Showing papers on "Knudsen number published in 1992"

Slip Length in a Dilute Gas

Abstract: We study the phenomenon of slip length using molecular dynamics and direct simulation Monte Carlo simulations of a dilute gas. Our work extends the range of Knudsen numbers that have been previously studied. Bhattacharya and Lie suggest a logarithmic dependence of slip length on Knudsen number. By a simple redefinition of the mean free path, we obtain good agreement between simulation results and Maxwell theory for slip length. The anomalies seen by Bhattacharya and Lie appear to be due to their definition of the mean free path

Numerical analysis of steady flows of a gas evaporating from its cylindrical condensed phase on the basis of kinetic theory

Abstract: Steady evaporating flows from a cylindrical condensed phase in an infinite expanse of its vapor gas are investigated numerically on the basis of the Boltzmann–Krook–Welander equation. Not only the mass flow rate and the energy flow rate from the cylinder, but also the local variables of the gas over the whole flow field are obtained for a wide range of the Knudsen number and the pressure ratio, which is defined by the pressure at infinity divided by the saturation gas pressure at the temperature of the condensed phase. The acceleration of gas flows, especially to a supersonic flow, near the cylinder and the deceleration to the stationary state at infinity are clarified. The discontinuity of the velocity distribution function in the gas, a typical behavior of a gas around a convex body, is analyzed accurately with the difference scheme devised for this purpose, and its relation to the S layer [Phys. Fluids 16, 1422 (1973)] is discussed.

On the need for and the possibility of a unified description of kinetic and hydrodynamic processes

Abstract: The aim of the paper is to demonstrate the possibility of a unified description of kinetic and hydrodynamic processes on the basis of a generalized kinetic equation without the use of perturbation theory with respect to the Knudsen number The derivation of the generalized kinetic equation is based on a concrete definition of a continous medium in the kinetic and hydrodynamic description of nonequilibrium processes in a Boltzmann gas and in a fully ionized plasma The concept of a “point” of a continuous medium is introduced through the definition of corresponding physically infinitesimally small volumes On this basis we also give a definition of a Gibbs ensemble to the description of nonequilibrium processes in statistical theory Besides the usual “collision integral,” which takes into account the dissipation through the redistribution of the particles with respect to the velocities, the generalized kinetic equation contains in the case of the physical definition of the “continuous medium” an additional term of diffusion type For this reason, it becomes possible to describe kinetic and hydrodynamic processes at all admissible Knudsen numbers Boltzmann'sH theorem is proved for the generalized kinetic equation The entropy production is determined by the sum of two positive contributions, which are due, respectively, to the redistribution of the particles in the velocity space and in ordinary space The entropy flux also consists of two terms, one proportional to the entropy and one proportional to the entropy gradient The presence of the second term makes it possible to give a general definition of a heat flux for arbitrary Knudsen numbers For small Knudsen numbers and slow processes, it reduces to Fourier's law The equations of gas dynamics follow from the generalized kinetic equation without the use of perturbation theory with respect to the Knudsen number They take into account not only processes of viscosity and heat conduction but also self-diffusion The region of applicability of the equations of gas dynamics is discussed Generalized kinetic equations are obtained for the distribution functions of the states of the electrons and ions of a partly ionized plasma Kinetic equations for active media, and also in the theory of Brownian motion, are discussed

Gaseous mixture slit flow at intermediate Knudsen numbers

Abstract: The problem of a two‐dimensional binary gaseous mixture slit flow is investigated on the basis of numerical calculations of the linearized kinetic model proposed by McCormack [Phys. Fluids 16, 2095 (1973)]. The mass flow rate and the field flow of both components are calculated for several Knudsen numbers between 10 and 0.1. Kinetic coefficients satisfying Onsager’s reciprocity relations are found.

