Relaxation processes in hypersonic rarefied binary gas mixture flow around a cylinder

Using the direct simulation Monte Carlo method, the hypersonic flow of a binary gas mixture around a cylinder is investigated over a wide rarefaction range: from an almost continuum regime (at the Knudsen number Kn = 0.01) to free-molecular flow. The effect of a small admixture of heavy diatomic particles in a light gas flow on the relaxation processes near the cylinder and the heat flux is studied.     

Stochastic Effects of 2D Random Arrays of Cylinders on Rarefied Gas Permeability Using the Lattice Boltzmann Method

A large set of 2D random arrays of circular cylinders is generated to perform a statistical study on rarefied gas flow through micro-porous media. The flow regimes in this work lie for Knudsen numbers (Kn) ranging from the continuum to the transition regimes. Arrays are built by randomly placing cylinders with constant diameter with a uniform distribution without overlapping, and are generated for three target porosities. Fluid flow is assumed to be incompressible and isothermal. A modified lattice Boltzmann model is adopted to account for discrete effects, with slip-velocity boundary conditions and a Kn-dependent multi-relaxation time collision operator. The apparent permeability is modeled with Darcy’s law with a Klinkenberg-type relationship and compared with existing correlations. Velocity fields highlight the increasing contribution of fluid flow through small pores with increasing Kn. Numerical results show that porous media randomness leads to an uncertainty on rarefied gas permeability calculation despite the same structural characteristics and may not strictly follow a specific correlation. The influence of a local collision operator based on a local Kn instead of a global one in the numerical model is also studied. Results show that the permeability in rarefied regimes undergoes significant deviation when applying the local collision operator compared to the global one. These differences could result from a more accurate capture of the pore-scale behavior with a local Kn. Thus, it emphasizes the sensitivity of the model and the apparent permeability calculation to the appropriate definition of Kn.

     

Noncondensable gas effect on condensation in a separate type two-phase closed thermosyphon

An analytical study is presented of the noncondensable gas effect on vapor condensation in a separate type two-phase closed thermosyphon. The appreciable detrimental effect is observed from the results obtained by investigating condensation of steam-air and steam-hydrogen mixtures. It is shown that the effect of hydrogen on condensation is more remarkable than that of air for the same operating conditions of the thermosyphon. By examining four working temperature levels, the noncondensable gas effect is found to be accentuated at high system pressures. The computed results are all corresponding to the parameter range of engineering interest, and intended to have practical applications to designing separate type heat pipe heat exchangers.Eine analytische Studie des Einflusses eines nicht kondensierbaren Gases auf die Dampfkondensation in einem separaten, geschlossenen Zwei-Phasen-Thermosyphon wird vorgestellt. Der merklich schädliche Einfluß wird beim Untersuchen der Kondensation von Dampf-Luft- und Wasserstoff-Dampf-Gemischen beobachtet. Es wird gezeigt, daß für die gleichen Arbeitsbedingungen des Thermosyphons der Einfluß von Wasserdampf auf die Kondensation beachtlicher ist, als der von Luft. Bei der Untersuchung von vier Arbeitstemperatur-Niveaus wurde herausgefunden, daß der Einfluß des nichtkondensierbaren Gases bei hohen Systemdrücken herausragend ist. Die errechneten Ergebnisse sollen bei der Auslegung von einem separaten geschlossenen Zwei-Phasen-Thermosyphon Verwendung finden.     

Propagation of Elastic Waves in a Gas-Filled Poroelastic Medium: The Influence of the Boundary Conditions

We consider the propagation of elastic waves in gas-filled porous media at small but non-zero values of Knudsen numbers \( {\text{Kn}} \), where \( {\text{Kn}} = \lambda /l \), \( \lambda \) is the mean free path of gas molecules; \( l \) is the characteristic size of inclusion (the so-called slip regime). In this case, it is possible to apply the classic equations of hydrodynamics with modified boundary conditions at solid walls. We have assumed that the gas molecules distribution function is satisfied at the modified Maxwell boundary conditions (Struchtrup 2013; Mohammadzadeh and Struchtrup 2015). We have obtained the expressions for drag and added mass coefficients for the Biot equations of poroelasticity for a system of randomly oriented gas-filled cylindrical capillaries. Our calculations have shown that the drag and added mass coefficients depend considerably on the Knudsen number and the properties of the surface. The influence of the interfacial slip effect on the velocities of the compressional wave of the first kind and shear wave is small, but the velocity and attenuation of the compressional wave of the second kind are considerably influenced by this effect. The results obtained show the fundamental possibility of the determination of the accommodation coefficient by measuring the velocity of the compressional wave of the second kind for different values of the Knudsen number.     

