Knudsen Regime Research Articles

Unified gas-kinetic wave-particle methods. II. Multiscale simulation on unstructured mesh

In this paper, we present a unified gas-kinetic wave-particle (UGKWP) method on unstructured mesh for the multiscale simulation of continuum and rarefied flow. Inheriting from the multiscale transport in the unified gas-kinetic scheme (UGKS), the integral solution of the kinetic model equation is employed in the construction of the UGKWP method to model the flow physics on the scales of cell size and time step. A novel wave-particle adaptive formulation is introduced in the UGKWP method to describe the flow dynamics in each control volume. The local gas evolution is constructed through the dynamical interaction of the deterministic hydrodynamic wave and the stochastic kinetic particle. To model the gas dynamics on the scales of cell size and time step, the decomposition, interaction, and evolution of the hydrodynamic wave and the kinetic particle depend on the ratio of time step to local collision time. In the rarefied flow regime, the UGKWP method recovers the nonequilibrium flow physics by discrete particles and performs as a stochastic particle method. In the continuum flow regime, the UGKWP method captures the flow behavior solely by macroscopic variable evolution and becomes a gas-kinetic hydrodynamic flow solver, the same as the gas-kinetic scheme, for viscous and heat-conducting Navier–Stokes solutions. In the transition regime, both kinetic particle and hydrodynamic wave contribute adaptively in the UGKWP to capture the peculiar nonequilibrium flow physics in a most efficient way. In different flow regimes, the Sod shock tube, lid-driven cavity flow, laminar boundary layer, and high-speed flow around a circular cylinder are computed to validate the UGKWP method on unstructured mesh. The UGKWP method obtains the same UGKS solutions in all Knudsen regimes. However, with an automatic wave-particle decomposition, the UGKWP method becomes very efficient. For example, at Mach number 30 and Knudsen number 0.1, the UGKWP has several-order-of-magnitude reductions in computational cost and memory requirement in comparison with UGKS. Overall, the UGKWP can capture the gas dynamics in all flow regimes efficiently and accurately from the free molecular transport to the Navier-Stokes flow evolution.

Design and Simulation of a Wireless SAW-Pirani Sensor with Extended Range and Sensitivity.

Pressure is a critical parameter for a large number of industrial processes. The vacuum industry relies on accurate pressure measurement and control. A new compact wireless vacuum sensor was designed and simulated and is presented in this publication. The sensor combines the Pirani principle and Surface Acoustic Waves, and it extends the vacuum sensed range to between 10−4 Pa and 105 Pa all along a complete wireless operation. A thermal analysis was performed based on gas kinetic theory, aiming to optimize the thermal conductivity and the Knudsen regime of the device. Theoretical analysis and simulation allowed designing the structure of the sensor and its dimensions to ensure the highest sensitivity through the whole sensing range and to build a model that simulates the behavior of the sensor under vacuum. A completely new design and a model simulating the behavior of the sensor from high vacuum to atmospheric pressure were established.

Effects of correlated morphological and topological heterogeneity of pore network on effective transport and reaction parameters

The effective diffusivity and the effective Thiele modulus of reactive porous matrices are determined by using a hybrid discrete and continuum model. The model describes static heterogeneity with weak noise on both morphology and topology of the matrix. Variations in diffusivity with respect to pore size (radius and length) and network connectivity up to second order are analyzed to assess the effect of stochastic media on catalytic reaction and transition of a single species. The porous medium is simulated by several building blocks (BB) which are regular in topology and morphology. A methodology is presented to construct a BB such that it fits the true matrix. Permutation of several regular pore networks in series constructs the porous media. Local mass balances over each BB yields model equations for reaction and transport in the matrix. Correlated random walk theory is used in each BB in order to relate the diffusivity to the coordination number.The Gaussian correlation function is assumed to model the autocorrelation of independent variables which are used to characterize the correlated porous media. The contribution of the fluctuating parameters in the mass conservation of the matrix yields a new closure equation in which ensemble average concentration of the reactant depends on different cross-correlations. The model is stable as long as the Thiele modulus at the correlation length scale is small. It predicts that geometrical heterogeneities lead to correction in effective diffusivity while the reaction rate is unaffected due to small Thiele modulus at correlation length scale. The model is verified by assessing the Thiele modulus of a pore network under Knudsen regime during a first order catalytic reaction; a good agreement was found especially for moderate Thiele modulus of correlated network. However, it fails to simulate random networks. Our systematic study revealed that the pore radius statistics impact significantly the diffusivity and the Thiele modulus while the topological parameters are important in weakly connected networks. The model predictions show that the diffusion constant decreases sharply as pore radius variance increases but increases smoothly with Thiele modulus.