Computation of transition and molecular diffusivities in fibrous media

Abstract: The tortuosities of fibrous media in the heretofore unexplored transition and ordinary regimes are computed using a Monte Carlo scheme based on the Einstein equation for random walkers. The model structure is that of fully penetrable cylinders (FPC) in a unit simulation volume. The mean square displacement technique is combined with the first passage time distribution to accelerate the progress of the walkers at low Knudsen number. The results include the computation of transition regime transport coefficients for the first time. The calculated ordinary tortuosities are approximately equal to the reciprocal of the porosity over a wide range, while the transition tortuosities are shown to deviate from the reciprocal porosity with a simple dependence on Knudsen number. The limits of the transition regime are shown to correspond roughly to Knudsen numbers of 0.50 and 100, respectively. The calculated Knudsen tortuosities are shown to improve on earlier results obtained by the authors using a flux-based technique.

Knudsen layers from a computational viewpoint

Abstract: Boundary layer equations arising in the kinetic theory of gases are solved approximately by a simple variational procedure. This method is inspired from Pomraning's computations for radiative transfer problems.

Prediction of in-flight particle parameters during plasma spraying of ceramic powders

Abstract: A theoretical prediction model is presented to estimate the in-flight velocity, temperature, and size of a ceramic particle traversing through a plasma flame. The model accounts for the various phenomena that can influence the transport rate calculations in plasma spraying operations, which typically involve very fine particles (<50 μm) subjected to an extremely non-isothermal environment. The mathematical formulation simultaneously considers internal heat conduction in particles, accounts for the steep temperature gradients that prevail in plasma–particle systems, and incorporates the Knudsen discontinuum effects on both heat and momentum transfer. The significance of these factors is illustrated through some example calculations performed under typical plasma spraying conditions. Comparison with experimental data reveals that the present method, although simple and easy to use, enables accurate predictions to be made and can be a very useful tool in the model assisted development of various plas...

The moment method and rarefied gas flow in channels. General relations

Abstract: The relation between mass and heat fluxes and the distribution function moments at the channel walls have been established by an example of pure gas flow in a plane channel. The bulk and the Knudsen contributions to the fluxes have been separated. Phenomenological equations of non-equilibrium thermodynamics for boundary fluxes valid for arbitrary Knudsen numbers have been obtained. The validity of the Onsager relations for coefficients both in classical and in boundary phenomenological equations of non-equilibrium thermodynamics has been shown.

Hypersonic rarefied flow about a delta wing - direct simulation and comparison with experiment

Abstract: Three-dimensional simulations of hypersonic rarefied flow about a delta wing are made using the direct simulation Monte Carlo (DSMC) method of Bird, and the results of the computations are compared with recent experimental data obtained in a vacuum wind tunnel at the DLR in Gottingen, Germany. The present study considers Mach 8.89 nitrogen flow for a range of conditions that include Knudsen numbers of 0.016 to 3.505 for an incidence angle of 30 deg, and angles of incidence of 15 to 60 deg for a constant Knudsen number of 0.389. The calculations provide details concerning the flowfield structure and surface quantities. Comparisons between the calculations and the available experimental measurements are made for aerodynamic and overall heat-transfer coefficients and recovery temperature. The agreement between the measured and calculated data are very good, well within the estimated measurement uncertainty. Comparisons are also made with modified Newtonian and free-molecule theories.

Thermophoresis in gases at small Knudsen numbers

Abstract: Theoretical and experimental studies characterizing the status of the science of thermophoresis in gases at small Knudsen numbers (relative to the dimensions of the object) are analyzed. The known modifications of the theory of thermophoresis amount to an approximation linear in the Knudsen number, while the zero-order approximation corresponds to the result of Epstein. Analysis of the experimental data on thermophoresis of aerosols of high heat conductivity shows that, among all the experiments described in the literature, not one satisfies the requirements that allow one to perform a correct comparison with the theory. Also the results of experiments with aerosols (objects) of low heat conductivity are unsatisfactory. An original method of studying thermophoresis in gases is proposed.

