Ordinary molecular diffusion

Transfer of products from the interior catalyst pores to the gross external surface of the catalyst by ordinary molecular diffusion and/or Knudsen diffusion. 

Obviously, there will be a range of pressures or molecular concentrations over which the transition from ordinary molecular diffusion to Knudsen diffusion takes place. Within this region both processes contribute to the mass transport, and it is appropriate to utilize a combined diffusivity (Q)c). For species A the correct form for the combined diffusivity is the following. 

A narrow pore-size distribution and a tortuosity factor of three may be assumed. Using the methods suggested by Reid and Sherwood (7), the ordinary molecular diffusivity DAB is found to be 0.150 cm2/sec. 

If the dominant mode of transport within the catalyst pores is ordinary molecular diffusion, the analysis becomes somewhat more complex. The ordinary molecular diffusivity is inversely proportional to the pressure so that in this case... 

If ordinary molecular diffusion is the dominant mass transfer process, the kinetic theory of gases indicates that the diffusivity is proportional to T3/2 and it is easily shown that... 

The only instances in which external mass transfer processes can influence observed conversion rates are those in which the intrinsic rate of the chemical reaction is so rapid that an appreciable concentration gradient is established between the external surface of the catalyst and the bulk fluid. The rate at which mass transfer to the external catalyst surface takes place is greater than the rate of molecular diffusion for a given concentration or partial pressure driving force, since turbulent mixing or eddy diffusion processes will supplement ordinary molecular diffusion. Consequently, for porous catalysts one... 

In Chapter 11, we indicated that deviations from plug flow behavior could be quantified in terms of a dispersion parameter that lumped together the effects of molecular diffusion and eddy dif-fusivity. A similar dispersion parameter is usefl to characterize transport in the radial direction, and these two parameters can be used to describe radial and axial transport of matter in packed bed reactors. In packed beds, the dispersion results not only from ordinary molecular diffusion and the turbulence that exists in the absence of packing, but also from lateral deflections and mixing arising from the presence of the catalyst pellets. These effects are the dominant contributors to radial transport at the Reynolds numbers normally employed in commercial reactors. 

For naphthalene in air the ordinary molecular diffusivity may be evaluated using a pseudo binary approach and the methods outlined in Reid and Sherwood (10). For an inlet temperature of 630 °K and an inlet pressure of 25 psia, the result is... 

The Schmidt and Prandtl numbers must be evaluated in order to be able to determine concentration and temperature differences between the bulk fluid and the external surface of the catalyst. The Schmidt number for naphthalene in the mixture may be evaluated using the ordinary molecular diffusivity employed earlier, the viscosity of the mixture, and the fluid density. 

Macropore diffusion Diffusion in macropores —pores that are large compared with the molecular diameter. Several different mechanisms contribute to macropore diffusion, notably ordinary molecular diffusion in larger macropores at higher pressures or in liquids and Knudsen diffusion in smaller macropores at low pressures. Also referred to as intraparticle diffusion. 

Fick s law of molecular diffusion states that, for a binary mixture of components A and B, the molar flux of component A by ordinary molecular diffusion relative to the molar average velocity of the mixture in the positive z direction, is proportional to the concentration gradient dcA/dz, which is negative in the direction of ordinary molecular diffusion ... 

Why can not atmospheric evaporation rates be calculated with ordinary molecular-diffusion equations ... 

Several possibilities are summarized in Figure 14—30. In the trivial case (case homogeneous mixture, there will be no mass transfer by molecular diffusion or convection since there is no concentration gradient or bulk motion. The next case (case b) corresponds to the flow of a well-mixed fluid mixture tlvrough a pipe. Note that there is no concentration gradients and thas molecular diffusion in this case, and all species move at the bulk flow velocity of Vt The mixture in the thud case (case c) is stationary (17= 0) and thus it corresponds to ordinary molecular diffusion in sfationary mediums, which we discussed before. Note that the velocity of a species at a location in this... 

The theory of seismic pumping proposed by Panov and co-workers at the Donetzk Polytechnical Institute (pers. commun., 1981 Panov et al., 1980) attributes anomalous Rn occurrences over fault zones to micropulsations in active fault zones that increase the emanation efficiency of soils. They base their theory partly on the fact that Tn activities rise simultaneously with Rn activities over active faults and partly on the fact that there are no unusual Ra accumulations in the soils where the anomalous Rn values are observed. The theory is attractive and warrants further investigation. Zverev et al. (1980) have also observed increased Rn and Tn emanation over faults and in laboratory experiments soil samples under treatment with ultrasound do indeed emanate more Rn than samples not so treated. Wilkening (1980) has reviewed the processes by which Rn is transported from the soil to the Earth s surface and concludes that " Rn transport by ordinary molecular diffusion appears to be the dominant process". 