Ethanol dehydration pathways in H-ZSM-5: Insights from temporal analysis of products

Abstract Ethanol dehydration to ethene via direct and ether mediated paths is mechanistically investigated via transient experiments in a Temporal analysis of products, TAP-3E reactor over a temperature range of 473–573 K in Knudsen regime conditions. Pulse experiments of ethanol over H-ZSM-5 do not yield diethyl ether as a gas phase product. Cofeed experiments with diethyl ether and C-13 labeled ethanol show that ethene formation from diethyl ether is the preferential route. Kinetic parameters from ab initio based microkinetic modelling of ethanol dehydration are compared to the experimental data of the TAP reactor by an in-house developed reactor model code TAPFIT. Rate coefficients in ethene adsorption are in agreement with the ab initio based microkinetic modelling parameters. The experimental data from a diethyl ether feed are compared to the simulated responses from the ab initio based kinetic parameters and further optimized by regression analysis. Reaction path analysis with the optimized kinetic parameters identifies the preference of an ether mediated path under the applied transient experimental conditions.

An implicit unified gas-kinetic scheme for unsteady flow in all Knudsen regimes

The unified gas-kinetic scheme (UGKS) is a direct modeling method for multiple scale transport. Based on the ratio of time step to the particle collision time, the local evolution solution on the mesh size and time step scales is used in the construction of the multiscale method. For a flow problem covering multiple flow regimes, such as the hypersonic flow around a flying vehicle in near space, the UGKS is able to capture the highly compressed Navier-Stokes solution in one region and fully expanded free molecular flow in another region, with significant variations of the ratio between the time step and the local particle collision time around the vehicle. For an explicit UGKS, the time step in the whole computational domain is determined by the CFL condition. With implicit and multigrid techniques, the efficiency of the UGKS [1,2] has been improved by two orders of magnitude for steady state computation. However, for unsteady flow computation, due to the CFL condition the global time step used in the explicit UGKS may be limited by the smallest cell size in the computational domain. As a result, for a largely stretched non-uniform mesh the global time step becomes very small and the ratio of the time step to the local particle collision time may get a very small value. Under such a circumstance, even though the UGKS is a multiscale method, the real physics represented in the explicit UGKS may be constrained to the kinetic scale transport only, and the advantage of the multiscale nature in UGKS has not been fully utilized. In order to solve the multiscale unsteady flow problem efficiently, the time step restriction from a global CFL condition has to be released. In this paper, we will develop an implicit UGKS (IUGKS) for unsteady flows by alternatively solving the macroscopic and microscopic governing equations within a time step iteratively. With a pre-defined uniform large evolution time step, the local CFL number varies greatly in different region, such as on the order 1 in the large numerical cell size region, and 100 in the small cell size region. In order to preserve coherent flow evolution and keep the multiscale nature, the time averaged numerical flux across a cell interface is still evaluated by the explicit UGKS under the local CFL condition. Therefore, the multiscale property of the UGKS modeling has been kept over non-uniform meshes. With improved temporal discretization, the current IUGKS can automatically go back to the explicit UGKS and obtain identical solutions when the time step of the implicit scheme gets to that of an explicit one. Many numerical examples are included to validate the scheme for both continuum and rarefied flows with a large variation of artificially generated mesh size. The IUGKS has a second order accuracy and presents reasonably good results for unsteady flow computation, and its efficiency has been improved by dozens of times in comparison with the explicit UGKS.

Ceramic Membranes for the Separation of Plasma Enhancement Gases

Low and medium Z impurity seeding gases, or plasma enhancement gases (PEGs) will be injected into the plasma of a Tokamak to convert the plasma thermal energy to ultraviolet and soft X-ray radiation. Possible PEGs are nitrogen, neon, argon, xenon and other inert gases. for the exhaust processing system of the demonstration fusion reactor (DEMO), the use of inorganic membranes was recently taken into consideration. Two commercial ceramic membranes (produced by Atech Innovations GmbH, Germany) were tested at a permeation apparatus built at ENEA Casaccia laboratories. Single gas permeances for H2, He, Ar and N2 were measured with a pressure drop across the membranes between 10 Pa up to 130 kPa at room temperature. For both membranes the permeance order follows the Knudsen regime. The Knudsen permeance measured for the two membranes are on the order of 10−7–10−8. The selectivities found are low (similar to ideal Knudsen).