Transport and Related(Gas and Vapour) Separation in Ceramic Membranes

Abstract: The main transport mechanisms in pores with a size of 3 to 100 nm are Knudsen and surface diffusion. According to Knudsen diffusion the transport of gases is inversely proportional to the square root of the molar mass. Surface diffusion is an additional mechanism to Knudsen diffusion and decreases, in most cases, the separation possibilities. Only surface diffusion of hydrogen (on metals) can increase the separation factors. Multilayer adsorption and capillary condensation can be very selective mechanisms for the separation of condensable gases from gas mixtures in pores of 2 to 10 nm. The application is limited because of the temperature/pressure regime which is needed for condensation and which may lead to technical inconveniences. In pores smaller than 2 nm and in zeolites micropore diffusion occurs. Transport of gases is activated with activation energies smaller than 25 kJ/mol for amorphous microporous systems. In silica-type of microporous toplayers H2,He and CO2 are transported much faster than any other type of gas. At 200°C a separation factor of over 200 is found for the mixture H2/C3H6. The transport mechanism in amorphous microporous materials is still under discussion. In this contribution the mentioned transport mechanisms will be discussed and their possibilities in gas and vapour separation will be illustrated.

Three dimensional particle simulation of high altitude rocket plumes

Abstract: The interaction of two nozzles exhausting into vacuum generates a complex three-dimensional shock structure. The shock structure and resulting plume flowfield is characterized by the nozzle separation distance. For the appropriate range of penetration Knudsen numbers, the analysis of this shock structure can be suitably accomplished through a Monte Carlo simulation. This paper describes the application of a general three-dimensional Monte Carlo simulation on the Connection Machine CM-2 to the analysis of the plume self-interaction shock in the near field. Results are presented for two cases, corresponding to a small and a large nozzle separation distance. The results correctly reproduce the expected flow features and demonstrate the ability of this method to properly simulate the start of the plume self-interaction shock. This has significance not only for allowing analysis of the self-interacting plume in the near field, but also for allowing the subsequent simulation of the far field flow through the use of a continuation downstream exit boundary.

Kinetic theory based nonlinear treatment for motions of a binary gas mixture involving slightly strong evaporation and condensation processes: a system of macroscopic equations, the boundary conditions and the Knudsen-layer corrections

Abstract: An analysis for the steady behaviour of a binary mixture of a vapour and an inert gas in a general domain has been carried out based on the Boltzmann equation of BGK type subject to the diffusive boundary condition under the following situations: i) the amount of inert gas is dense enough so that the mean free path of the inert gas molecules is small compared with the characteristic length of the system. However, the collision between the molecules of the vapour and those of the inert gas is so frequent that the mean free path of the vapour molecules is comparable to that of the inert gas molecules; hence, the mean free path of the vapour molecules is adopted for the definition of the Knudsen number of the system, which is small compared with unity; and ii) the deviation of the system from a reference stationary equilibrium state is small, but its magnitude is of the order of the Knudsen number. In this case, the problem becomes nonlinear, and the kinetic equation and its boundary condition cannot be linearized. Since the Mach number of the system is a measure of the degree of the deviation of the system and is proportional to the Knudsen number times the Reynolds number, the Reynolds number in the present case is finite. By the singular perturbation method, we have derived, as an asymptotic solution to the Boltzmann equation of BGK type for small Knudsen numbers, the macroscopic equations governing the fluid dynamic quantities and the appropriate boundary conditions for them at the interface together with the Knudsen-layer corrections near the interface up to the second order of approximation in the analysis. The equations obtained are essentially of Navier-Stokes type at each order; the macroscopic boundary conditions are those of no slip and no jump at the first order of approximation but, at the second order, they are slip and jump conditions and comprise several terms, each of which is expressed by one of the set of solutions of the macroscopic equations at the previous approximation times aconstant; the Knudsen-layer corrections are of the same type as the boundary conditions with theconstant replaced by a rapidly decreasingfunction of the distance from the interface. Theseconstants andfunctions are universal in the sense that they are totally independent of the geometry of the problems. The derived system of the macroscopic equations and boundary conditions makes possible at the level of ordinary fluid dynamics the treatment of various flow problems of a binary gas mixture, giving an adequate description of the behaviour of the mixture and its component gases for moderate values of the concentration of inert gas. The vapour-mass transfer for these values of the concentration is quite small because it is governed by the diffusional ability of the vapour through the inert gas.

