Coefficients of internal diffusion

Kinetic studies of ion exchange on partially ion-exchanged type A zeolites of Mg Ca and Mn " revealed that mini-mums and maximums characterize the differential coefficients of internal diffusion for every exchange of 2 Na " ions for one divalent cation per unit cell of the zeolite. On the basis of these observations, assuming definite interactions between the cations and the zeolite lattice, predictions can be made concerning the distribution and arrangement of cations in the unit cells of a type A zeolite. Research on liquid phase adsorption of n-alkanes on partially ion-exchanged type A zeolites indicated that the differential diffusion coefficients for alkane adsorption are influenced likewise by cation distribution in the unit cells of the zeolite. 

Figure 1. Coefficients of internal diffusion in exchange between divalent cations and ions on type A zeolite...

The effective diffusion coefficients were calculated from the experimentally observed data (time, amount of cation exchanged, temperature), using Paterson s solution of Fick s second law, or published approximate solutions (8, 16). Taking into consideration particle shape and particle size distribution, the differential coefficients of internal diffusion in ion exchange can be ascertained by a method previously described (9). 

But cation diffusion always can occur by the interchange mechanism, whereas in pore—fissure diffusion it depends on the presence of unoccupied cation positions in a certain number and a definite distribution. Accordingly, the coefficients of internal diffusion are maximal when both diffusion mechanisms are operating at the same time, and minimal when diffusion is occurring only by the interchange mechanism. 

Figure 3. Coefficients of internal diffusion in adsorption of n-decane from n-decane-toluene solution by type A magnesium zeolites and calcium zeolites at a zeolite uptake of qt in grams n-decane/grams zeolite = 0.120...

Inspection of Fig. 15.3 reveals that while for jo 0.1 nAcm , the effectiveness factor is expected to be close to 1, for a faster reaction with Jo 1 p,A cm , it will drop to about 0.2. This is the case of internal diffusion limitation, well known in heterogeneous catalysis, when the reagent concentration at the outer surface of the catalyst grains is equal to its volume concentration, but drops sharply inside the pores of the catalyst. In this context, it should be pointed out that when the pore size is decreased below about 50 nm, the predominant mechanism of mass transport is Knudsen diffusion [Malek and Coppens, 2003], with the diffusion coefficient being less than the Pick diffusion coefficient and dependent on the porosity and pore stmcture. Moreover, the discrete distribution of the catalytic particles in the CL may also affect the measured current owing to overlap of diffusion zones around closely positioned particles [Antoine et ah, 1998]. 

Under these conditions, the reaction rate is first order with respect to sulfur dioxide and the rate coefficient per unit volume of catalyst, including the effect of internal diffusion, is equal to kvs = 0.3 m3/m3 s. 

Other results also confirm the important role of internal diffusion. Experimental activation energies (67—75 kJ mol"1) of the sucrose inversion catalysed by ion exchangers [506—509] were considerably lower than those of a homogeneously catalysed reaction (105—121 kJ mol"1) [505, 506,508] and were close to the arithmetic average of the activation energy for the chemical reaction and for the diffusion in pores. The dependence of the rate coefficient on the concentration in the resin of functional groups in the H+-form was found to be of an order lower than unity. A theoretical analysis based on the Wheeler—Thiele model for a reaction coupled with intraparticle diffusion in a spherical bead revealed [510,511] that the dependence of the experimental rate coefficient on acid group concentration should be close to those found experimentally (orders, 0.65 and 0.53 for neutralisation with Na+ and K+ ions respectively [511] or 0.5 with Na+ ions [510]). 

Kholyavenko and Rubanik (143) investigated the effect of internal diffusion on the ethylene oxidation rate, using the diaphragm method, and calculated the effective diffusion coefficients for ethylene and carbon dioxide diffusing through a silver diaphragm. 

The basic requirement in any study of internal diffusion is an understanding of the various modes of transport in a straight capillary bulk, Knudsen, configurational, and surface. Then this knowledge is extended to diffusion in the porous matrix of a pellet to formulate expressions for an effective diffusion coefficient. We confine our treatment in this book to listing in Table 7.3 the more important equations for direct use in estimation. Given in this table are the equations for bulk (for macropores with > 200 A), Knudsen (for micro-... 

