Knudsen theory

For pores having a mean radius between 10 and l(h m, we explain the porous body flow by the Knudsen theory here the diffusion coefficient and, consequently, the flow, strongly depend on... 

According to the Hertz-Knudsen theory, the drop of vapor pressure depends on the rate of evaporation v (cm/s) ... 

Though by no means a complete theory, this is at least a reasonable explanation of the Knudsen minimum, and it then remains to explain why the minimum is not observed for flow through porous media. Pollard and Present attributed this to the limited length/diameter ratio of the channels in a typical porous medium and gave a plausible argument in favor of this view. 

At very low densities It Is quite easy Co give a theoretical description of thermal transpiration, alnce the classical theory of Knudsen screaming 9] can be extended to account for Che Influence of temperature gradients. For Isothermal flow through a straight capillary of circular cross-section, a well known calculation [9] gives the molar flux per unit cross-sectional area, N, In the form... 

Pressure drop and heat transfer in a single-phase incompressible flow. According to conventional theory, continuum-based models for channels should apply as long as the Knudsen number is lower than 0.01. For air at atmospheric pressure, Kn is typically lower than 0.01 for channels with hydraulic diameters greater than 7 pm. From descriptions of much research, it is clear that there is a great amount of variation in the results that have been obtained. It was not clear whether the differences between measured and predicted values were due to determined phenomenon or due to errors and uncertainties in the reported data. The reasons why some experimental investigations of micro-channel flow and heat transfer have discrepancies between standard models and measurements will be discussed in the next chapters. 

We consider the problem of liquid and gas flow in micro-channels under the conditions of small Knudsen and Mach numbers that correspond to the continuum model. Data from the literature on pressure drop in micro-channels of circular, rectangular, triangular and trapezoidal cross-sections are analyzed, whereas the hydraulic diameter ranges from 1.01 to 4,010 pm. The Reynolds number at the transition from laminar to turbulent flow is considered. Attention is paid to a comparison between predictions of the conventional theory and experimental data, obtained during the last decade, as well as to a discussion of possible sources of unexpected effects which were revealed by a number of previous investigations. 

We begin the comparison of experimental data with predictions of the conventional theory for results related to flow of incompressible fluids in smooth micro-channels. For liquid flow in the channels with the hydraulic diameter ranging from 10 m to 10 m the Knudsen number is much smaller than unity. Under these conditions, one might expect a fairly good agreement between the theoretical and experimental results. On the other hand, the existence of discrepancy between those results can be treated as a display of specific features of flow, which were not accounted for by the conventional theory. Bearing in mind these circumstances, we consider such experiments, which were performed under conditions close to those used for the theoretical description of flows in circular, rectangular, and trapezoidal micro-channels. 

The main aim of the present chapter is to verify the capacity of conventional theory to predict the hydrodynamic characteristics of laminar Newtonian incompressible flows in micro-channels in the hydraulic diameter range from dh = 15 to db = 4,010 pm, Reynolds number from Re = 10 up to Re = Recr, and Knudsen number from Kn = 0.001 to Kn = 0.4. The following conclusions can be drawn from this study ... 

The subject of this chapter is single-phase heat transfer in micro-channels. Several aspects of the problem are considered in the frame of a continuum model, corresponding to small Knudsen number. A number of special problems of the theory of heat transfer in micro-channels, such as the effect of viscous energy dissipation, axial heat conduction, heat transfer characteristics of gaseous flows in microchannels, and electro-osmotic heat transfer in micro-channels, are also discussed in this chapter. 

In bulk diffusion, the predominant interaction of molecules is with other molecules in the fluid phase. This is the ordinary kind of diffusion, and the corresponding diffusivity is denoted as a- At low gas densities in small-diameter pores, the mean free path of molecules may become comparable to the pore diameter. Then, the predominant interaction is with the walls of the pore, and diffusion within a pore is governed by the Knudsen diffusivity, K-This diffusivity is predicted by the kinetic theory of gases to be... 

As described above, the magnitude of Knudsen number, Kn, or inverse Knudsen number, D, is of great significance for gas lubrication. From the definition of Kn in Eq (2), the local Knudsen number depends on the local mean free path of gas molecules,, and the local characteristic length, L, which is usually taken as the local gap width, h, in analysis of gas lubrication problems. From basic kinetic theory we know that the mean free path represents the average travel distance of a particle between two successive collisions, and if the gas is assumed to be consisted of hard sphere particles, the mean free path can be expressed as... 