Perspectives on hypersonic viscous and nonequilibrium flow research

Abstract: An attempt is made to reflect on current focuses in certain areas of hypersonic flow research by examining recent works and their issues. Aspects of viscous interaction, flow instability, and nonequilibrium aerothermodynamics pertaining to theoretical interest are focused upon. The field is a diverse one, and many exciting works may have either escaped the writer's notice or been abandoned for the sake of space. Students of hypersonic viscous flow must face the transition problems towards the two opposite ends of the Reynolds or Knudsen number range, which represents two regimes where unresolved fluid/gas dynamic problems abound. Central to the hypersonic flow studies is high-temperature physical gas dynamics; here, a number of issues on modelling the intermolecular potentials and inelastic collisions remain the obstacles to quantitative predictions. Research in combustion and scramjet propulsion will certainly be benefitted by advances in turbulent mixing and new computational fluid dynamics (CFD) strategies on multi-scaled complex reactions. Even for the sake of theoretical development, the lack of pertinent experimental data in the right energy and density ranges is believed to be among the major obstacles to progress in aerothermodynamic research for hypersonic flight. To enable laboratory simulation of nonequilibrium effects anticipated for transatmospheric flight, facilities capable of generating high enthalpy flow at density levels higher than in existing laboratories are needed (Hornung 1988). A new free-piston shock tunnel capable of realizing a test-section stagnation temperature of 10(exp 5) at Reynolds number 50 x 10(exp 6)/cm is being completed and preliminary tests has begun (H. Hornung et al. 1992). Another laboratory study worthy of note as well as theoretical support is the nonequilibrium flow experiment of iodine vapor which has low activation energies for vibrational excitation and dissociation, and can be studied in a laboratory with modest resources (Pham-Van-Diep et al. 1992).

Kinetic analysis of the processes of evaporation and condensation on a surface

Abstract: Evaporation and condensation processes on a plane surface are analyzed for the one-dimensional steady-state case. Previous work is reviewed, starting with that of Knudsen. The results obtained in the analysis of high-rate condensation by solving the Boltzmann kinetic equation with the method of moments are presented. The region of the existence of supersonic strong condensation modes and the interrelation between density, temperature and vapor flow velocity for subsonic flows are determined

Reactions at the gas-phase-porous-solid interface: Mechanisms and effects of surface geometry around room temperature

Abstract: The luminescence quenching of Ru(bpy)2+3 by molecular oxygen on a porous silica and controlled porous glass was studied at the 253–353 K temperature range. The reaction was observed to take place by a mixture of an Eley-Rideal (ER) and a Langmuir-Hinshelwood mechanism. The relative contribution of the two mechanisms was calculated and the ER component was found to dominate at higher temperatures. The lack of effect of the average pore diameter (apd) on the rate of the quenching reaction supports our contention that the Knudsen diffusion controlled reaction theory developed elsewhere is incorrect. In addition we show that, for porous solids, diffusion controlled reaction theory is not applicable in principle below the Knudsen regime.

The Finite Pointset Method for hypersonic flows in the rarefied gas regime

Abstract: There seems to be no doubt any longer, that it is not recommendable to use the Navier-Stokes equation for the description of the flow around aspace vehicle in altitudes above 80 km. One has to go one step up in the hierarchy and use a kinetic equation, which holds even if the gas is far away from the thermodynamic equilibrium.

Fluid dynamic limit for a modified Broadwell system

Abstract: This report discusses a new approach to the resolution of the fluid dynamic limit for the Broadwell system modeling gas dynamics. The main idea is to replace the Knudsen number e in the Broadwell model by et , t the time variable to obtain self similar solutions in ξ=x/ t and then let e → 0+ . The limit thus obtained is a solution of the Riemann problem for the fluid dynamic limit equations.