Since in the macroscale model, the reaction rate and diffusion coefficient are effective ones that are obtained on an ensemble-averaged basis, the internal diffusion will not appear in the controlling equations explicitly. The effective reaction rate already includes the influence of internal diffusion inside catalyst pellets. The external mass transfer term, which mainly accounts for the species transport outside catalyst pellets, is used in the controlling equations in macroscale models. So, the diffusion mentioned in macroscale model normally represents species diffusion outside catalyst pellets. In fluidized bed, species diffusion is closely related to the flow regime in the reactor (Abba et al., 2003). Abba et al. (2003) summarized the formulae for calculating diffusion coefficients in different flow regimes in fluidized bed. 

There are many factors influencing internal diffusion, such as the size of particles and pore of catalysts, molecular diffusion coefficient, temperature, pressure and other parameters in reaction kinetics etc. Among these factors, the size of catalyst particles and reaction temperature are the most important and easily adjustable parameters. The estimation and elimination of internal diffusion effect can usually use the ways as follows ... 

Obviously, system pressure has serious influence on internal diffusion resistance. Firstly, molecular diffusion coefficient decreases proportionally with the increase of pressure. Secondly, high pressure leads to the enhancement of productivity of catalysts which would lead to the increase of internal diffusion resistance. 

Perron, J. R. (1990). Diffusion coefficients of internal states for the calculation of thermal conductivity. Physica, A 166, 25-231. 

Here Pf and are the fluid and solid densities, and is a volumetric mass transfer coefficient for internal diffusion based on the "long-time solution" of Pick s equation. A relation due to Glueckauf augments the factor of 10 by 50% to account for short-time contact, and this relation is commonly used in percolation processes ... 

It should be kept in mind that all transport processes in electrolytes and electrodes have to be described in general by irreversible thermodynamics. The equations given above hold only in the case that asymmetric Onsager coefficients are negligible and the fluxes of different species are independent of each other. This should not be confused with chemical diffusion processes in which the interaction is caused by the formation of internal electric fields. Enhancements of the diffusion of ions in electrode materials by a factor of up to 70000 were observed in the case of LiiSb [15]. 

The experimental value for Agl is 1.97 FT cirT1 [16, 3], which indicates that the silver ions in Agl are mobile with nearly a thermal velocity. Considerably higher ionic transport rates are even possible in electrodes, by chemical diffusion under the influence of internal electric fields. For Ag2S at 200 °C, a chemical diffusion coefficient of 0.4cm2s, which is as high as in gases, has been measured... 

Checking the absence of internal mass transfer limitations is a more difficult task. A procedure that can be applied in the case of catalyst electrode films is the measurement of the open circuit potential of the catalyst relative to a reference electrode under fixed gas phase atmosphere (e.g. oxygen in helium) and for different thickness of the catalyst film. Changing of the catalyst potential above a certain thickness of the catalyst film implies the onset of the appearance of internal mass transfer limitations. Such checking procedures applied in previous electrochemical promotion studies allow one to safely assume that porous catalyst films (porosity above 20-30%) with thickness not exceeding 10pm are not expected to exhibit internal mass transfer limitations. The absence of internal mass transfer limitations can also be checked by application of the Weisz-Prater criterion (see, for example ref. 33), provided that one has reliable values for the diffusion coefficient within the catalyst film. 

Further to aspects discussed in Section 23.5.1.2, the simplest strategy, if it is practical, is slow decompression. If the pressure can be reduced sufficiently slowly, the gas which is dissolved in the elastomer can diffuse out of the mbber without buUding up sufficient internal pressure to cause damage. The rate of decompression required may be roughly calculated if the diffusion coefficient of the gas and its solubility in the elastomer are known— but in practice it will probably require a laboratory simulation. 

Adsorption equilibrium of CPA and 2,4-D onto GAC could be represented by Sips equation. Adsorption equilibrium capacity increased with decreasing pH of the solution. The internal diffusion coefficients were determined by comparing the experimental concentration curves with those predicted from the surface diffusion model (SDM) and pore diffusion model (PDM). The breakthrough curve for packed bed is steeper than that for the fluidized bed and the breakthrough curves obtained from semi-fluidized beds lie between those obtained from the packed and fluidized beds. Desorption rate of 2,4-D was about 90 % using distilled water. 

Information about the kinetics of interconversion of the species in Scheme 12 has been obtained (Smith et al., 1981). The values of the rate coefficients for external protonation of ii to give io+ and o+o+ are probably close to the diffusion-controlled limit. However, the rate of internal monoprotonation of ii to ii+ is quite low and the reaction can be followed by observing the change in nmr signals with time. At pH 1 and 25°C the half-life is 7 min. Under these conditions, insertion of the second proton into the cavity takes several weeks to reach completion, but can be observed in convenient times at higher... 

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