Diffusion interparticle transport

Interparticle transport may be possible either by surface diffusion across the support or by vapor phase transport. Depending on the supported systems and on the sintering conditions the particles may grow predominantly via one of these possible routes. Supported PcFe deposits have been sintered in experimental conditions close to that of the condensation process (T = 235°C, residual pressure 10 torr). The surface areas of PcFe after 2 and 3 hours of sintering are shown in Table IV. It is seen that the sintering effect is more pronounced with a homogeneous carbon support. 

Under these circumstances, the interparticle transport resistances can be neglected. What are left are the intraparticle resistances, i.e. the heat and mass transfer effects inside the catalyst particles. Since the current case reflects the situation that few reactant and product molecules exist in an environment of solvent molecules, the simplest Fick s law approach with effective diffusion coefficients can be considered as sufficient for the description of molecular diffusion. 

This kinetic zero average contrast (KZAC) experiment [100-102] is an extension to the static zero average contrast (ZAC) described in Sect. 3.1.7. ZAC is used to effectively remove the structure factor such that interparticle correlations are eliminated and the single entities are visible, whereas in KZAC the trick is used to render mixing processes hence, diffusion and transport become observable without perturbing the system in any substantial way. 

When multiple scattering is discarded from the measured signal, DLS can be used to study the dynamics of concentrated suspensions, in which the Brownian motion of individual particles (self-diffusion) differs from the diffusive mass transport (gradient or collective diffusion), which causes local density fluctuations, and where the diffusion on very short time-scales (r < c lD) deviates from those on large time scales (r c D lones and Pusey 1991 Banchio et al. 2000). These different diffusion coefficients depend on the microstructure of the suspension, i.e. on the particle concentration and on the interparticle forces. For an unknown suspension it is not possible to state a priori which of them is probed by a DLS experiment. For this reason, a further concentration limit must be obeyed when DLS is used for basic characterisation tasks such as particle sizing. As a rule of thumb, such concentration effects vanish below concentrations of 0.01-0.1 vol%, but certainty can only be gained by experiment. 

In Chapter 4, two different transport regimes were identified transport inside the particles and transport between the bulk fluid and the surface of the catalyst particles. Transport inside the catalyst particles is known as internal or intraparticle transport, or as pore diffusion. Transport between the bulk fluid stream and the external surface of the catalyst particles is known as external or interparticle transport. The mechanisms of transport are different fiar these two regimes, and the rates of transport are influenced by different variables. Internal transport will be treated first, followed by extmial tranqiort. These discussions will be preceded by a brief overview iff the physical nature of heterogeneous catalysts. 

Interpretation of pubhshed data is often comphcated by the fact that rather complex catalytic materials are utilized, namely, poly disperse nonuniform metal particles, highly porous supports, etc., where various secondary effects may influence or even submerge PSEs. These include mass transport and discrete particle distribution effects in porous layers, as confirmed by Gloaguen, Antoine, and co-workers [Gloaguen et al., 1994, 1998 Antoine et al., 1998], and diffusion-readsorption effects, as shown by Jusys and co-workers for the MOR and by Chen and Kucemak for the ORR [Jusys et al., 2003 Chen and Kucemak, 2004a, b]. Novel approaches to the design of ordered nanoparticle arrays where nanoparticle size and interparticle distances can be varied independently are expected to shed hght on PSEs in complex multistep multielectron processes such as the MOR and the ORR. 

Some additional complexity arises from the possibility of different adsorption sites and the presence of pores, which reflect in nonideal adsorption isotherms and mass-transfer problems. The mass transport can be relatively slow in pores and interparticle spaces [13], as it is the case of P25, for which, in suspension, there are particles ranging from 0.2 to 2 p,m, formed by 30-nrn-sizcd primary particles. In such spaces, the diffusion coefficient is comparable to liquid diffusion in zeolites. 

Let us now consider coagulation of particles in the absence of any repulsive barrier. In addition, we assume that, although there are no interparticle forces that contribute to the transport of particles toward each other, there is sufficient attraction between the particles on contact for them to form a permanent bond. As early as 1917, Smoluchowski formulated the equations for the collision rate for particles transported by diffusion alone (Smoluchowski 1917), and we develop the same idea here. 

Because of the high surface free energy at the liquid-solid interface, it is suggested that the stages of nucleation, transport of species by surface diffusion, and crystallization occur at the interface in the boundary layer. Culfaz and Sand in this volume (48) propose a mechanism with nucleation at the solid-liquid interface. This mechanism should be most evident in more concentrated gel systems where interparticle contact is maximized for aggregation, coalescence, or ripening processes. The epitaxy observed by Kerr et al. (84) in cocrystallization of zeolites L, offretite, and erionite further supports a surface nucleation mechanism. 