Slow gas motions and the negative drag of a strongly heated spherical particle

In the slow flows of a strongly and nonuniformly heated gas, in the continuum regime (Kn → 0) thermal stresses may be present. The theory of slow nonisothermal continuum gas flows with account for thermal stresses was developed in 1969–1974. The action of the thermal stresses on the gas results in certain paradoxical effects, including the reversal of the direction of the force exerted on a spherical particle in Stokes flow. The propulsion force effect is manifested at large but finite temperature differences between the particle and the gas. This study is devoted to the thermal-stress effect on the drag of a strongly heated spherical particle traveling slowly in a gas for small Knudsen numbers (M ～ Kn → 0), small but finite Reynolds numbers (Re ≤ 1), a linear temperature dependence of the transport coefficients µ ∝ T, and large but finite temperature differences ((T _w ? T _∞ )/T _M8 ～ 1). Two different systems of equations are solved numerically: the simplified Navier-Stokes equations and the modified Navier-Stokes equations with account for the thermal stresses.     

A two-phase boundary-layer model for laminar mixed-convection condensation with a noncondensable gas on inclined plates

A finite-volume-based numerical model for mixed-convection laminar film condensation from a flowing mixture of a vapor and a heavier noncondensable gas on inclined isothermal flat plates is presented. The full boundary layer equations for the liquid film and the vapor-gas mixtures (including liquid inertia and energy convection terms) are solved implicitly with appropriate liquid-mixture interface conditions. Results were obtained for three mixtures, covering wide ranges of liquid Prandtl number and free-stream gas concentration in the forced-convection, mixed-convection and free-convection flow regimes. The effects of liquid inertia were found to be significant only for low-Prandtl-number fluids and lower gas concentrations. The effects of liquid energy convection were found to be significant only for high-Prandtl-number fluids and to be most significant for mixed-convection condensation. Received on 3 March 1998     

Intense evaporation of a gas from a two-dimensional periodic surface

Evaporation (or condensation) of a gas is said to be intense when the normal component of the velocity of the gas in the Knudsen layer has a value of the order of the thermal velocity of a molecule, c_T=(2kT/m)^1/2. In this case the distribution function of the molecules with respect to their velocities in the Knudsen layer differs from the equilibrium (Maxwellian) value by its own magnitude. As a result of this, over the thickness of the Knudsen layer the macroparameters also vary by their own magnitudes. So in order to obtain the correct boundary conditions for the Euler gas dynamic equations, it is necessary to solve the nonlinear Boltzmann equation in the Knudsen layer. The problem of obtaining such boundary conditions for the case of a plane surface was considered in [1–11]. In the present study this problem is solved for a two-dimensional periodic surface in the case when the dimensions of the inhomogeneities are of the order of the mean free path of the molecules and the inhomogeneities have a rectangular shape. The flow in the Knudsen layer becomes two-dimensional, and this leads to a considerable complication of the solution of the problem.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 132–139, March–April, 1985.In conclusion the author would like to express his gratitude to V. A. Zharov for his valuable advice, and also V. S. Galkin, M. N. Kogan, and N. K. Makashev for discussion of the results obtained.     

Analysis of the Onsager Relations by Analytic Methods in Kinetic Gas Theory

The well-known Onsager relations L ₁₂=L ₂₁ are verified within the framework of kinetic gas theory with allowance for the mass and heat fluxes localized in the Knudsen layer. On the basis of an analytic solution of the BGK (Bhatnagar-Gross-Krook) equation, it is shown that the Onsager relations are fulfilled correct to at least exponential corrections (–1/Kn) in the Knudsen number Kn.     

