A mean free path kinetic theory of void diffusion in a porous medium with surface diffusion. Asymptotic expansion in the Knudsen number

1981 ◽  
Vol 74 (8) ◽  
pp. 4742-4744 ◽  
Author(s):  
Fun Gau Ho ◽  
William Strieder
Keyword(s):  
Porous Medium ◽  
Asymptotic Expansion ◽  
Kinetic Theory ◽  
Free Path ◽  
Surface Diffusion ◽  
Knudsen Number ◽  
Mean Free Path

A Note on Modeling of Nano-Scale Thermal Flow via the Lattice Boltzmann Method

2012 ◽  
Author(s):  
P. Lopez ◽  
Y. Bayazitoglu
Keyword(s):  
Distribution Function ◽  
Free Path ◽  
Knudsen Number ◽  
Reference Data ◽  
Mean Free Path ◽  
Knudsen Layer ◽  
High Order ◽  
Distance Functions ◽  
Transition Flow ◽  
Wall Distance

Lattice Boltzmann (LB) method models have been demonstrated to provide an accurate representation of the flow characteristics in rarefied flows. Conditions in such flows are characterized by the Knudsen number (Kn), defined as the ratio between the gas molecular Mean Free Path ( MFP, λ) and the device characteristic length (L). As the Knudsen number increases, the behavior of the flow near the walls is increasingly dominated by interactions between the gas molecules and the solid surface. Due to this, linear constitutive relations for shear stress and heat flux, which are assumed in the Navier-Stokes-Fourier (NSF) system of equations, are not valid within the Knudsen Layer (KL). Fig. 1 illustrates the characteristics of the velocity field within the Knudsen layer in a shear-driven flow. It is easily observed that although the NSF equations with slip flow boundary conditions (represented by dashed line) can predict the velocity profile in the bulk flow region, they fail to capture the flow characteristics inside the Knudsen layer. Slip flow boundary conditions have also been derived using the integral transform technique [1]. Various methods have been explored to extend the applicability of LB models to higher Knudsen number flows, including using higher order velocity sets, and using wall-distance functions to capture the effect of the walls on the mean free path by incorporating such functions on the determination of the local relaxation parameters. In this study, a high order velocity model which contains a two-dimensional, thirteen velocity direction set (e.g., D2Q13), as shown in Fig. 2, is used as the basis of the current LB model. The LB model consists of two independent distribution functions to simulate the density and temperature fields, while the Diffuse Scattering Boundary Condition (DSBC) method is used to simulate the fluid interaction with the walls. To further improve the characterization of transition flow conditions expected in nano-scale heat transfer, we explored the implementation of two wall-distance functions, derived recently based on an integrated form of a probability distribution function, to the high-order LB model. These functions are used to determine the effective mean free path values throughout the height of the micro/nano-channel, and the resulting effect is first normalized and then used to determine local relaxation times for both momentum and energy using a relationship based on the local Knudsen number. The two wall-distance functions are based on integral forms of 1) the classical probability distribution function, ψ(r) = λ0−1e−r/λ0, derived by Arlemark et al [2], in which λ0represents the reference gas mean free path, and 2) a Power-Law probability distribution function, derived by Dongari et al [3]. Thus, the probability that a molecule travels a distance between r and r+dr between two successive collisions is equal to ψ(r)dr. The general form of the integral of the two functions used can be described by ψ(r) = C − f(r), where f(r) represents the base function (exponential or Power Law), and C is set to 1 so that the probability that a molecule will travel a distance r+dr without a collision ranges from zero to 1. The performance of the present LB model coupled with the implementation of the two wall-distance functions is tested using two classical flow cases. The first case considered is that of isothermal, shear-driven Couette flow between two parallel, horizontal plates separated by a distance H, moving in opposite directions at a speed of U0. Fig. 3 shows the normalized velocity profiles across the micro-channel height for various Knudsen numbers in the transition flow regime based on our LB models as compared to data based on the Linearized Boltzmann equation [4]. The results show that our two LB models provide results that are in excellent agreement with the reference data up to the high end of the transition flow regime, with Knudsen numbers greater than 1. The second case is rarefied Fourier flow within horizontal, parallel plates, with the plates being stationary and set to a constant temperature (TTop > TBottom), and the Prandtl number is set to 0.67 to match the reference data based on the Direct Simulation Monte Carlo (DSMC) method [5]. Fig. 4 shows the normalized temperature profiles across the microchannel height for various Knudsen numbers in the slip/transition How regime. For the entire Knudsen number range studied, our two LB models provide temperature profiles that are in excellent agreement with the non-linear profile seen in the reference data. The results obtained show that the effective MFP relationship based on the exponential function improves the results obtained with the high order LB model for both shear-driven and Fourier flows up to Kn∼1. The results also show that the effective MFP relationship based on the Power Law distribution function greatly enhances the results obtained with the high order LB model for the two cases addressed, up to Kn∼3. In conclusion, the resulting LB models represent an effective tool in modeling non-equilibrium gas flows expected within micro/nano-scale devices.