By comparing the relative magnitude of the mean free path (z) and the pore diameter (27), it is possible to determine whether bulk diffusion or Knudsen diffusion may be regarded as negligible. Using the principles of the kinetic theory... 

According to Knudsen if a small circular orifice of diameter less than the mean free path of the molecules in a container, is opened in the wall of the container to make a connection to a high vacuum surrounding the container, the mass of gas effusing through the orifice, of area A, is given by an equation derived from the kinetic theory, where the pressure is in atmospheres. 

Vaporisation. The maximum theoretical vaporisation rate v (kg/m2 s) from the surface of a pure liquid or solid is limited by its vapour pressure and is given by the Hertz-Knudsen equation1l04, which can be derived from the kinetic theory of gases ... 

In pores that are appreciably smaller than the mean free path, molecules tend to collide with the pore walls rather than with other molecules. Having collided with the wall, the molecules are momentarily retained and then released in a random direction. The coefficient, >, which controls this Knudsen diffusion, considered by Satterheld(31), and in Volume 1, Chapter 3, may be derived from the kinetic theory to give ... 

Knudsen, M. The Kinetic Theory of Gases, Methuen London, 1934. 

The species diffusivity, varies in different subregions of a PEFC depending on the specific physical phase of component k. In flow channels and porous electrodes, species k exists in the gaseous phase and thus the diffusion coefficient corresponds with that in gas, whereas species k is dissolved in the membrane phase within the catalyst layers and the membrane and thus assumes the value corresponding to dissolved species, usually a few orders of magnitude lower than that in gas. The diffusive transport in gas can be described by molecular diffusion and Knudsen diffusion. The latter mechanism occurs when the pore size becomes comparable to the mean free path of gas, so that molecule-to-wall collision takes place instead of molecule-to-molecule collision in ordinary diffusion. The Knudsen diffusion coefficient can be computed according to the kinetic theory of gases as follows... 

Continuum theory applies when the mean free path of the vapor A,- is small compared with the droplet radius, that is, when the Knudsen number Kn is small (Kn = A,/a 1). From the kinetic theory of gases (Jeans, 1954), the mean free path of the vapor in a binary system is given by... 

Halides and Oxyhalides. Molecules of VCI2, prepared by Knudsen cell techniques, have been isolated in solid inert-gas matrices and their i.r. spectra indicate a linear structure. However, similar studies suggested that VF2 molecules are non-linear. The d d spectrum of gaseous VCI2 has been discussed in terms of ligand field theory, and the Tanabe-Sugano matrix for a linear d system presented. ... 

The parameters D and Dk > whether for macro (denoted by subscript m) or for micro (denoted by subscript ju) regions, are normal bulk and Knudsen diffusion coefficients, respectively, and can be estimated from kinetic theory, provided the mean radii of the diffusion channels are known. Mean radii, of course, are obtainable from pore volume and surface area measurements, as pointed out in Sect. 3.1. For a bidisperse system, two peaks (corresponding to macro and micro) would be expected in a differential pore size distribution curve and this therefore provides the necessary information. Macro and micro voidages can also be determined experimentally. 

Let the flow of molecules into the Knudsen cell be F (molecules s l). In the absence of the reactive surface, these molecules are removed when they strike the escape aperture into the mass spectrometer. Let kCM. be the effective first-order rate constant (s ) for escape of the gas from the cell through this orifice, which can be measured experimentally. Alternatively, kcsc can be calculated from kinetic molecular theory since the number of collisions per second, Js, of a gas on a... 

The Knudsen diffusion coefficient may be computed according to the kinetic theory of gases ... 

To use this formula, the assumption has been made that the fuel consists of a binary mixture of hydrogen and water, while the cathodic gas is a binary mixture of oxygen and nitrogen. The diffusion coefficient for binary mixtures D y eff is estimated by the equation proposed by Hirschfelder, Bird and Spotz [12], and the Knudsen diffusion coefficient for species i is given by free molecule flow theory [11], Finally, combining Equations (6.15-6.18) the anodic and the cathodic concentration overvoltages are given by (see also Equations (A3.20) and (A3.21)) ... 

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