurements. This dimensionless location of the maximum deviation increases with the increasing pressure ratios in the slip flow regime. The effect of several parameters such as the channel aspect ratio and the Knudsen number on the locations of maximum deviation from linearity are investigated. The nonlinearity of the pressure distribution is also discussed.

Homogenization of the Generalized Reynolds Equation for Ultra-Thin Gas Films and Its Resolution by FEM

Abstract: We address the numerical modeling of roughness or texture effects in ultra-thin gas films. Rarefaction (high Knudsen number) effects are dealt with using the Generalized Reynolds Equation, and a homogenization procedure is proposed to rigorously account for arbitrary roughness/texture shapes. The presentation is focused on head-disk magnetic storage devices, but the techniques proposed are general. Some details of the implementation, along with numerical tests, are included. By removing the small space and time scales from the problem, the methodology allows for efficient modeling of slider bearings with small-scale features.

Kinetic theory of turbulence modeling: smallness parameter, scaling and microscopic derivation of Smagorinsky model

Abstract: A mean-field approach (filtering out subgrid scales) is applied to the Boltzmann equation in order to derive a subgrid turbulence model based on kinetic theory. It is demonstrated that the only Smagorinsky type model which survives in the hydrodynamic limit on the viscosity time scale is the so-called tensor-diffusivity model. Scaling of the filter-width with Reynolds number and Knudsen number is established. This sets the first rigorous step in deriving turbulence models from kinetic theory.

Molecular gas dynamics observations of Chapman-Enskog behavior and departures therefrom in nonequilibrium gases.

Abstract: Bird's direct simulation Monte Carlo method is used to compute the molecular velocity distribution of a gas with heat flow. At continuum nonequilibrium conditions (small heat flux), Chapman-Enskog behavior is obtained for inverse-power-law molecules (hard-sphere through Maxwell): the Sonine-polynomial coefficients away from walls (i.e., the normal solution) agree with theory. At noncontinuum nonequilibrium conditions (large heat flux), these coefficients differ systematically from their continuum values as the local Knudsen number (nondimensional heat flux) is increased.

Plane Couette flow of binary gaseous mixture in the whole range of the Knudsen number

Abstract: The Couette flow of binary gaseous mixtures is studied on the basis of the McCormack model of the Boltzmann equation, which was solved numerically by the discrete velocity method. The calculations were carried out for three mixtures of noble gases: neon–argon, helium–argon, and helium–xenon. The stress tensor and bulk velocity of both species were calculated for several values of the gas rarefaction in the range from 0.01 to 40 for three values of the molar concentrations: 0.1,0.5 and 0.9. The numerical solution together with an analytical solution based on the slip boundary condition cover the whole range of the gas rarefaction. It was showed that the Couette flow is weakly affected by the intermolecular interaction law.

Charging of aerosol particles in the near free–molecule regime

Abstract: The charging of small neutral and charged particles suspended in weakly ionized plasma is investigated under the assumption that the Coulomb + image forces give rise to the ion transport in the carrier plasma and define the rate of charging processes. Our approach is based on a BGK version of the kinetic equation [1,2] describing the ion transport in the presence of force fields created by the particle charge and the image force. A special type of the perturbation theory (with respect to the reciprocal Knudsen number) is used for calculating the rate of ion deposition onto neutral and charged particles. As the starting approximation, the free-molecule ion distribution with a floating ion flux is used for evaluating the collision term in the Boltzmann equation. The value of the ion flux as a function of the particle size is then fixed self-consistently from the solution of the Boltzmann equation with the approximated collision term. The expression for the ion flux J(a) to the spherical particle of radius a is derived in the form $J = \xi(a) J_{fm}$ , where J fm is the free-molecule flux (no carrier plasma) and $\xi(a)$ is a correction factor taking into account the ion-molecular collisions. The latter is shown to never exceed unity and to depend weakly on the particle-ion interaction.