Gas mean free path

The mean free path, A is another fundamental scale necessary for describing the dynamical properties of gases. Mean free path is defined as the average distance traveled by the molecule between two successive collisions. Kinetic theory of gases establishes the following two expressions for the mean free path of gases. 

For a flying height around 2 nm, collisions between the molecules and boundary have a strong influence on the gas behavior and lead to an invalidity of the customary definition of the gas mean free path. This influence is called a "nanoscale effect" [46] and will be discussed more specifically in Chapter 6. 

Zhang, Y. R. and Meng,Y. G., Direct Simulation Monte Carlo [32] Method on Nanoscale Effect of Gas Mean Free Path," Lubr. 

The light-scattering experiment showed that the particle size was on the order of 50 nm, significantly smaller than the gas mean-free path. In this limit the thermophoretic velocity is [136]... 

Low-pressure impactors utilize the fact that Cc is a function of not only particle diameter but also, through the gas mean free path, pressure. Therefore at low pressures Cc can be substantially larger than for the same size particle at atmospheric pressure. 

The Knudsen number, Kn is the ratio of the gas mean free path, X, to the characteristic dimension in the flow field, D, and, it determines the degree of rarefaction and the degree of the validity of the continuum approach. As Kn increases, rarefaction become more important, and eventually the continuum approach breaks down. The following regimes are defined based on the value of Kn. ... 

Fig. 31. Heat transfer coefficient between the wafer backside and the electrode for different He gas pressures as a function of the gap width. The point on each isobaric curve gives the gap width which is equal to the corresponding gas mean free path. After [163].

Knudsen number - ratio of gas mean free path (A.) to a characteristic dimension (d) Kn... 

Unfortunately, even though (9.5) provides valuable insights into the dependence of X.BB on the gas concentration and molecular size, it is not convenient for the estimation of the mean free path of a pure gas, because one needs to know the diameter of the molecule ob, a rather ill-defined quantity as most molecules are not spherical. To make things even worse, the mean free path of a gas cannot be measured directly. However, the mean free path can be theoretically related to measurable gas microscopic properties, such as viscosity, thermal conductivity, or molecular diffusivity. One therefore can use measurements of the above gas properties to estimate theoretically the gas mean free path. For example, the mean free path of a pure gas can be related to the gas viscosity using the kinetic theory of gases... 

Gas transport in nano-confinements can significantly deviate from the kinetic theory predictions due to surface force effects. Kinetic theory-based approaches based on the assumption of dynamic similarity between nanoscale confined and rarefied flows in low-pressure environments by simply matching the Knudsen and Mach numbers are incomplete. Molecular dynamics simulations of nanoscale gas flows in the early transition and free-molecular flow regimes reveal that the wall force field penetration depth should be considered as an important length scale in nano-confined gas flows, in addition to the channel dimensions and gas mean free path. 

This entry presents the deviations of nanoscale confined shear-driven gas flows from kinetic theory predictions. Subsequently, results proved that the dynamic similarity between the rarefied and nanoscale confined gas flows is incomplete. Importance of wall force field effects is clearly indicated, and the wall force field penetration depth is introduced as an important length scale in addition to the channel dimension and gas mean free path in nano-confined gas flows. 

Nanoaerosol particles interact with the carrier gas molecules and consequently affect their dynamics. Nanoaerosol particles are small enough to approach the mean free path of air, which is about 67 nm under standard conditions. For nanoaerosol the continuum assumption is no longer valid and can attain free molecular flow there is a noncontinuum interaction between the particles and the carrier gas. The corresponding slipping effect is quantified by the Cunningham coefficient in terms of gas mean free path (A) and particle diameter ... 

The critical value S of S, above which impaction arises depends on the geometrical and fluid dynamic parameters of the impactor, and takes typically values near 0.1. S can be determined by calibration, so that the experiment conditions at which the transition is observed for a well defined cluster do fix tp in equation 11. Accordingly, impactors measure the product mpD, which is also proportional to the ratio between mass and drag. For objects much smaller than the gas mean-free-path, tp is proportional to the ratio of mp over the cluster s cross sectional area. In the... 

Note that the kinematic viscosity, speed of sound, and gas mean free path are related. That is,... 

For very small particles, when the particle size becomes comparable with the gas mean free path, slip occurs and the expression for drag must be modified accordingly. Cunningham obtained the needed correction to the Stokes drag force ... 

Therefore, it is necessary to establish the thermodynanuc flow regime in a specimen while analyzing the gas flow through fibrous media. This can be done by calculating an empirical parameter known as the Knudson number Kn, which is defined as the ratio of the gas mean-free path to the fiber radius and is expressed mathematically as shown by Eqn (8.4). 

The gas-mean free path A can be calculated based on the thermodynamic properties of the permeating gas at test conditions as shown in Eqn (8.5). 

The gas mean free path for oxygen denoted by A was calculated using values of oxygen properties available in literature and listed in Table 8.19, and the value was used to calculate the Knudsen number for each test sample. 

We are now going to use this distribution fiinction, together with some elementary notions from mechanics and probability theory, to calculate some properties of a dilute gas in equilibrium. We will calculate tire pressure that the gas exerts on the walls of the container as well as the rate of eflfiision of particles from a very small hole in the wall of the container. As a last example, we will calculate the mean free path of a molecule between collisions with other molecules in the gas. 

If these assumptions are satisfied then the ideas developed earlier about the mean free path can be used to provide qualitative but useful estimates of the transport properties of a dilute gas. While many varied and complicated processes can take place in fluid systems, such as turbulent flow, pattern fonnation, and so on, the principles on which these flows are analysed are remarkably simple. The description of both simple and complicated flows m fluids is based on five hydrodynamic equations, die Navier-Stokes equations. These equations, in trim, are based upon the mechanical laws of conservation of particles, momentum and energy in a fluid, together with a set of phenomenological equations, such as Fourier s law of themial conduction and Newton s law of fluid friction. When these phenomenological laws are used in combination with the conservation equations, one obtains the Navier-Stokes equations. Our goal here is to derive the phenomenological laws from elementary mean free path considerations, and to obtain estimates of the associated transport coefficients. Flere we will consider themial conduction and viscous flow as examples. 

At the present time there exist no flux relations wich a completely sound cheoretical basis, capable of describing transport in porous media over the whole range of pressures or pore sizes. All involve empiricism to a greater or less degree, or are based on a physically unrealistic representation of the structure of the porous medium. Existing models fall into two main classes in the first the medium is modeled as a network of interconnected capillaries, while in the second it is represented by an assembly of stationary obstacles dispersed in the gas on a molecular scale. The first type of model is closely related to the physical structure of the medium, but its development is hampered by the lack of a solution to the problem of transport in a capillary whose diameter is comparable to mean free path lengths in the gas mixture. The second type of model is more tenuously related to the real medium but more tractable theoretically. 

---