In this study the ratio of the particle sizes was set to two based on the average value for the two samples. As a result, if the diffusion is entirely controlled by secondary pore structure (interparticle diffusion) the ratio of the effective diffusion time constants (Defl/R2) will be four. In contrast, if the mass transport process is entirely controlled by intraparticle (platelet) diffusion, the ratio will become equal to unity (diffusion independent of the composite particle size). For transient situations (in which both resistances are important) the values of the ratio will be in the one to four range. Diffusional time constants for different sorbates in the Si-MCM-41 sample were obtained from experimental ZLC response curves according to the analysis discussed in the experimental section. Experiments using different purge flow rates, as well as different purge gases... 

Early experience also showed that the induced plasma current in a tokamak generates a magnetic field that loops die minor axis nf Ihe torus. The field lines form helices along the toroidal surface the plasma must cross the lines to escape. It does so through the cumulative action of many random displacements caused by interparticle collisions, tin effect diffusing across the field lines and out of the system). Thermal energy is transported by much the same process. 

A two-site model has been used (18) to model cesium transport in soils. In this model a Langmuir-type model, Equation 3,was used to represent surface sorption and a first-order model, Equation 1, was used to approximate interparticle diffusion. Extraction with CaC was used to verify the exchangeable site inventory. 

Available reaction-transport models describe the second regime (reactant transport), which only requires material balances for CO and H2. Recently, we reported preliminary results on a transport-reaction model of hydrocarbon synthesis selectivity that describes intraparticle (diffusion) and interparticle (convection) transport processes (4, 5). The model clearly demonstrates how diffusive and convective restrictions dramatically affect the rate of primary and secondary reactions during Fischer-Tropsch synthesis. Here, we use an extended version of this model to illustrate its use in the design of catalyst pellets for the synthesis of various desired products and for the tailoring of product functionality and molecular weight distribution. 

Here (3Br(ij), Psll(i,j), and PDS(ij) are the transport coefficients for interparticle contacts between particles of diameters d, and dj by Brownian diffusion, fluid shear, and differential sedimentation, respectively kB is Boltzmanns constant T is the absolute temperature p, is the viscosity of the liquid G is the mean velocity gradient of the liquid g is the gravity acceleration and pp and p, are the densities of the particles and the liquid, respectively. 

More generally, any force could be used to move the particles, so a more general definition of this type of transport coefficient will be the mobility diffusion coefficient, Du = ksTulci. Eq. (11.66)). Note that while this relationship between the conductivity and the diffusion coefficient was derived for noninteracting carriers, we now use this equation as a definition also in the presence of interparticle interactions, when o is given by Eq. (11.76). 

The coefficients a(p, c) and tj(p, c) describe chemical and physical effects on the kinetics of deposition. The transport of particles from the bulk of the flowing fluid to the surface of a collector or media grain by physical processes such as Brownian diffusion, fluid flow (direct interception), and gravity are incorporated into theoretical formulations for fj(p, c), together with corrections to account for hydrodynamic retardation or the lubrication effect as the two solids come into close proximity. Chemical effects are usually considered in evaluating a(p, c). These include interparticle forces arising from electrostatic interactions and steric effects originating from interactions between adsorbed layers of polymers and polyelectrolytes on the solid surfaces. 

Three particle transport processes that bring about interparticle contacts are considered here Brownian diffusion (thermal effects), fluid shear (flow effects), and differential settling (gravity effects). Following Smoluchowski s approach, the appropriate individual transport coefficients for these three processes arc as... 

For the flocculation tank, the significant transport mechanisms leading to interparticle collisions are assumed to be Brownian diffusion and fluid shear. Expressions for the collision frequency functions for these mechanisms were derived by Smoluchowski and are as follows ... 

In the simplest case of a fixed bed of adsorbent particles, the following mass transport processes are considered axial dispersion in the interparticle fluid phase, fluid-to-particle mass transfer, intrapaitide diffusion, and a first-order, reversible adsorption in the interior of the particle. The last step corresponds to a linear adsorption isotherm with a finite adsorption rate. This assumption includes the case of inflnitdy fast adsorption rate. 

In summary, we have shown that interaction between the surfaces of clay particles and water modifies the structure of the water and that this modification 1) affects the swelling of the clay and, thereby, the permeability of the system, 2) affects the viscosity and yield point of the interparticle water, and 3) affects adsorption of solutes by the clay. Hence, it is clear that clay/water interaction will affect the convective and diffusive transport of solutes through a soil, especially one that is rich in clay. 

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