Film condensation of multicomponent mixtures

The classic Nusselt model is generalized for the condensation process of a multicomponent vapor mixture. The condensed vapors may be miscible in their liquid state or contain noncondensable gases.The reduction in the condensation rate owing to the accumulation of a noncondensable gas or the more volatile components near the condensate interface is demonstrated for three component systems of methanol—water—air and acetone—methanol—water. Also the effects of interfacial suction and forced convection are included.The analytical solution incorporates Diffusion Law for a multicomponent system and both exact and approximate integral method solutions are applied. The accuracy of the integral method turns out to be remarkably good.     

Influence of excitation of internal degrees of freedom of molecules on the process of weak evaporation or condensation

A study is made of the process of weak evaporation (or condensation) with allowance for excitation of vibrational and rotational degrees of freedom of diatomic molecules. The solution to the corresponding Knudsen layer problem is obtained on the basis of a model kinetic equation of the type of the Morse equation [1]. A relation is obtained that establishes the connection between the rate of evaporation (or condensation) and the parameters of the surface and the gas above it. The boundary conditions of slip for the equations of gas dynamics are analyzed. The results are compared with the evaporation or condensation in the case of a monatomic gas. The introduction of accommodation coefficients for an evaporating surface is considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6. pp. 98–110, November–December, 1979.     

Gas flow in cylindrical capillaries under intermediate conditions

At present, there are sufficient solutions of the problem of free-molecular gas flow through a short cylindrical channel, for example, [1–3]. In intermediate flow conditions, for Knudsen number Kn 1, solutions have been obtained for the limiting cases: an infinitely long channel [4] and a channel of zero length (an aperture) [5]. However, no solution is known for short channels for Kn 1. The present work reports a calculation by the Monte Carlo method of the macroscopic characteristics of the gas flow through a short cylindrical channel (for various length—radius ratios), taking into account intermolecular collisions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 187–190, January–February, 1977.     

Asymptotic Solutions of the Equations for a Viscous Heat-Conducting Compressible Medium

Small nonstationary perturbations in a viscous heat-conducting compressible medium are analyzed on the basis of the linearization of the complete system of hydrodynamic equations for small Knudsen numbers (Kn ≪ 1). It is shown that the density and temperature perturbations (elastic perturbations) satisfy the same wave equation which is an asymptotic limit of the hydrodynamic equations far from the inhomogeneity regions of the medium (rigid, elastic or fluid boundaries) as M _a = v/a → 0, where v is the perturbed velocity and a is the adiabatic speed of sound. The solutions of the new equation satisfy the first and second laws of thermodynamics and are valid up to the frequencies determined by the applicability limits of continuum models. Fundamental solutions of the equation are obtained and analyzed. The boundary conditions are formulated and the problem of the interaction of a spherical elastic harmonic wave with an infinite flat surface is solved. Important physical effects which cannot be described within the framework of the ideal fluid model are discussed.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, 2005, pp. 76–87.Original Russian Text Copyright © 2005 by Stolyarov.     

The effects of Knudsen-dependent flow velocity on vibrations of a nano-pipe conveying fluid

In this paper, we investigate the effect of nano-flow on vibration of nano-pipe conveying fluid using Knudsen (Kn). We use Euler–Bernoulli plug-flow beam theory. We modify no-slip condition of nano-pipe conveying fluid based on Kn. We define a Kn-dependent flow velocity. We consider effect of slip condition, for a liquid and a gas flow. We reformulate Navier–Stokes equations, with modified versions of Kn-dependent flow velocity. We observe that for passage of gas through nano-pipe with nonzero Kn, the critical flow velocities decreased considerably as opposed to those for zero Kn. This can show that ignoring Kn effect on a gas nano-flow may cause non-conservative design of nano-devices. Furthermore, a more impressive phenomenon happens in the case of clamped-pinned pipe conveying gas fluid. While we do not observe any coupled-mode flutter for a zero Kn, we can see the coupled-mode flutter, accompanying the second-mode divergence, for a nonzero Kn.     

Gas-kinetic theory of lubrication

An equation of the gas-kinetic theory of lubrication is obtained under the assumption of incompressibility of the gas on the basis of solution of the Boltzmann equation by the moment method with a special approximating function. In the limit of a small Knudsen number calculated using the minimal gap, the equation goes over into Reynolds's wellknown equation. Reynolds's problem of a lubricating layer of gas between two closely spaced planes is considered. In the limit of a small Knudsen number, agreement with the well-known solution of the hydrodynamic theory is obtained. A comparison is made with the solution obtained by the hydrodynamic method with slip boundary conditions under neglect of the compressibility of the gas.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 161–166, January–February, 1984.I thank L. P. Smirnov for constant interest in the work, and also the participants of G. I. Petrov's seminar for helpful discussions.     

