Parallel diffusion

The section discusses diffusion across a number of diffusion barriers in parallel. Diffusion across the skin represents one of the best examples to illustrate steady diffusion involving two or more independent diffusional pathways in parallel [6],... 

Figure 5.22 Formation of metal oxide films (a) parallel diffusion of cations and electrons (b) counterdiffusion of cations and holes (c) counterdiffusion of anions and electrons (d) parallel diffusion of anions and holes and (e) counterdiffusion of anions and cations.

Parallel diffuse reflectance UV (DRUV) and EPR spectroscopic investigations (51,52,54) have provided evidence that the nature of the oxo intermediates formed on contact with H202 depends on the intrinsic local structure and environment of the Ti ions. The tetrapodal structures seem to generate oxo species the concentrations of which correlate with selectivity in the epoxidation of alkenes. 

An analog expression can be found for parallel diffusion in Chapter 8. By integration of this expression over the entire sphere volume the fraction / (r) of initial gas remaining at dimensionless time r becomes... 

Scheuplein RJ (1966) Analysis of permeability data for the case of parallel diffusion pathways. Biophys J 6 1-17. 

We now consider a simple extension of the presentations in Secs. 8-10 and 8-11 to analyze a medium where reflection, transmission, and absorption modes are all important. As in Sec. 8-10, we shall analyze a system consisting of two parallel diffuse planes with a medium in between which may absorb, transmit, and reflect radiation. For generality we assume that the surface of the transmitting medium may have both a specular and a diffuse component of reflection. The system is shown in Fig. 8-58. 

If local equilibrium is established instantaneously the ionic composition of parallel diffusion fluxes in the bead pores (i.e., free ions) and the gel phase of the bead (i.e., condensed and bound ions) is determined in particular by the selectivity factor, i.e., by the shape of the ex-... 

SIMPLIFIED EXPERIMENTAL METHOD TO ANALYSE INTRA-ACTIVATED CARBON PARTICLE DIFFUSION BASED ON PARALLEL DIFFUSION... 

The rate of contaminant adsorption onto activated carixm particles is controlled by two parallel diffusion mechanisms of pore and surface diffusion, which operate in different manners and extents depending upon adsorption temperature and adsorbate concentration. The present study showed that two mechanisms are separated successfully using a stepwise linearization technique incorporated with adsorption diffusion model. Surface and pore diffiisivities were obtained based on kinetic data in two types of adsorbers and isothermal data attained from batch bottle technique. Furthermore, intraparticle diffiisivities onto activated carbon particles were estimated by traditional breakthrough curve method and final results were compared with those obtained by more rigorous stepwise linearization technique. 

An intraparticle diffusion model including svurface and pore diffusion has been presented by other researchers [2,3,4,6]. The mathematical equation for parallel diffusion model is expressed as... 

Aqueous-phase Ni concentration, however, was found to have a pronounced effect on ion exchange kinetics, which is not consistent with prediction from the IPDC model based on the Nernst-Planck equation. Experimental results could, however, be accounted for with the aid of a macroporous model, originally developed by Yoshida and Kataoka based on the assumption of parallel diffusion of counterions in the solid-gel phase and in the macro-... 

Gaseous formaldehyde Indoor and ambient air UV-Vis 0.08 ppbv 300 Alternated gas collection (STOP)/sample handling (GO) involving two parallel diffusion scrubbers [144]... 

Lee (1995) has proposed a model for fast-track diffusion to explain the effects of combined lattice diffusion and diffusion along fast diffusion pathways through the lattice such as defects. Lee proposed that the combined diffusion could be modeled as two parallel diffusion mechanisms with argon atoms partitioning between the two. The mathematical model produces realistic release patterns, but does not currently take account of the distances between fast track pathways and the time taken for atoms to reach one (cf. Arnaud and Kelley 1995). Future development of the fast track model may provide very fruitful avenues for research. 

Optical techniques have also been used to measure the minority carrier diffusion lengths in both n- and p-type GaP/GaAsj[-xPx SLSs (29). The measured diffusion lengths perpendicular to the SLS interfaces were 0.1 pm and the parallel diffusion lengths were 1.5 pm. The parallel diffusion lengths are similar to those measured in bulk GaP. The small perpendicular diffusion length is consistent with the existence of potential barriers in both the conduction and valence bands which inhibit transport normal to the SLS interfaces. 

The reaction may occur by a number of mechanisms, provided electroneutrality is maintained. An extreme assumption could be, for example, that the only mobile cation is AP. Electro-neutrality involves a parallel diffusion of three 0 ions, with every two AP ions. An important consequence of such a diffusion mechanism is that the marker would be displaced both alumina ions would be transported towards the spinel-MgO boundary while the spinel-Al203 boundary progresses into the alumina side. The displacement of the marker is known as the Kirkendal effect. 

There have been surprisingly few simulations of the OTTLE response [135]. Recently, exphcit finite difference digital simulation of an OTTLE with a platinum minigrid has shown that the critical ratio of the parallel diffusion path inside the mesh hole (half the hole size) and the vertical diffusion path (thin-layer thickness) is greater than 1.33 then there wiU be a... 

Onyest5ftak, G., Valyon, J., Bota, A., and Rees, L.V.C., Frequency-response evidence for parallel diffusion processes in the bimodal micropore system of an activated carbon, Helvetica Chim. Acta, 85, 2463-2468, 2002. 

In nonlinear adsorption systems where parallel diffusion mechanisms hold, the mass balance equation given in eq. (9.2-lb) is still valid. The difference is in the functional relationship between the concentrations of the two phases, that is the local adsorption isotherm. In general, this relationship can take any form that can describe well equilibrium data. Adsorption isotherm such as Langmuir, Unilan, Toth, Sips can be used. In this section we present the mathematical model for a general isotherm and then perform simulations with a Langmuir isotherm as it is adequate to show the effect of isotherm nonlinearity on the dynamics behaviour. The adsorption isotherm takes the following functional form ... 

This chapter has presented a number of adsorption models for homogeneous particles where parallel diffusion mechanism is operating. This type of mechanism is applicable to solids such as activated carbon. We will present in the next chapter a number of models for zeolite type solids where a bimodal diffusion mechanism is operating. 

The macropore diffusion case is basically dealt with in Chapter 9 where we dealt with parallel diffusion in homogeneous solids. Since surface diffusion on the exterior surface of the zeolite microparticle is almost negligible due to the very low capacity on those surfaces, we can ignore the surface diffusion contribution in the analysis of the last chapter when we apply such analysis to a zeolite-type particle. 

Parallel Diffusion Model with Local Equilibrium... 

Pore diffusion model with local equilibrium Parallel diffusion model with local equilibrium Pore diffusion with finite adsorption kinetics Bi-dispersed model with local equilibrium... 

Parallel diffusion model with local equilibrium... 

There are essentially two types of diffusion in the zeolite bed which occur in parallel diffusion in macropores (intercrystalline space) and in micropores (intraciystalline pores). Each type of diffusion has a specific characteristic time, Tjnter or Tintra, which depends on the intercrystalline (Djnter) or intraciystalline (Dintra) diffusion coefficient and the distance to cover, the sample length, I, or the radius, R, of the crystallites assumed to be spherical (20 pm for our samples). Tinter and Tjntra are given by [4] ... 

The fast parallel diffusion was explained on the basis that in this direction, between adjacent Te layers, the Cu moved through a region of relatively weak electrostatic bonding forces and large layer spacing. 

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