Free Molecular Regime Research Articles

Power laws for the establishment of a stationary concentration in a binary solution droplet growing in a vapor-gas medium

The kinetic laws are found for establishing a stationary concentration in solution droplets growing under diffusion and free-molecular regimes in a vapor-gas medium composed of two condensing vapors and a passive gas. Parameters of these laws are related to the thermodynamic and kinetic characteristics of condensing substances.

Model of thermal conductivity in powder beds

Heat transfer between contacting solid particles in gas is localized about the points of contact because of the low thermal conductivity of gas. This suggests the model of a network of discrete thermal resistances. Thermal interaction of two particles in the zone of contact between them includes thermal conductivity in the solid phase as well as heat transfer through a gas gap between the particles, where the conductive heat transfer at higher distances from the center of the contact changes to the transition transfer and free molecular transfer near the contact center. In the framework of this approach the effective thermal conductivity depends on three dimensionless morphological parameters: the volume fraction of solid phase, the mean coordination number, and the ratio of the size of necks between particles to the size of particles. The physical properties of the phases are specified by the thermal conductivities of the solid and the gas, the adiabatic exponent of the gas, and the Knudsen number. The proposed model agrees with the experimental S curves of the effective thermal conductivity versus the logarithm of gas pressure. It also describes the experimental tendencies of increasing the effective thermal conductivity with the particle size and the volume fraction of solid. The model can be applied to powder beds with micron-sized particles as well as to packed beds with millimeter-sized particles in gas. Free molecular and transient regimes of heat transfer in the gas filling pores can be important even for millimeter-sized particles at atmospheric pressure when the Knudsen number is as low as ${10}^{\ensuremath{-}5}$. These rarefied gas phenomena are responsible for such a complicated behavior of the considered heterogeneous media. Their quantitative description gives the model, which does not contain fitting parameters.

Blunted-Cone Heat Shields of Atmospheric Entry Vehicles

Spacecraft entry into the atmosphere of a planet requires protection against the extreme temperatures that result from aerodynamic heating. This is normally achieved through use of a heat shield, which also provides the necessary aerodynamic braking and stability. The shape of the heat shield used varies considerably between spacecraft, and spherical and blunted-cone geometries are often employed. Theblunted-cone' heat shield has been developed through experimental design and computational simulation. Here, we demonstrate that this generic shape can be derived mathematically and yields the maximum stabilizing aerodynamic torque of all possible shapes (1). The derived single shape is universal, depending only on the center-of- mass, and provides invariance in static stability due to minor heat shield damage. The derived universal shape is obtained by requiring the aerodynamic torque be insensitive to small shape changes. Importantly, this heat shield shape gives the maximum stabilizing torque in both the continuum and free molecular (FM) regimes. For continuum flows, the obtained shape is always stable, regardless of the position of the center-of-mass. For FM flows, stability is dependent on the nature of the gas interactions with the surface and the position of the center-of-mass - even though the heat shield shape remains unchanged. This establishes that static stability may not be possible for some vehicle configurations, since the solution gives the shape that yields the maximum torque and this is unstable in some cases. Importantly, the design of practical heat shields involves numerous competing factors, which include the expected heat load and the craft volumetric efficiency, in addition to aerodynamic stability (2, 3). We thus emphasize that the presented results focus on only one component of this multi-objective problem. Nonetheless, the derived shape shows good agreement with the heat shields of previous entry vehicles, a comparison of which shall be given.

The laws of establishing stationary composition in a droplet condensing in a binary vapor–gas environment

It is shown that the mole fractions of components within a droplet growing in an atmosphere of two condensing gases and a carrier gas approach their stationary values with a power-law behavior in time on a large scale and with exponential behavior on a small scale for both diffusion-controlled and free-molecular regimes of isothermal condensation. The parameters of the power and the exponential laws are specified for each regime of binary condensation and are linked to the thermodynamic and kinetic characteristics of condensing vapors and to the stationary mole fractions of the components in a growing binary droplet. The stationary composition of the solution within the droplet is shown to be established at a comparatively small relative increase of the droplet radius. A relaxation equation for the droplet composition at arbitrary initial deviations of mole fractions from their stationary values has been solved, and the limitations on the initial deviations allowing monotonic establishment of stationary composition in solution within a growing droplet have been considered.

