Networks, with pore diffusion

Internal diffusion ofreactants. This step depends on the porosity of the catalyst and the size and shape of the catalyst particles, and occurs together with the surface reaction. The active catalyst component is usually highly dispersed within the three-dimensional porous support. The reactant molecules have to diffuse through the network of pores toward the active sites. The activation energy for pore diffusion li2 may represent a substantial share of the activation energy of the chemical reaction itself. 

Internal diffusion of products. The desorbed products then diffuse through the network of pores to reach the outer surface of the catalyst with kinetics and activation energy similar to those in step 2. 

Smith and coworkers recently proposed a specific and novel mineral-based solution to the problem of dilution and diffusion of prebiotic reactants. They have suggested [132-134] the uptake of organics within the micron-sized three-dimensional cross-linked network of pores found to exist within the top 50 xm, or so, of alumina-depleted, silica-rich weathered feldspar surfaces. These surfaces incorporate cavities typically about 0.5 pm in diameter along with cross inter-connections of about 0.2 pm. The nominal area of the weathered feldspar surface is apparently multiplied by a factor of about 130 arising from this network. The similarity of these pores to the catalytic sites in zeolite-type materials is pointedly mentioned. 

Figure 5 shows the simulation of the reaction kinetic model for VO-TPP hydro-demetallisation at the reference temperature using a Be the network with coordination 6. The metal deposition profiles are shown as a function of pellet radius and time in case of zero concentration of the intermediates at the edge of the pellet. Computer simulations were ended when pore plugging occurred. It is observed that for the bulk diffusion coefficient of this reacting system the metal deposition maximum occurs at the centre of the catalyst pellet, indicating that the deposition process is reaction rate-determined. The reactants and intermediates can reach the centre of the pellet easily due to the absence of diffusion limitations. 

Using these equations for heat conduction in the porous network emd the pseudo-steady state emalysis described for pore diffusion, the time to dry a spherical green body with pore heat conduction as the rate determining step is given by... 

Fig. 31. Computer simulation of the random walk in a two-dimensional network with obstacles distributed statistically over the pore segments (dashed lines) and over the channel intersections (solid lines). Different transition probabilities h to pass the barrier are assumed b = 0.004, 0.020, and 0.120. The experimental results of the methane self-diffusion in ZSM-5 containing coadsorbed benzene (O, cf. Fig. 30) are included 86).

The rate processes of diffusion and catalytic reaction in simple square stochastic pore networks have also been subject to analysis. The usual second-order diffusion and reaction equation within individual pore segments (as in Fig. 2) is combined with a balance for each node in the network, to yield a square matrix of individual node concentrations. Inversion of this 2A matrix gives (subject to the limitation of equimolar counterdiffusion) the concentration profiles throughout the entire network [14]. Figure 8 shows an illustrative result for a 20 X 20 network at an intermediate value of the Thiele modulus. The same approach has been applied to diffusion (without reaction) in a Wicke-Kallenbach configuration. As a result of large and small pores being randomly juxtaposed inside a network, there is a 2-D distribution of the frequency of pore fluxes with pore diameter. 

One approach to minimize mass-transfer resistance in a stagnant mobile phase employs specially designed particles with a bimodal network of pores. The larger pores (>1000 A) facilitate convective transport of the mobile phase inside the particles, whereas the small pores (<500 A) are explored by the sample components by diffusion only and provide the necessary surface area for adequate sorption capacity. 

It has long been known that, when the network of pores is fractal, diffusion by molecular movement in this network differs from the transport in media with properties independent of scale. In particular, diffusion of solutes in a fractal pore network does not obey Fick s law, and anomalous, or non-Fickian diffusion takes place instead (Gefen at al., 1983). When Fick s second law... 

The effective diffusion coefficient accounts for the rate of diffusion of the protein through the complex, porous polymer matrix. Incorporation of an effective diffusion coefficient is necessary because protein molecules do not diffuse through the pure polymer phase, but must find a path out of the slab by diffusing through a tortuous, water-filled network of pores. Z>eff is assumed to be independent of position in the slab Equation 9-14 is justified in this case by local averaging over a volume that is large compared to a single pore. Characteristic desorption or protein release curves are represented by Equation 9-18, with substitution of Dgn- for Z),.p ... 

To understand how microscopic properties of the material influence these phenomena, it is necessary to develop more complex models of protein release. When detailed information on the microgeometry of the porous network in the polymers is available (as it is for protein-loaded EVAc matrices [37]) or can be estimated accurately, detailed models of protein release can be developed. For example, percolation models of pore network topology (such as those described in Chapter 4, see Figure 4.20) were coupled with analytical models of pore-to-pore diffusion rates to predict the rate of diffusion of proteins from EVAc matrices [16]. Effective diffusion coefficients predicted using this approach agree with those estimated by measuring rates of protein release from the matrix (Figure 9.13). 

Microporous membranes in general (pore diameter < 2 nm), and zeolite membranes in particular, have pores whose dimensions are similar to those of many molecules. This means that often molecules cannot pass each other in a restrictive pore medium, and single file diffusion occurs. Such a molecular queuing (see Figure 11.24) may provide a new scenario for avoiding secondary reactions, that is, to increase selectivity in consecutive reaction networks with a valuable intermediate... 

Let us start with the derivation of the mass balance equations for a batch adsorber containing spherical adsorbent particles. The solute molecules diffuse from the bulk reservoir into the particles through a network of pores within the particle. The diffusion process is generally described by a Fickian type equation... 

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