Ballistic and Collisional Flow Contributions to Anti-Fourier Heat Transfer in Rarefied Cavity Flow

This paper investigates anti-Fourier heat transfer phenomenon in a rarefied gas confined within a lid-driven cavity using a novel flow decomposition technique in the direct simulation Monte Carlo (DSMC) method proposed by Stefanov and co-workers. An isothermal cavity with different degrees of flow rarefaction from near continuum to mid transition regimes was considered to investigate cold-to-hot heat transfer from ballistic/collision flow decomposition viewpoint. A new cold-to-hot heat transfer indicator in the form of a scalar product of normalized heat flow vector and normalized temperature gradient vector has been introduced for the overall, ballistic and collision parts of these vectors. Using the new indicator, contributions of ballistic and collision flow parts to temperature and heat flux components was investigated with a specific emphasis on the cold-to-hot heat transfer phenomenon. We demonstrated that both ballistic and collision flow parts contribute to the occurrence of cold-to-hot heat transfer. However, it was found out that considered separately both ballistic and collision parts of heat transfer, when related to corresponding ballistic and collision temperature fields, they are ever hot-to-cold for all degrees of flow rarefaction. Thus, cold-to-hot heat transfer is a result of a subtle interplay between ballistic and collision parts in the slip and transition Knudsen regimes.

Gas Flow at the Ultra-nanoscale: Universal Predictive Model and Validation in Nanochannels of Ångstrom-Level Resolution.

Gas transport across nanoscale pores is determinant in molecular exchange in living organisms as well as in a broad spectrum of technologies. Here, we report an unprecedented theoretical and experimental analysis of gas transport in a consistent set of confining nanochannels ranging in size from the ultra-nanoscale to the sub-microscale. A generally applicable theoretical approach quantitatively predicting confined gas flow in the Knudsen and transition regime was developed. Unlike current theories, specifically designed for very simple channel geometries, our approach can be applied to virtually all geometries, for which the probability distribution of path lengths for particle-interface collisions can be computed, either analytically or by numerical simulations. To generate a much needed benchmark experimental model, we manufactured extremely reproducible membranes with two-dimensional nanochannels. Channel sizes ranged from 2.5 to 250 nm, and angstrom level of size control and interface tolerances were achieved using leading-edge nanofabrication techniques. We then measured gas flow in the Knudsen number range from 0.2 to 20. Excellent agreement between theoretical predictions and experimental data was found, demonstrating the validity and potential of our approach.

Correction to: A Numerical Investigation of the Mean Free Path Distribution in the Knudsen Regime

Correction to: A Numerical Investigation of the Mean Free Path Distribution in the Knudsen Regime

A Numerical Investigation of the Mean Free Path Distribution in the Knudsen Regime

In this research, numerical simulations are used to analyze the mean free path distribution of a (highly) rarefied gas hosted in a constant temperature model cylinder. In the Knudsen regime, gas–gas collisions do not occur and molecular transport can be fully attributed to gas–wall collisions. It is commonly accepted that the mean free path probability distribution then becomes highly nontrivial. Therefore, we propose to analyze this function in terms of its first two moments. We show numerically that both the first and second moments of the path distribution are important in describing the transport of a rarefied gas. In case of the Smoluchowski thermostat, a thermal wall mechanism for the cylindrically shaped pore, the obtained values can be compared with the results that follow from kinetic gas theory.

Assessment of the LD-DSMC hybrid method for hypersonic rarefied flow

Hypersonic aerothermodynamics for a probe entering a planetary atmosphere is an important issue in space exploration. The probe experiences various Knudsen number regimes, ranging from rarefied to continuum, due to density variation in the planet’s atmosphere. To simulate such multiscale flows, a novel hybrid particle scheme is employed in the present work. The hybrid particle scheme employs the direct simulation Monte Carlo (DSMC) method in rarefied flow regions and the low diffusion (LD) particle method in continuum flow regions. Numerical procedures in the low diffusion particle method are implemented within an existing DSMC algorithm. The hybrid scheme is assessed using two hypersonic, blunt body flow problems: Mach 10 nitrogen flow over a sphere, and Mach 40 carbon dioxide flow over the Mars Path Finder each with a global Knudsen number of 0.002. Standard DSMC and CFD results are compared with the LD-DSMC hybrid simulation results. The hybrid scheme results show good overall agreement with results from standard DSMC computation, while CFD is inaccurate especially in the wake where a highly rarefied region exists. The LD-DSMC hybrid solution is able to increase computational efficiency upto 50% in run time in comparison to DSMC. And sensitivity to numerical parameters of the LD-DSMC method is also studied.

