Molecular mean free path

When Che diameter of the Cube is small compared with molecular mean free path lengths in che gas mixture at Che pressure and temperature of interest, molecule-wall collisions are much more frequent Chan molecule-molecule collisions, and the partial pressure gradient of each species is entirely determined by momentum transfer to Che wall by mechanism (i). As shown by Knudsen [3] it is not difficult to estimate the rate of momentum transfer in this case, and hence deduce the flux relations. 

As a consequence of these simple deductions, Graham s experiments c effusion through an orifice came to be regarded as one of the earliest direct experimental checks on the kinetic theory of gases. However, a closer examination of his experimental conditions reveals that this view is mistaken. As mentioned earlier, his orifice diameters ranged upwards from 1/500 in., while the upstream pressure was never very much less thai atmospheric. Under these circumstances the molecular mean free path len ... 

For gas-phase diffusion in small pores at lowpressure, the molecular mean free path may be larger than the pore diameter, giving rise to Knudsen diffusion. Satterfield (Ma.s.s Tran.sfer in Heterogeneous Catalysis, MIT, Cambridge, MA, 1970, p. 43), gives the following expression for the pore dimisivity ... 

Burgdorfer, A., The Influence of the Molecular Mean Free Path on the Performance of Hydrodynamic Gas Lubricated Bearings,"ASMC/. Basic Eng., Vol. 81,1959, pp. 94-100. 

A wavelength (molecular) mean free path ca/cm, equation 9.2-37... 

For all practical purposes, isothermal flows with Ma < 0.2 can be treated as incompressible, i.e., density variations in the fluid around the particle are negligible. Compressibility effects become important as Ma is increased, especially for Ma approaching and in excess of unity. The Knudsen number is defined as the ratio of the molecular mean free path in the fluid to some characteristic particle dimension. For a spherical particle... 

Note that in this procedure the effect of molecular mean free path, that is, of molecular size is neglected. As an example we estimate diffusivity of toluene (Mk) uene = 92 gmol-1) from diffusivity of benzene (Afbenzene = 78 gmol-1) and get Dtoluene a (0.096 cm s-1) [92/ 78]-1/2 = 0.088 cm s-1. The experimental value is 0.086 cm s 1 (Gilliland, 1934). 

In the free molecular flow regime, where the molecular mean free path X is much larger than the orifice diameter, the speed of molecules exiting the reactor can be taken as the mean molecular speed, (v) = SRT/nMW, and the flux, FM, is given by... 

Since the width of this zone is significantly greater than the length of the molecular mean free path, we should speak not of energy transfer by direct impact, but of heat conduction and other dissipative processes in the gas—diffusion and viscosity—related to the gradients of the temperature, concentration and the velocity along the normal to the wave front. 

When the size of the pores is much smaller than the molecular mean free paths in the gas mixture, collisions of gas molecules with the pore wall are more frequent than inter-molecular collisions. This type of gas transport is also known as Knudsen diffusion. In this case each molecule acts independently of all the others and each component in a mixture behaves as though it were present alone. The movement of each molecule can be conveniently pictured as a random walk between the walls of pores. This leads to the following expression for the molar flux of species A ... 

X molecular mean free path, 0.1 JJ.ni for room temperature ambient air m... 

With representative values for A, Cp, and po with Vq 50 cm/s, equation (4) gives S 10 cm. Therefore 5 is large compared with a molecular mean free path (about 10 cm), and the continum equations of fluid dynamics are valid within the deflagration wave but 3 is small compared with typical dimensions of experimental equipment (for example, the diameter of the burner mouth, and hence the radius of curvature of the flame cone, for experiments with Bunsen-type burners), and laminar deflagration waves may be approximated as discontinuities in many experiments. Since equations (3) and (4) imply that 3 at constant temperature, experimental... 

In solid or liquid explosives, reactive molecules are continually interacting, and limitations on detonation structures associated with molecular mean free paths no longer apply. It becomes entirely possible for significant release of chemical energy to occur within the structure of the leading shock. This fact motivates new approaches to studies of detonation structure on the basis of molecular dynamics [189], [190]. Although the fundamental complexities that are encountered make the problem difficult, further pursuit of these lines of investigation seems desirable. 

Unless the pressure is so low that the molecular mean free path (Appendix E) is comparable to the dimensions of the chamber enclosing the gas, the rates of the first and last steps, 1 and 5, are governed by molecular diffusion rates (or by rates of mass transfer through a boundary layer, if the gas is in motion). Although these macroscopic transfer processes, which lie within the realm of fluid mechanics (Appendixes C-E), often are the ratedetermining steps [62] (see also Chapter 12), we shall assume here that they are so fast that they can be neglected. 

Pick diffusivity for species i relative to mixture molecular mean-free-path (we assume Rp /) latent heat of fuel vaporization (per g) interparticle separation dimensionless interparticle separation, L/Rp Lewis number (mass/heat diflFusivity ratio) individual particle gasification rate fuel particle number density total number of particles in cloud total pressure... 

For molecular mean free paths much less than the mean free diameters of the intercrystalline void space in the zeolite bed, Din, is controlled by the same mechanism as in the gas phase, with a self-diffusion coefficient Dg. Due to the steric confinement, Dimer is reduced with respect to Dg by a tortuosity factor, Tb, with values typically of the order of 2-3. The mean free path can be estimated through the relation... 

Experimental results have been published for micro tubes [1], micro channels [2], and micro heat pipe [3]. The micro scale experimental results differ from the prediction of conventional models. Some neglected phenomena must be taken into account in micro scale convection. One of them is the Knudsen number defined as the ratio of the molecular mean free path to characteristic length of the micro conduit. 

In conventional parallel plate channel the intermolecular collisions dominate, because the characteristic length 2 y 1 is much larger than the molecular mean free path. The velocity of the fluid at the surface is zero u[yl]=0. 

In micro parallel plate channel the interactions between the fluid and the wall become significant, because the molecular mean free path is comparable to 2 y 1. The gas slip along the wall with a finite velocity in the axial direction as described by Maxwell in 1890 [13]. The kinetic theory of gases gives the following boundary condition at the surface of the channel [28] ... 

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