Numerical investigation of turbomolecular pumps using the direct simulation Monte Carlo method with moving surfaces

A new approach for performing numerical direct simulation Monte Carlo (DSMC) simulations on turbomolecular pumps in the free molecular and transitional flow regimes is described. The chosen approach is to use surfaces that move relative to the grid to model the effect of rotors and stators on a gas flow. The current article describes the method and compares the results to experimental and theoretical data by Sawada [Bull. JSME 22, 362 (1979)]. The agreement between our results and Sawada’s results are excellent.

Squeeze film damping in the free molecular flow regime with full thermal accommodation

We introduce an analytical model for the gas damping of a MEMS resonator in the regime of free molecular flow. Driving force in this model is the change in density in the gap volume due to the amplitude of the oscillating microstructure, which is counteracted by the random walk diffusion in the gap that tries to restore the density to its equilibrium value. This results in a complex-valued force that contributes to both the damping as well as the spring constant, depending on the value of ω τ with ω the resonance frequency and τ the random walk diffusion time. The diffusion time is calculated analytically using the model for random walk Brownian motion and numerically by a Monte Carlo simulation of the ballistic trajectories of the molecules following Maxwell–Boltzmann statistics and full thermal accommodation in gas-surface collisions. The model is verified by comparison to accurate data on the pressure dependency of the damping of three MEMS resonators, showing agreement within 10%.

Low pressure evaporative cooling of micron-sized droplets of solutions and its novel applications

For free molecular regime the mathematical model of low pressure evaporative cooling of binary droplets in gas flow is developed. The model includes five ordinary differential equations and takes into account effects such as the release of the latent heat of condensation of both components and the release of the latent heat of dissolution. Simulations were made for weak aqueous solutions of ammonia. It was discovered that compositions of gas flow and the aqueous solution affect the rate of evaporative cooling of droplets. The ratio of mass flow of solution and gas flow is also an important parameter. The cooling rate of such binary droplets can reach the value of about 2 × 10 5 K/s. As first applications we consider the air cooler based on evaporative cooling of droplets. For pressure of 20–80 Torr in aerosol reactor, it is shown that in the cooler with length of about 1 m temperature of air flow may drop to about 10–15 °C. The second application is the formation of nanoparticle in evaporating multicomponent droplet with two volatile components. Simulation was made for aqueous solution of ammonia which is widely used by experimentalists and engineers now. Effects of the number of precursors in droplet and supersaturation in droplet on the final size of nanoparticles were investigated.

Rarefied gas flow through a thin slit into vacuum simulated by the Monte Carlo method over the whole range of the Knudsen number

A rarefied gas flow through a thin slit into vacuum is studied on the basis of the direct simulation Monte Carlo method. The mass flow rate and flow field are calculated over the whole range of the gas rarefaction from the free-molecular regime to the viscous one. A comparison to other results on the same problem available in literature is performed. An interpolating formula for the reduced flow rate is obtained.

Compact model of squeeze-film damping based on rarefied flow simulations

A new compact model of squeeze-film damping is developed based on the numerical solution of the Boltzmann kinetic equation. It provides a simple expression for the damping coefficient and the quality factor valid through the slip, transitional and free-molecular regimes. In this work, we have applied statistical analysis to the current model using the chi-squared test. The damping predictions are compared with both Reynolds equation-based models and experimental data. At high Knudsen numbers, the structural damping dominates the gas squeeze-film damping. When the structural damping is subtracted from the measured total damping force, good agreement is found between the model predictions and the experimental data.