A Fractal Model for Gas–Water Relative Permeability in Inorganic Shale with Nanoscale Pores

A reliable gas–water relative permeability model in shale is extremely important for the accurate numerical simulation of gas–water two-phase flow (e.g., fracturing fluid flowback) in gas-shale reservoirs, which has important implication for the economic development of gas-shale reservoir. A gas–water relative permeability model in inorganic shale with nanoscale pores at laboratory condition and reservoir condition was proposed based on the fractal scaling theory and modified non-slip boundary of continuity equation in the nanotube. The model not only considers the gas slippage in the entire Knudsen regime, multilayer sticking (near-wall high-viscosity water) and the quantified thickness of water film, but also combines the real gas effect and stress dependence effect. The presented model has been validated by various experiments data of sandstone with microscale pores and bulk shale with nanoscale pores. The results show that: (1) The Knudsen diffusion and slippage effects enhance the gas relative permeability dramatically; however, it is not obviously affected at high pressure. (2) The multilayer sticking effect and water film should not be neglected: the multilayer sticking would reduce the water relative permeability as well as slightly decrease gas relative permeability, and the film flow has a negative impact on both of the gas and water relative permeability. (3) The increased fractal dimension for pore size distribution or tortuosity would increase gas relative permeability but decrease the water relative permeability for a given saturation; however, the effect on relative permeability is not that notable. (4) The real gas effect is beneficial for the gas relative permeability, and the influence is considerable when the pressure is high enough and when the nanopores of bulk shale are mostly with smaller size. For the stress dependence, not like the intrinsic permeability, none of the gas or water relative permeability is sensitive to the net pressure and it can be ignored completely.

Directed Self-Assembly of Dipeptide Single Crystal in a Capillary.

Controlled growth of one-dimensional nanostructures is playing a key role in creating types of materials for functional devices. Here, we report procedures for controlled assembly of the dipeptide diphenylalanine (FF) into aligned and ultralong single crystals in a capillary. With the evaporation of solvent, nucleation of the crystal occurred in the confined region, and the crystal grew continuously with a supply of molecules from the concentration gradient system inside the capillary. Based on the "Knudsen regime", an ultralong aligned individual FF single crystal possessing an active optical waveguide property at macroscopic length scale could be obtained. Moreover, capillary is also an effective microdevice to investigate the disassembly process of the FF single crystals. This strategy has potentials to broaden the range of applications of aligned organic nanomaterials.

Analytical expression for the rotational friction coefficient of DLCA aggregates over the entire Knudsen regime

We apply a self-consistent field method (Corson et al. 2017c) to calculate the rotational friction coefficient for fractal aerosol particles in the transition flow regime. Our method considers hydr...

Liquid slip over gas nanofilms

We propose the rarefied gas-cushion model (r-GCM), as an extended version of the gas-cushion model (GCM), to estimate the apparent slip of water flowing over a gas layer trapped at a solid surface. Nanobubbles or gas nanofilms may manifest rarefied gas effects, and the r-GCM incorporates kinetic boundary conditions for the gas component in the slip Knudsen regime. These enable an apparent hydrodynamic slip length to be calculated given the gas thickness, the Knudsen number and the bulk fluid viscosities. We assess the r-GCM through non-equilibrium molecular dynamics (NEMD) simulations of shear-driven liquid flow over an infinite gas nanofilm covering a solid surface, from the gas slip regime to the early transition regime, beyond which NEMD is computationally impractical. We find that, over the flow regimes examined, the r-GCM provides better predictions of the apparent liquid slip, and retrieves both the GCM and the free-molecular behaviour in the appropriate limits.