ASYMPTOTIC ANALYSIS FOR A PARTICLE TRANSPORT EQUATION WITH INELASTIC SCATTERING IN EXTENDED KINETIC THEORY

1998 ◽  
Vol 08 (05) ◽  
pp. 851-874 ◽  
Author(s):  
JACEK BANASIAK ◽  
GIOVANNI FROSALI ◽  
GIAMPIERO SPIGA
Keyword(s):  
Kinetic Theory ◽  
Asymptotic Analysis ◽  
Transport Equation ◽  
Free Path ◽  
Null Space ◽  
Collision Operator ◽  
Mean Free Path ◽  
Molecular Flow ◽  
Asymptotic Equation ◽  
Linear Transport

In this paper we perform the asymptotic analysis for a linear transport equation for test particles in an absorbing and inelastically scattering background, when the excited species can be considered as non-participating. This model is derived in the frame of extended kinetic theory and rescaled with the Knudsen number ∊. After examining the main properties of the collision model and of the scattering operator in the case with an infinite interval of energy as well as the case with a finite interval, the modified (compressed) Chapman–Enskog expansion procedure is applied to find the asymptotic equation for small mean free path. A specific feature of this model is that the collision operator has an infinite-dimensional null-space. The main result is that in the small mean free path approximation on [Formula: see text] level we obtain a free molecular flow for a suitable hydrodynamic quantity, rather than the diffusion which is typical for linear transport problems.

scholarly journals Conduction of heat through rarefied gases

1910 ◽  
Vol 83 (562) ◽  
pp. 254-264 ◽  
Keyword(s):  
Kinetic Theory ◽  
Free Path ◽  
Mean Free Path ◽  
Thermal Method ◽  
Rarefied Gases ◽  
Dewar Flask ◽  
Narrow Tube ◽  
Rate Of Cooling ◽  
Rate Of Evaporation

In a previous paper we used a thermal method of indicating the degree of vacuum by measuring the rate of evaporation of liquid air in a Dewar flask exhausted in various ways. In the present work we have attempted to obtain information of the conduction of heat through twelve gases at pressures so low that the actual path of the molecule is comparable with its mean free path. It is to be expected that this condition will hold good over a range of pressure the greater the smaller the diameter of the containing vessel, and for this reason we worked in a long narrow tube. Previous investigations of this character have been carried out by Sir William Crookes and C. F. Brush, by measuring the rate of cooling of heated mercury thermometers placed inside globes exhausted by the Sprengel pump. The observations of the latter bear most closely upon the present work, and partly anticipate them. He points out that in the five gases he examined, at pressures up to a few millionths of an atmosphere, the heat-transmitting power of the gas varies directly as the pressure. This is to be expected from the kinetic theory, as pointed out by Smoluchowski de Smolan.

scholarly journals The motion of electrons in gases

1923 ◽  
Vol 103 (720) ◽  
pp. 53-57 ◽  
Keyword(s):  
Kinetic Theory ◽  
Electric Field ◽  
Free Path ◽  
Mean Free Path ◽  
Theoretical Formula ◽  
Kinetic Theory Of Gases ◽  
Thermal Agitation ◽  
Mean Velocity ◽  
Gas Molecules

The velocity ( v ) of an electron in a gas, due to an electric field of strength X, is given approximately by theoretical formula v = 0·815 X e λ/ m V. where e denotes the charge on the electron, λ its mean free path, m its mass, and V its mean velocity of thermal agitation. Townsend has made many determinations of this velocity v , and also of V, in several gases at different pressures ( p ) and finds that v is a function of X/ p , and that the values of λ given by the above equation are of the same order, in most cases, as those deduced from the viscosity by means of the kinetic theory of gases. The equation v = 0·815X e λ/ m V is obtained by assuming that there is no persistence of velocities when electrons collide with gas molecules.

Clausius: the kinetic theory of gases

2020 ◽  
pp. 543-551
Author(s):  
Robert T. Hanlon
Keyword(s):  
Heat Capacity ◽  
Kinetic Theory ◽  
Free Path ◽  
Mean Free Path ◽  
Kinetic Theory Of Gases ◽  
Path Distance ◽  
The Mean ◽  
Monatomic Gas

Rudolf Clausius developed the first modern version of the kinetic theory of gases. His derivation provided the means to predict the heat capacity of a monatomic gas and to quantify the mean free path distance traveled by atoms between collisions.

scholarly journals On the calculation of the velocity and temperature distributions for flow along a flate plate

1936 ◽  
Vol 154 (882) ◽  
pp. 364-377 ◽  
Keyword(s):  
Kinetic Theory ◽  
Turbulent Flow ◽  
Free Path ◽  
Mean Free Path ◽  
Mixing Length ◽  
Shearing Stress ◽  
Mean Velocity ◽  
Temperature Distributions ◽  
The Mean

Kármán and Prandtl were the first investigators to publish theoretical ults for problems of turbulent flow involving plane boundaries. Before nsidering any particular problem the general considerations of these iters will be outlined. Prandtl's is, perhaps, the easier method to follow. He considered a bulent motion in which the mean velocity u remains parallel to a tain direction—O x , say,—and is a function of y only, O y being perpendicular to O x , and he arrived at the result τ = ρ l 2 | du / dy | du / dy (1) the shearing stress, where ρ is the density of the fluid and l is a length, led the mixing length; it is the analogue of the mean free path in the etic theory of gases. The conception of the mixing length of the sent problem is physically much less surely grounded than the mean e path of the kinetic theory.

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