Taylor-expansion moment method for agglomerate coagulation due to Brownian motion in the entire size regime

Through applying the Taylor-expansion technique to the particle general dynamic equation, the newly proposed Taylor-expansion moment method (TEMOM) is extended to solve agglomerate coagulation due to Brownian motion in the entire size regime. The TEMOM model disposed by Dahneke's solution (TEMOM–Dahneke) is proved to be more accurate than by harmonic mean solution (TEMOM–harmonic) through comparing their results with the reference sectional model (SM) for different fractal dimensions. In the transition regime, the TEMOM–Dahneke gives the more accurate results than the quadrature method of moments with three nodes (QMOM3). The mass fractal dimension is found to play an important role in determining the decay of agglomerate number and the spectrum of agglomerate size distribution, but the effect decreases with decreasing agglomerate Knudsen number. The self-preserving size distribution (SPSD) theory and linear decay law for agglomerate number are only applicable to be in the free molecular regime and continuum plus near-continuum regime, but not perfectly in the transition regime.

On coagulation mechanisms of charged nanoparticles produced by combustion of hydrocarbon and metallized fuels

The contributions of van der Waals, Coulomb, and polarization interactions between nanometersized particles to the particle coagulation rate in both free-molecular and continuum regimes are analyzed for particle charges of various magnitudes and signs. Analytical expressions are obtained for the coagulation rate constant between particles whose interaction in the free-molecular regime is described by a singular potential. It is shown that van der Waals and polarization forces significantly increase the coagulation rate between a neutral and a charged particle (by a factor of up to 10) and can even suppress the Coulomb repulsion between like-charged particles of widely different sizes.

Computation of TPB length, surface area and pore size from numerical reconstruction of composite solid oxide fuel cell electrodes

A numerical technique for generating the solid oxide fuel cell composite electrode structure based on measurable starting parameters has been developed. This method can be used to calculate all relevant electrode microstructural parameters. The structures are formed of randomly distributed overlapping spheres with particle size distributions that match the particle sizes of actual ceramic powders. In this work, the triple phase boundary length, internal surface area, pore size and the percolation of all relevant phases within the porous composite electrode were computed. It was found that the triple phase boundary length computed for particles with real particle size distributions was as much as 40% lower compared to mono-sized particles with the same mean particle sizes. Percolation thresholds were in the range of 30–40 vol% for solid phases and over 99% of pores were found to belong to a percolating network at porosities as low as 25%. In addition, the average pore size was found to be 0.19 μ m for typical electrode compositions, smaller than what is commonly assumed and in certain circumstances, the distribution of pore sizes may place gas transport in the free molecular flow regime.

Computer modeling of stationary particles transport in open cylindrical nanosystems by Monte Carlo method

The research carried out allows us to define the escapes of monatomic particles of open cylindrical-type systems in a free molecular flow regime. The calculations have been made using the direct Monte Carlo method. The probabilities of particles escapes and alongside distributions on the walls of the systems and sprayed surfaces have been defined for various particle bounding energy values with systems' walls, different temperatures and various system wall heights ratios to particles sizes. The amount of additional energy taken away by the flowing out particles has been calculated.

Stability Analysis of Beagle2 in the Free-Molecular and Transition Regimes

Covers advancements in spacecraft and tactical and strategic missile systems, including subsystem design and application, mission design and analysis, materials and structures, developments in space sciences, space processing and manufacturing, space operations, and applications of space technologies to other fields.

Pressure-driven diffusive gas flows in micro-channels: from the Knudsen to the continuum regimes

Despite the enormous scientific and technological importance of micro-channel gas flows, the understanding of these flows, by classical fluid mechanics, remains incomplete including the prediction of flow rates. In this paper, we revisit the problem of micro-channel compressible gas flows and show that the axial diffusion of mass engendered by the density (pressure) gradient becomes increasingly significant with increased Knudsen number compared to the pressure driven convection. The present theoretical treatment is based on a recently proposed modification (Durst et al. in Proceeding of the international symposium on turbulence, heat and mass transfer, Dubrovnik, 3-18 September, pp 25-29, 2006) of the Navier-Stokes equations that include the diffusion of mass caused by the density and temperature gradients. The theoretical predictions using the modified Navier-Stokes equations are found to be in good agreement with the available experimental data spanning the continuum, transition and free-molecular (Knudsen) flow regimes, without invoking the concept of Maxwellian wall-slip boundary condition. The simple theory also results in excellent agreement with the results of linearized Boltzmann equations and Direct Simulation Monte Carlo (DSMC) method. Finally, the theory explains the Knudsen minimum and suggests the design of future micro-channel flow experiments and their employment to complete the present days understanding of micro-channel flows.