Effect of cathode porosity on the Lithium-air cell oxygen reduction reaction – A rotating ring-disk electrode investigation

The kinetics of the oxygen reduction reaction (ORR) on the practical air cathode in a Lithium-air cell, which is conventionally composed of porous carbon with or without catalysts supported on it, was investigated. The mechanism and kinetics of the oxygen reduction reaction (ORR) was studied on a porous carbon electrode in an oxygen saturated solution of 0.1M Lithium bis-trifluoromethanesulfonimide (LiTFSI) in Dimethoxyethane (DME) using cyclic voltammetery (CV) and the rotating ring-disk electrode (RRDE) technique. The oxygen reduction and evolution reactions were found to occur at similar potentials to those observed on a smooth, planar glassy carbon (GC) electrode. The effect of porosity and the resultant increase in surface area were readily observed in the increase in the transient time required for the intermediates to reach the ring and the much larger disk currents (compared to smooth, planar GC) recorded respectively. The RRDE data was analyzed using a kinetic model previously developed by us and the rate constants for the elementary reactions were calculated. The rates constant for the electrochemical reactions were found to be similar in magnitude to the rate constants calculated for smooth GC disks. The porosity of the electrode was found to decrease the rate of desorption of the intermediate and the product and delay their diffusion by shifting it from a Fickian regime in the electrolyte bulk to the Knudsen regime in the film pores. Thus, it is shown that the effect of the electrode porosity on the kinetics of the ORR is physical rather than electrochemical.

Mean stop paths and diffusion regimes of molecules in one-dimensional zeolite channels

Molecular dynamics methods were used to simulate the diffusion of noble gas molecules, Ne, Ar, and Kr, in the one-dimensional zeolite channels of VET, AFI, VFI, and MCM-41. The gaseous, Knudsen, and configurational diffusion regimes were concisely described by a diffusion regime map. The boundary between the gaseous and the Knudsen diffusion regimes was determined by the probability, Pmm, of bimolecular collision, which was a function of both diameter ratio (dc/dm) and volume density ρV. For a molecule in a channel surrounded by other molecules, the concept of mean stop path (MSP) was proposed as the average distance between collisions, either with another molecule or with the channel wall. The MSP depends on both the mean free path and the difference in channel and molecular diameters (dc-dm). A theoretical model for calculating MSP of noble gas molecules in one-dimensional zeolite channels was derived based on the additivity of diffusion resistances. The mean stop path can describe the characteristics of molecular motion in a zeolite channel in the same way that mean free path does in an infinite space.

Simulation approach of atomic layer deposition in large 3D structures

We present a new simulation method predicting thicknesses of thin films obtained by atomic layer deposition in high aspect ratio 3D geometries as they appear in MEMS manufacturing. The method features a Monte-Carlo computation of film deposition in free molecular flow, as well as in the Knudsen and diffusive gas regime, applicable for large structures. We compare our approach to analytic and simulation results from the literature. The capability of the method is demonstrated by a comparison to experimental film thicknesses in a large 3D structure. Finally, the feasability to extract process parameters, i.e. sticking coefficients is shown.

Thermal diode in gas-filled nanogap with heterogeneous surfaces using nonequilibrium molecular dynamics simulation

A thermal diode serves as a basic building block to design advanced thermal management systems in energy-saving applications. However, the main challenges of existing thermal diodes are poor steady-state performance, slow transient response, and/or extremely difficult manufacturing. In this study, the thermal diode is examined by employing an argon gas-filled nanogap with heterogeneous surfaces in the Knudsen regime, using nonequilibrium molecular dynamics simulation. The asymmetric gas pressure and thermal accommodation coefficients changes are found due to asymmetric adsorptions onto the heterogeneous nanogap with respect to the different temperature gradient directions, and these in turn result in the thermal diode. The maximum degree of diode (or rectification) is Rmax ∼ 7, at the effective gas-solid interaction ratio between the two surfaces of ε* = 0.75. This work could pave the way to designing advanced thermal management systems such as thermal switches (transistors).

Phonon focusing and electron–phonon drag in semiconductor crystals with degenerate charge-carrier statistics

We study the effect of anisotropy in elastic properties on the electron–phonon drag and thermoelectric phenomena in gapless semiconductors with degenerate charge-carrier statistics. It is shown that phonon focusing leads to a number of new effects in the drag thermopower at low temperatures, when diffusive phonon scattering from the boundaries is the predominant relaxation mechanism. We analyze the effect of phonon focusing on the dependences of the thermoelectromotive force (thermopower) in HgSe:Fe crystals on geometric parameters and the heat-flow directions relative to the crystal axes in the Knudsen regime of the phonon gas flow. The crystallographic directions that ensure the maximum and minimum values of the thermopower are determined and the role of quasi-longitudinal and quasi-transverse phonons in the drag thermopower in HgSe:Fe crystals at low temperatures is analyzed. It is shown that the main contribution to the drag thermopower comes from slow quasi-transverse phonons in the directions of focusing in long samples.