Coefficient bulk diffusion

The limiting cases of greatest interest correspond to conditions in which the mean free path lengths are large and small, respectively, compared with the pore diameters. Recall from the discussion in Chapter 3 that the effective Knudsen diffusion coefficients are proportional to pore diameter and independent of pressure, while the effective bulk diffusion coefficients are independent of pore diameter and inversely proportional to pressure. 

It may seem curious that Knudsen diffusion coefficients still appear in equations (5.18) and (5.19), which supposedly give the flux relations at the limit of bulk diffusion control. However, inspection reveals that only ratios of these coefficients are effectively present, and from equation (2,11) it follows that... 

There is no doubt that the coefficient of the third term on the right hand side of this equation is much larger than the coefficient of the second term at the bulk diffusion limit, and this justifies our original form (5,29). However, on constructing from and using equation (5,32) and its... 

The first case corresponds to a situation in which all Knudsen diffusion coefficients are equal, and all binary pair bulk diffusion coefficients are equal ... 

The second case follows from a suggestion of Bird [27] to Che effect that binary pair bulk diffusion coefficients might be approximated by expressions of the form... 

Here L is the thickness of the porous septum and jS the length of each dead-end micropore, the effective binary bulk diffusion coefficient... 

The first thing to notice about these results is that the influence of the micropores reduces the effective diffusion coefficient below the value of the bulk diffusion coefficient for the macropore system. This is also clear in general from the forms of equations (10.44) and (10.48). As increases from zero, corresponding to the introduction of micropores, the variance of the response pulse Increases, and this corresponds to a reduction in the effective diffusion coefficient. The second important point is that the influence of the micropores on the results is quite small-Indeed it seems unlikely that measurements of this type will be able to realize their promise to provide information about diffusion in dead-end pores. 

As a particular case of this result, it follows that the stoichiometric relations are always satisfied in a binary mixture at the limit of bulk diffusion control and Infinite permeability (at least to the extent that the dusty gas equations are valid), since then all the binary pair bulk diffusion coefficients are necessarily equal, as there is only one of them. This special case was discussed by Hite and Jackson [77], and the reasoning set out here is a straightforward generalization of their treatment. 

In Chase equaclons Che symbols Z u, A, b and id are noc defined quite as In equations (12.39), since Che bulk diffusion coefficient muse now be replaced by a Knudsen diffusion coefficient. Thus... 

Effective diffusion coefficient, in porous medium at bulk diffusion limit, 14... 

Diffusivity and tortuosity affect resistance to diffusion caused by collision with other molecules (bulk diffusion) or by collision with the walls of the pore (Knudsen diffusion). Actual diffusivity in common porous catalysts is intermediate between the two types. Measurements and correlations of diffusivities of both types are Known. Diffusion is expressed per unit cross section and unit thickness of the pellet. Diffusion rate through the pellet then depends on the porosity d and a tortuosity faclor 1 that accounts for increased resistance of crooked and varied-diameter pores. Effective diffusion coefficient is D ff = Empirical porosities range from 0.3 to 0.7, tortuosities from 2 to 7. In the absence of other information, Satterfield Heterogeneous Catalysis in Practice, McGraw-HiU, 1991) recommends taking d = 0.5 and T = 4. In this area, clearly, precision is not a feature. 

Rate of Formation of Primary Precursors. A steady state radical balance was used to calculate the concentration of the copolymer oligomer radicals in the aqueous phase. This balance equated the radical generation rate with the sum of the rates of radical termination and of radical entry into the particles and precursors. The calculation of the entry rate coefficients was based on the hypothesis that radical entry is governed by mass transfer through a surface film in parallel with bulk diffusion/electrostatic attraction/repulsion of an oligomer with a latex particle but in series with a limiting rate determining step (Richards, J. R. et al. J. AppI. Polv. Sci.. in press). Initiator efficiency was... 

In order to verify the conditions of this averaging process, one has to relate the displacements during the encoding time - the interval A between two gradient pulses, set to typically 250 ms in these experiments - with the characteristic sizes of the system. Even in the bulk state with a diffusion coefficient D0, the root mean square (rms) displacement of n-heptane or, indeed, any liquid does not exceed several 10 5 m (given that = 2D0 A). This is much smaller than the smallest pellet diameter of 1.5 mm, so that intraparticle diffusion determines the measured diffusion coefficient (see Chapter 3.1). This intrapartide diffusion is hindered by the obstades of the pore structure and is thus reduced relative to D0 the ratio between the measured and the bulk diffusion coeffident is called the tortuosity x. More predsely, the tortuosity r is defined as the ratio of the mean-squared displacements in the bulk and inside the pore space over identical times ... 

