Single-file conditions

Another deviation from the pattern of ordinary diffusion must be expected if the reactant and product molecules are subjected to single-file conditions, i.e. if (i) the zeolite pore system consists of an array of parallel channels and if (ii) the molecules are too big to pass each other. In this case, the molecular mean-square displacement z t)) is found to be proportional to the square root of the observation time, rather than to the observation time itself. First PFG NMR studies of such systems are in agreement with this prediction [8]. By introducing a mobility factor F, in analogy to the Einstein relation for ordinary diffusion. 

The effect of single-file diffusion limitation on the rate of an irreversible first order catalytic reaction was studied both theoretically and experimentally. A rate equation was derived using a relation between the effective diffusion constant and the concentration of adsorbed molecules under single-file conditions which is valid on the time scale of catalytic reactions. The hydroisomerization of 2,2-dimethylbutane on platinum loaded large crystallites of H-Mordenite was used as a test reaction to verify the theoretical results. 

It should be emphasized, however, that likely in none of these studies was the zeolite material of such an ideal structure as implied in data analysis. In this respect, experimental studies with artificially created single-file systems [127-129] may provide a much higher rehability of the pre-supposed structural features. A treatise on the substantial deviations of the real structure, with particular emphasis on the consequences for ideal host systems for single-file diffusion as evidenced by optical techniques, is given in [95]. Irrespective of these limitations, however, a number of peculiarities of catalytic reactions in zeolites with one-dimensional channel systems are most Ukely to be attributed to the special conditions of molecular transport and molecular arrangement imder single-file conditions. 

The depalletizer in a packaging line accepts stacked bottles on pallets that have dividers and discharges one container at a time in single file. Sizing of the depalletizer for unsorted containers must be analyzed with respect to the product mix ratios, type and speed of the filler, and the condition of the pallets, conveyors, and containers. 

It is worthwhile to note that under conditions of single-file diffusion, the definition of the Thiele modulus... 

Experiments with single-file systems of finite extension are thus found to be easily affected by the influence of the boundary conditions. Therefore, one should be aware of the fact that the observed displacements are sufficiently below the hmiting values for which the boimdary conditions start to become relevant. 

Since the two limiting cases of open and closed ends have been shown to lead, respectively, to an enhancement and a reduction of the mean square displacement in comparison to an infinite single-file system, it may be anticipated that, imder the influence of boundary conditions intermediate between these two hmiting cases, molecular propagation in a finite single-file system may even proceed as in a single-file system of infinite extension. 

Eq. 21 with Eq. 23 results as the solution of the corresponding differential equation of normal diffusion with the appropriate initial and boundary conditions. These relations hold with the adequate interpretation of D as a self-diffusivity or a transport diffusivity, respectively, for both tracer exchange between the initially adsorbed species A by species B and the relative uptake in an adsorption experiment. It should be noted that Eq. 21 also describes the molecular uptake by single-file systems, since with respect to adsorption/desorption there are no differences between single-file systems and systems which permit normal diffusion. 

Most importantly, combining the expressions for the intracrystalline mean life time Eq. 23 and the effective self-diffusivity (Eq. 19), in the case of singlefile diffusion the mean time of molecular exchange is found to scale with L, rather than with the dependence typical of normal diffusion. Therefore, under the conditions of single-file diffusion, the exchange rate with zeoUte crystallites decreases even more pronounced with increasing crystal sizes as in the case of normal diffusion. 

SO far only been attained by Monte Carlo simulations. Figure 5 illustrates the situation due to the combined effect of diffusion and catalytic reaction in a single-file system for the case of a monomolecular reaction A B [1]. For the sake of simplicity it is assumed that the molecular species A and B are completely equivalent in their microdynamic properties. Moreover, it is assumed that in the gas phase A is in abimdance and that, therefore, only molecules of type A are captured by the marginal sites of the file. Figure 5 shows the concentration profile of the reaction product B within the singlefile system imder stationary conditions. A parameter of the representation is the probabiUty k that during the mean time between two jump attempts (t), a molecule of type A is converted to B. It is related to the intrinsic reactivity k by the equation... 

In complete agreement with the fact that the product molecules in a singlefile system are prevented from leaving the system by their file neighbors, the concentration profiles in the single-file cases show a much more pronounced tendency of accumulation of the reaction products in the file center than in the case of normal diffusion. Under stationary conditions, the effective reactivity k is related to the intrinsic reactivity k by the equation... 

Fig. 5 Concentration profiles of the molecules of species B within the single-file system under stationary conditions and comparison with the dependence to be expected for ordinary diffusion (broken line, Eq. 29). The quantity 2L(klDy (the Thiele modulus 0) in Eq. 29 has been chosen to coincide with the generalized Thiele modulus (cf. Eq. 30) of the single-file reaction for k = 1.27 x 10 (0 = 2.77). z denotes the distance from the middle of the file and L = NX its length. From [1] with permission...

Thiele moduli. These also appear in the smaller area below the single-file profile curve for /c = 1.27 x 10 in Fig. 5, in comparison to the area under the profile resulting from the conditions of normal diffusion with the same Thiele modulus. By comparing single-file diffusion with ordinary diffusion the prevailing effect is clearly the dramatic enhancement of rintra and hence of the Thiele modulus, leading to a correspondingly dramatic reduction of the effectiveness factor. 

The basis for the national or international standards (lEC, UL, VDE, MDD [the European Medical Device Directive]) is to reduce file risk of hazardous currents reaching the patient under normal conditions. Even under a single fault condition, patient safety shall be secured. 

Exact solutions. It is possible to obtain some exact results for mean residence times even for channels with large numbers of particles although the results are typically cumbersome [90, 91]. Here, we briefly sketch the main points of the derivation for the case of single-file transport in a uniform channel in equilibrium with a solution of particles [90]. Most generally, the system of multiple particles in a channel is described by the multi-particle probability function P(x,t y) that the vector of particles positions is x at time t, starting from the initial vector y [53, 90, 92]. The crucial insight is that because the particles cannot bypass each other, the initial order of the particles is conserved if y < y for any two particles at the initial time, it implies that x < for all future times. That is, the parts of the phase space accessible to these particles are bounded by the planes defined by the condition = x in the vector space x. This implies a reflective boundary condition at the x = plane for any two different particles m and n,... 

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