Non-isothermal flow of rarefied gas through a long pipe with elliptic cross section

A non-isothermal rarefied gas flow trough a long tube with an elliptical cross section due to pressure and temperature gradients is studied on the basis of the S-model kinetic equation in the whole range of the Knudsen number covering both free molecular regime and hydrodynamic one. A wide range of the pipe section aspect ratio is considered. The mass flow rate is calculated as a function of the pressures and temperatures on the tube ends. The thermomolecular pressure effect has been modeled and the coefficient of the thermomolecular pressure difference has been calculated in whole range of the Knudsen number and in wide range of the pipe section aspect ratio.

Gas flow through a slit into a vacuum in a wide range of rarefaction

Gas flow through a two-dimensional slit into a vacuum is investigated by a direct simulation Monte Carlo method. Results for the mass flow rate are obtained as a function of the rarefaction parameter, which is inversely proportional to the Knudsen number. The distributions of density, temperature and mass velocity, and streamlines are presented. In the free molecular flow regime and in the hydrodynamic limit, our results agree with theoretical asymptotes, and in the transition regime, they compare well with numerical simulations by other authors.

Case Studies of Air Filtration at Microscales: Micro- and Nanofiber Media

In this work, 3–D fibrous geometries are developed to resemble the microstructure of spun-bonded and electrospun filters media and used here to simulate their filtration efficiency and pressure drop. For the sake of simplicity, a continuum flow theory was considered to prevail for the case of spun-bonded media (microfiber media) whereas our electrospun media (nanofiber media) were assumed to be in a free molecular flow regime. Our simulations results are in good general agreement with the experimental data. Especially, in predicting media's pressure drop, our results show better predictions when compared to some of the existing models. We also quantitatively demonstrated that by decreasing the fiber diameter, the minimum collection efficiency of the media having identical pressure drops increases. This effect is accompanied by a decrease in the particle diameter associated with these minimum efficiencies – the most penetrating particle diameter. Studying the influence of the gas temperature, we showed that filter's efficiency increases as the gas temperature increases. Conversely, the filter's pressure drop decreases by increasing the gas temperature.

Wafer level sealing characterization method using Si micro cantilevers

In this study, a wafer level sealing characterization method using the pressure dependence of the mechanical quality factor of Si micro cantilevers has been developed. Since the equation for design of a cantilever near fixed walls in free molecular flow regime has not been derived, a new equation has been proposed. We determine that the minimum measurable pressure necessary for the characterization of MEMS sealing is 0.1 Pa, and the suitable sensor type is the resonant type using the optical measurement method. The devices are fabricated from silicon-on-insulator (SOI) and pyrex glass wafers and have many cavities in which there are three cantilever structures. A measurable pressure range of 10 −2 to 10 3 Pa is achieved. The errors in the quality factors calculated by the proposed equation are influenced by the cantilever dimensions; the energy dissipation of the cantilever (180 μm × 40 μm × 5 μm) is calculated with an error of less than 22%.

Photophoretic Structuring of Circumstellar Dust Disks

We study dust accumulation by photophoresis in optically thin gas disks. Using formulae for the photophoretic force that are applicable for the free molecular regime and for the slip-flow regime, we calculate dust accumulation distances as a function of particle size. It is found that photophoresis pushes particles (smaller than 10 cm) outward. For a Sun-like star, these particles are transported to 0.1-100 AU, depending on their size, and form an inner disk. Radiation pressure pushes small particles (≲1 mm) out further to form an extended outer disk. Consequently, an inner hole opens inside ~0.1 AU. The radius of the inner hole is determined by the condition that the mean free path of the gas molecules equal the maximum size of the particles that photophoresis effectively works on (100 μm-10 cm, depending on the dust properties). The dust disk structure formed by photophoresis can be distinguished from the structure of model gas-free dust disks, because the particle sizes in the outer disk are larger, and the inner hole radius depends on the gas density.