Because of the intricate composition, various transport modes contribute to the supply of O2 to the reaction sites, involving viscous flow, Knudsen diffusion and ordinary molecular diffusion [103]. The relative importance of the distinct mechanisms depends decisively on the psd. Dominating transport though micropores favors Knudsen flow, whereas molecular diffusion with considerably larger diffusion constants will... 

The diffusivity in gas permeation may include ordinary molecular diffusion... 

Ordinary molecular diffusion is generally recognized as the primary mechanism for gas transport in the unsaturated zone (19,, . Fundamental theory for ordinary diffusion according to Fick s first and second laws is presented in (22) more extensive theoretical discussions of gas transport in porous media are presented in (2 ... 

Ordinary molecular diffusion through pores, which present tortuous paths and hinder the movement of large molecules when their diameter is more than 10% of the pore diameter. 

When treating diffusion of solutes in porous materials where diffusion is considered to occur only in the fluid inside the pores, it is common to refer to an effective diffusivity, DABeg, which is based on (1) the total cross-sectional area of the porous solid rather than the cross-sectional area of the pore and (2) on a straight path, rather than the actual pore path, which is usually quite tortuous. In a binary system, if pore diffusion occurs only by ordinary molecular diffusion, Fick s law can be used with an effective diffusivity that can be expressed in terms of the ordinary diffusion coefficient, DAB, as... 

For liquids, the mean free path is commonly a few angstroms, so the Knudsen number is almost always very small and diffusion inside the pores is usually only by ordinary molecular diffusion. In gases, the mean free path can be estimated from the following (Cussler, 1997) ... 

Knudsen and ordinary molecular diffusivities are inversely proportional to the square root of molecular weight. Hence, a fonrfold decrease in the molecular... 

ORDINARY MOLECULAR DIFFUSION IN BINARY AND PSEUDO-BINARY MIXTURES... 

The kinetic theory of dilute gases accounts for collisions between spherical molecules in the presence of an intermolecular potential. Ordinary molecular diffusion coefficients depend linearly on the average kinetic speed of the molecules and the mean free path of the gas. The mean free path is a measure of the average distance traveled by gas molecules between collisions. When the pore diameter is much larger than the mean free path, collisions with other gas molecules are most probable and ordinary molecular diffusion provides the dominant resistance to mass transfer. Within this context, ordinary molecular diffusion coefficients for binary gas mixtures are predicted, with units of cm /s, via the Chapman-Enskog equation (see Bird et al., 2002, p. 526) ... 

If the total pressure p is 1 atm and absolute temperature T is in the vicinity of 298 K, then kinetic theory predicts ordinary molecular diffusivities on the order of 0.1 cm /s, which are comparable to Knudsen diffusivities if the average pore radius is 0.1 xm (i.e., 10 A) and molecular weights are about 50 Da. When pore radii are larger than 1 xm, ordinary molecular diffusion provides the dominant resistance to mass transfer in porous catalysts at standard temperature... 

If an n-component gas mixture (i.e., n >2) can be treated as a pseudo-binary mixture, and the ordinary molecular diffusivity a. mix of component A is desired, then the formalism described above is applicable under the following conditions ... 

Addition of Resistances When Knudsen Diffusion and Ordinary Molecular Diffusion Are Operative in Binary and Multicomponent Gas Mixtures... 

When the average pore radius is between 50 and 10 A, it is necessary to add resistances from Knudsen flow and ordinary molecular diffusion to calculate the net diffusivity of component i in porous pellets. The procedure is (1) illustrated rigorously for binary mixtures, (2) extrapolated to multicomponent mixtures that... 

The net diffusivity of component A within the pores of a catalytic pellet is obtained by adding mass transfer resistances for Knudsen diffusion and ordinary molecular diffusion, where convection reduces the resistance due to ordinary molecular diffusion but Knudsen flow occurs over length scales that are much too small for convective mass transfer to be important. This addition of resistances is constructed to simulate resistances in series, not parallel. Consider the trajectory of a gas molecule that collides with the walls of a channel or other gas molecules. In the pore-size regime where Knudsen and ordinary molecular diffusion are equally probable, these collisions occur sequentially, which suggests that gas molecules encounter each of these resistances in series. Hence, for binary mixtures. 

Since convective mass transfer is negligible in porous catalytic pellets, it is reasonable to let /i = 0 in (21-47) and add the diffusivities inversely. If the pore diameter is much smaller than the mean free path of the gas, then colhsions with the walls are more frequent than collisions with other molecules, and Knudsen diffusion provides the dominant resistance. If the pores are much larger than the mean free path, then collisions with other molecules are more frequent than colhsions with the walls of the channel, and ordinary molecular diffusion provides the dominant resistance. Usually, one arrives at a sum of resistauces in series by following the trajectory of a single gas molecule within a catalytic pore. It is important to emphasize that one obtains the correct result by tracking a single molecule only if there are no other pathways by which diffusion supplies the... 

Comparison between Rigorous and Approximate Calculations Based on Ordinary Molecular Diffusion in Nonreactive Multicomponent Gas Mixtures... 

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