Direct simulation of low-density flow over airfoils

Abstract: The low-density aerodynamics about airfoils in the transition region is considered using a direct simulation Monte Carlo method. Numerical results are presented for two airfoils (the NACA-0009 and a 9% thick, circular- arc airfoil) traveling at Mach 4 and 5, at an angle of attack of 1.25 deg and altitudes 56 and 62 miles above sea level. The flow, having a Knudsen number range between 0.47-1.15 and a Reynolds number range between 15-114, departs considerably from the continuum theory. Results indicate that both lift and drag are very much penalized in the transition region. In contrast with previous work on slightly rarefied flow with smaller Knudsen numbers (slip flow regime), the effect of rarefaction becomes much more dominant than that due to viscosity. URRENT interest in high-altitude flight has prompted new exploration of low-density aerodynamics. Unfor- tunately, the theoretical models and computational capabil- ities developed to date have proven insufficient to fully char- acterize these flows. The difficulty has to do with the high degree of rarefaction of the flow. The low-density flows of interest occur in a region between the continuum flow region and the molecular flow region. In this "transition region," the concept of transport coefficients (which are the basis for the Navier-Stokes equations) becomes invalid. Thus, the tra- ditional Navier-Stokes analysis ceases to give accurate results. The objective of this research is to investigate the flow physics of high-altitude flight and to develop a new computational scheme for assessing vehicle performance in this important flow region. In order to achieve sufficient lift for level flight, a high- altitude vehicle must fly at high speed. A typical flight en- velope would include freestream Mach numbers between 3 and 15, at an altitude of 30-60 miles above sea level. This flight regime, although considerably beyond the capability of existing aircraft, lies within the flight envelope of the National Aerospace Plane (X-30). The aerodynamic challenge is to achieve sufficiently high-lift coefficients, at the prescribed al- titudes but at a relatively low Mach number, so that certain hypersonic flow problems (such as heat transfer, chemical reactions, and surface catalytic activities) can be circum- vented. In contrast with continuum flow, transitional flow requires a higher-order approximation to the Boltzmann equation. Nu- merical solution to the Boltzmann equation consists of 1) evaluation of the collision integral; and 2) integration of the differential equation. Although the integral form of the col- lision term causes much of the mathematical difficulty in solv- ing the Boltzmann equation, the use of the velocity space coordinates as independent variables in partial differential equations requires large amounts of computer time and stor- age. The advert of high-speed computers has spurred the development of numerical solutions to the Boltzmann equa- tion for several basic, one-dimensional cases.1

Direct Monte Carlo Simulations of Hypersonic Low-Density Flows about an ASTV Including Wake Structure

Abstract: Results of a numerical study concerning flow past a 70-deg blunted cone in hypersonic low-density flow environments are presented using the direct simulation Monte-Carlo method. The flow conditions simulated are those that can be obtained in existing low-density hypersonic wind tunnels. Results indicate that a stable vortex forms in the near wake at and below a freestream Knudsen number (based on cone diameter) of 0.01 and the size of the vortex increases with decreasing Knudsen number. The base region of the flow remains in thermal nonequilibrium for all cases considered herein.

Thermal desorption in the pressure gap and at high pressure using a Micro Knudsen Effusion Cell

Abstract: A new method and apparatus are described for performing thermal desorption at relatively high adsorbate pressure over single crystals and over porous adsorbents and catalysts. The apparatus, a Micro Knudsen Effusion Cell, encloses a sample in a small chamber or cell and allows only a limited pumping speed from the cell. Adsorbate pressures from ultrahigh vacuum to industrial reaction pressures are attainable depending on the ratio of surface area to pumping speed.