Here m(r, t) is the relative difference of the longitudinal magnetization M and its equilibrium value Mo m = (M — Mo)/Mo, D is the bulk diffusion coefficient and p is the bulk relaxation rate. The general solution to the Torrey-Bloch equation can be written as... 

The reaction is carried out in close-loop reactor connected to a mass spectrometer for 1S02, 180160 and 1602 analyses as a function of time [38], The gases should be in equilibrium with the metallic surface (fast adsorption/desorption steps 1 and f ) If the bulk diffusion is slow (step 6) and the direct exchange (step 5) does occur at a negligible rate, coefficients of surface diffusion Ds can be calculated from the simple relationship between the number of exchanged atoms Ne and given by the model of circular sources developed by Kramer and Andre [41] ... 

When surface diffusion is the only process of exchange, ag tends to an equilibrium value a at t - oo. In most cases, after a rapid step of surface diffusion, it can be observed that a% continues slowly decreasing. This phenomenon corresponds to a slow step of bulk diffusion (coefficient l)h). A model of bulk diffusion in spherical grains was developed by Kakioka et al. which led to the following equation [43] ... 

Ordinary or bulk diffusion is primarily responsible for molecular transport when the mean free path of a molecule is small compared with the diameter of the pore. At 1 atm the mean free path of typical gaseous species is of the order of 10 5 cm or 103 A. In pores larger than 1CT4 cm the mean free path is much smaller than the pore dimension, and collisions with other gas phase molecules will occur much more often than collisions with the pore walls. Under these circumstances the effective diffusivity will be independent of the pore diameter and, within a given catalyst pore, ordinary bulk diffusion coefficients may be used in Fick s first law to evaluate the rate of mass transfer and the concentration profile in the pore. In industrial practice there are three general classes of reaction conditions for which the bulk value of the diffusion coefficient is appropriate. For all catalysts these include liquid phase reactions... 

To evaluate the bulk diffusion coefficient, using Eq. (5.1) or (5.2), it is necessary to subtract the short-circuit diffusion contribution from the total concentration profile. Ideally, the concentration profile due to bulk diffusion will take on the shape of a bell, gradually flattening as the time of diffusion increases (Fig. 5.5). The solution to the... 

The likelihood that Fick s laws will be obeyed in a crystal containing dislocations is dependent upon the spacing between the defects. Provided that this spacing is much greater than the diffusion length (Dt)1/2, where D is the bulk diffusion coefficient, Fick s laws are obeyed, with an effective (measured) diffusion coefficient, Deff, given by... 

Radioactive 180 was diffused into a poly crystalline pellet of ZnO at 900°C for 48 h. The diffusion profile had a marked tail showing that extensive grain boundary diffusion had occurred. The variation of the concentration of the radioactive isotope with depth for the tail of the penetration profile is given in the following table. Calculate the grain boundary diffusion coefficient, Dgb, of 180 if the bulk diffusion coefficient at 900°C is 5.53 x 10-21 m2 s-1 and the grain boundary width is taken as 1 nm. 

If the transport process is rate-determining, the rate is controlled by the diffusion coefficient of the migrating species. There are several models that describe diffusion-controlled processes. A useful model has been proposed for a reaction occurring at the interface between two solid phases A and B [290]. This model can work for both solids and compressed liquids because it doesn t take into account the crystalline environment but only the diffusion coefficient. This model was initially developed for planar interface reactions, and then it was applied by lander [291] to powdered compacts. The starting point is the so-called parabolic law, describing the bulk-diffusion-controlled growth of a product layer in a unidirectional process, occurring on a planar interface where the reaction surface remains constant ... 

Differences in diffusion coefficients (bulk as well as surface)... 

Figure 28. Svensson s macrohomogeneous model for the i— 1/characteristics of a porous mixed-conducting electrode, (a) The reduction mechanism assuming that both surface and bulk diffusion are active and that direct exchange of oxygen vacancies between the mixed conductor and the electrolyte may occur, (b) Tafel plot of the predicted steady-state i— V characteristics as a function of the bulk oxygen vacancy diffusion coefficient. (Reprinted with permission from ref 186. Copyright 1998 Electrochemical Society, Inc.)...

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