Molecular diffusion processes

In this section we mov6 on to examine the behaviour of an open system in which the transport of reactants and products relies on molecular diffusion processes. The situation we envisage is that of a reaction zone in which various species diffuse and react. Outside this zone there exists an external reservoir in which the reactants have fixed concentrations. The reservoir provides a source of reactants which can diffuse across the boundary into the reaction zone, and either a source or a sink for the intermediates and products. The reaction-diffusion cell is sketched in Fig. 9.2. 

Often the flames are operated at reduced pressure (e.g., 20 Torr) to enhance molecular diffusive processes, leading to thicker flames that are more easily probed. 

Physical transport processes and mixing ratio. The concentration profile of a minor constituent in an atmosphere is often expressed as a mixing ratio by volume or a mole fraction rather than the concentration by atmospheric modelers. Physical transport processes involve vertical and horizontal mixing by turbulence and molecular diffusion. The molecular diffusion process can be ignored in the stratosphere since it is important only above about 40 km. 

The nature of dispersion. The effect which the solid packing has on the flow pattern within a tubular reactor can sometimes be of sufficient magnitude to cause significant departures from plug flow conditions. The presence of solid particles in a tube causes elements of flowing gas to become displaced randomly and therefore produces a mixing effect. An eddy diffusion coefficient can be ascribed to this mixing effect and becomes superimposed on the transport processes which normally occur in unpacked tubes—either a molecular diffusion process at fairly low Reynolds... 

In Section IV, the kinetics and mechanisms of catalytic HDM reactions are presented. Reaction pathways and the interplay of kinetic rate processes and molecular diffusion processes are discussed and compared for demetallation of nickel and vanadium species. Model compound HDM studies are reviewed first to provide fundamental insight into the complex processes occurring with petroleum residua. The effects of feed composition, competitive reactions, and reaction conditions are discussed. Since development of an understanding of the kinetics of metal removal is important from the standpoint of catalyst lifetime, the effect of catalyst properties on reaction kinetics and on the resulting metal deposition profiles in hydroprocessing catalysts are discussed. 

Hydrodemetallation reactions require the diffusion of multiringed aromatic molecules into the pore structure of the catalyst prior to initiation of the sequential conversion mechanism. The observed diffusion rate may be influenced by adsorption interactions with the surface and a contribution from surface diffusion. Experiments with nickel and vanadyl porphyrins at typical hydroprocessing conditions have shown that the reaction rates are independent of particle diameter only for catalysts on the order of 100 /im and smaller (R < 50/im). Thus the kinetic-controlled regime, that is, where the diffusion rate DeU/R2 is larger than the intrinsic reaction rate k, is limited to small particles. This necessitates an understanding of the molecular diffusion process in porous material to interpret the diffusion-disguised kinetics observed with full-size (i -in.) commercial catalysts. 

The balance between conduction and diffusion still operates for a much larger isolated wet object, provided radiation is excluded. This is the basis of the wet bulb thermometer method for measuring humidity. The actual rate of evaporation now is not as simply determined and is influenced by wind. The wet bulb temperature is almost independent of wind condition, owing to a convenient accident. Heat conduction is a diffusion process, and the diffusion coefficient for water vapor in air (0.24 sq. cm./sec.) is numerically close to the diffusion coefficient of temperature in air (thermal conductivity/specific heat = 0.20 sq. cm./sec.). Hence, the exact way in which each molecular diffusion process merges into the more rapid eddy diffusion process is not important because no matter how complex the transition is, it must be quantitatively similar for the two processes. 

It was shown that the principle of aqueous miniemulsions could be transferred to non-aqueous media [45]. Here, polar solvents, such as formamide or glycol, replace water as the continuous phase, and hydrophobic monomers are miniemulsified with a hydrophobic agent, which stabilizes the droplets against molecular diffusion processes. It turned out that steric nonionic surfactants based on poly(ethylene oxide) tails are far more efficient than ionic stabilizers,... 

Fig. 8.10. Grating wavenumber (q) dependence of the TG signals (broken lines) of a 50 pM photlLOV2 solution. The arrow indicates the increase of q. The q1 values are 4.5 x 1010, 7.3 x 1010, 3.4 x 1011, 6.3 X 1011, and 5.3 x 1012 m 2 in the order of the amplitude. The signals representing the molecular diffusion processes are shown, and these signals are normalized at the initial part of the diffusion signal...

The molecular diffusion term is inversely proportional to the flow rate, which means that the slower the flow rate, the longer component stays in the column and the molecular diffusion process has more time to broaden the peak. 

Absolute reaction rates can be affected by molecular diffusion processes that dictate the rates at which collisional encounter complexes occur before reaction. This affect usually shows up in the way reaction rates depend on the physical form of the reactants (gas, liquid, solid, solution, etc.), particularly on concentrations for reactants in gas or hquid phases. Adsorption of reactants onto surfaces can enhance the effective concentrations of reactive species and/or reduce the dimensionahty of the diffusion process. Classic work by Eigen and Richter (14) showed how restricting diffusion to one or two dimensions can dramatically increase potential reaction rates, and this principle has been applied to the kinetics of protein translocation along DNA chains, for example. See References 15 and 16 for more information. 

Mass transfer in real absorption equipment resembles a molecular diffusion process only in the basic idea of a concentration difference driving force. However, the two-film theory of Whitman can be used to construct a model similar in many respects to molecular diffusion equations. Fig. 1 is a schematic representing the Whitman two-film theory ... 

Having made the appropriate simplifications to the hydrodynamics, the relevant differential equations describing the molecular diffusion process in the diffusion layer may now be solved. The diffusion process is fully determined by... 

Alternatively an equation of the general form [l] can be obtained by treating penetration analogous to a molecular diffusion process (15,16). In this case the properties of the pore system and penetrating liquid are incorporated into a diffusion coefficient. However on the basis of experimental data obtained for one-dimensional flow in paper (17) there is still a need to develop more realistic models for the pore geometry. Despite this and primarily because swelling is a diffusion process,... 

A modified kinetic model based on the Rabinowitch [144] approach, taking into account the diffusion phenomena including the molecular diffusion processes and molecular size distribution, has been found to describe the conversion profile of Zn-catalyzed dicyanate cure for the entire range [98]. The average dif-fusivity decreased by several orders of magnitude during cure. The Rabinowitch model explains the diffusional limitations in reactions of small molecules as... 

The axial dispersion parameter T) accounts for mixing by both molecular diffusion processes and turbulent eddies and vortices. Because these two types of phenomena are to be characterized by a single parameter, and because we force the model to fit the form of Pick s law of diffusion,... 

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