Dipolar Energy Transfer and Knudsen Diffusion inside Disordered Porous Media: Interfacial Geometry and Chord Distributions

Abstract: The interfacial geometry of a disordered porous medium strongly influences Knudsen diffusion of gases and interfacial dipolar energy transfer. Some interesting comparison can be made between these two transport processes, involving statistical properties of chords belonging either to the solid matrix or to the pore network. A quantitative analysis of these two mechanisms is proposed, based on a chord distribution model. In a first part, we discuss how chord distribution functions contribute to the stastistical characterization of a porous medium. We show that a direct connection between imaging techniques and small angle scattering provides, in many cases, a reliable description of theses functions. In a second part, interfacial direct energy transfer is analyzed. This one step excitation transfer strongly relies on the interfacial autocorrelation function % MathType!MTEF!2!1!+- % feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaqGgpWdamaaDaaaleaapeGaae4uaaWdaeaapeGaaGOmaaaakiaa % cIcacaqGYbGaaiykaaaa!3BB8! ${{\varphi }}_{\text{S}}^2({\text{r}})$ . An analytic expression of % MathType!MTEF!2!1!+- % feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaqGgpWdamaaDaaaleaapeGaae4uaaWdaeaapeGaaGOmaaaakiaa % cIcacaqGYbGaaiykaaaa!3BB8! ${{\varphi }}_{\text{S}}^2({\text{r}})$ is given and theoretical predictions compared with available experiments. In a third part and following the seminal work of Derjaguin on the Knudsen diffusion, we critically examine how the self diffusion coefficient can be related to the two first moments of the pore chord distribution. A direct comparison with experimental results and numerical simulations is presented in the case of a model porous medium: the random packing of hard spheres. Finally, we analyze a trapping reaction where an excited gas molecule, diffusing in the Knudsen regime, relaxes primarily by wall effects. We show how the chord distribution of the pore network permits to compute the survival probability of the tagged molecules. Two situations are more closely analyzed: the strong and the very weak wall quenching efficiency.

Numerical investigations in the backflow region of a vacuum plume

Abstract: The objective of this research is to numerically simulate the vacuum plume flow field in the backflow region of a low thrust nozzle exit. In space applications, the low thrust nozzles are used as a propulsion device to control the vehicle attitude, or to maneuver the vehicle flight trajectory. When the spacecraft is deployed in the orbit or cruising in a planetary mission, the vacuum plume is created behind the nozzle exit (so called backflow region), by the exhausting gas of the propulsion system or by venting internal gas to the extremely low density ambient. The low density vacuum plume flow regions cover the continuum, transitional and free molecular flow regimes, which were characterized by the Knudsen number K(sub n), K(sub n) = lambda(sub m)/L where lambda(sub m) is the mean free path of the gas molecules and L is the characteristic length of the flow field. The transitional regime is defined by 0.01 is less than or equal to K(sub n) is less than or equal to 10. The conventional Navier-Stokes equations are valid only in the flow region close to the nozzle exit since the validity of the Navier-Stokes equations fails asymptotically as the Knudsen number increases. The vacuum plume characteristics prediction is primarily a problem of transitional aerodynamics.

Numerical investigations in the backflow region of a vacuum plume

Abstract: The objective of this research is to numerically simulate the vacuum plume flow field in the backflow region of a low thrust nozzle exit. In space applications, the low thrust nozzles are used as a propulsion device to control the vehicle attitude, or to maneuver the vehicle flight trajectory. When the spacecraft is deployed in the orbit or cruising in a planetary mission, the vacuum plume is created behind the nozzle exit (so called backflow region), by the exhausting gas of the propulsion system or by venting internal gas to the extremely low density ambient. The low density vacuum plume flow regions cover the continuum, transitional and free molecular flow regimes, which were characterized by the Knudsen number K(sub n), K(sub n) = lambda(sub m)/L where lambda(sub m) is the mean free path of the gas molecules and L is the characteristic length of the flow field. The transitional regime is defined by 0.01 is less than or equal to K(sub n) is less than or equal to 10. The conventional Navier-Stokes equations are valid only in the flow region close to the nozzle exit since the validity of the Navier-Stokes equations fails asymptotically as the Knudsen number increases. The vacuum plume characteristics prediction is primarily a problem of transitional aerodynamics.