Effect of vapor condensation on forced convection heat transfer of moistened gas

The forced convection heat transfer with water vapor condensation is studied both theoretically and experimentally when wet flue gas passes downwards through a bank of horizontal tubes. Extraordinarily, discussions are concentrated on the effect of water vapor condensation on forced convection heat transfer. In the experiments, the air–steam mixture is used to simulate the flue gas of a natural gas fired boiler, and the vapor mass fraction ranges from 3.2 to 12.8%. By theoretical analysis, a new dimensionless number defined as augmentation factor is derived to account for the effect of condensation of relatively small amount of water vapor on convection heat transfer, and a consequent correlation is proposed based on the experimental data to describe the combined convection–condensation heat transfer. Good agreement can be found between the values of the Nusselt number obtained from the experiments and calculated by the correlation. The maximum deviation is within ±6%. The experimental results also shows that the convection–condensation heat transfer coefficient increases with Reynolds number and bulk vapor mass fraction, and is 1∼3.5 times that of the forced convection without condensation.     

Asymptotic theory of the Boltzmann system,for a steady flow of a slightly rarefied gas with a finite Mach number: General theory

A steady rarefied gas flow with Mach number of the order of unity around a body or bodies is considered. The general behaviour of the gas for small Knudsen numbers is studied by asymptotic analysis of the boundary-value problem of the Boltzmann equation for a general domain. The effect of gas rarefaction (or Knudsen number) is expressed as a power series of the square root of the Knudsen number of the system. A series of fluid-dynamic type equations and their associated boundary conditions that determine the component functions of the expansion of the density, flow velocity, and temperature of the gas is obtained by the analysis. The equations up to the order of the square root of the Knudsen number do not contain non-Navier–Stokes stress and heat flow, which differs from the claim by Darrozes (in Rarefied Gas Dynamics, Academic Press, New York, 1969). The contributions up to this order, except in the Knudsen layer, are included in the system of the Navier–Stokes equations and the slip boundary conditions consisting of tangential velocity slip due to the shear of flow and temperature jump due to the temperature gradient normal to the boundary.     

Supersonic condensation of a monatomic gas

The problem of steady supersonic condensation of a monatomic gas on a plane evaporating surface is solved in the Knudsen layer by the direct statistical modeling method. The domain of existence of the solution of the problem is determined. The results of calculating the structure of the Knudsen layer near the surface are presented. A topological picture of the solutions of the strong evaporation and subsonic and supersonic strong condensation problems is given as a function of the Mach number, determined from the normal velocity component, and the other governing parameters.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 171–175, May–June, 1990.     

Boundary conditions for the equations of two-temperature gas dynamics of a binary mixture with strongly differing masses of the components

A study is made of the problem of the boundary conditions for the equations of two-temperature gas dynamics for a binary mixture with strongly differing masses when (m/M)^1/2 Kn 1 (Kn is the Knudsen number, m is the mass of the light molecules, and M the mass of the heavy molecules). The flow structure is established at velocities of the light and heavy components of the order of the velocity of sound of the heavy component. The formulation of the boundary conditions for the gas-dynamic equations is investigated. It is shown that the only closed boundary layer theory is Prandtl's theory taking into account terms of order Kn^1/2.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 93–98, September–October, 1982.I thank M. N. Kogan, N. K. Makashev, and E. S. Asmolov for assistance in the work and valuable discussions.     

Boundary kinetic effects in the problem of slow flow over an evaporating heat-conducting spherical particle

It is shown that for sufficiently large values of the thermal conductivity of the condensed phase' as compared with the thermal conductivity of the vapor (/' Kn) the effects associated with the presence of a Knudsen layer on the evaporating surface must be taken into account in order to obtain a solution of the problem of a spherical particle in a slow (Re, 1) continuum (Knudsen number Kn 1) flow of its own vapor. The drag is calculated for various types of boundary conditions on the particle surface.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 134–139, March–April, 1987.In conclusion the authors wish to thank V. S. Galkin and M. N. Kogan for useful discussions.