Medium effects diffusion

These must supplement the minimal set of experiments needed to determine the available parameters in the model-It should be emphasized here, and will be re-emphasized later, Chat it is important Co direct experiments of type (i) to determining Che available parameters of some specific model of Che porous medium. Much confusion has arisen in the past frcjci results reported simply as "effective diffusion coefficients", which cannot be extrapolated with any certainty to predict... 

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

Diffusion within the largest cavities of a porous medium is assumed to be similar to ordinary or bulk diffusion except that it is hindered by the pore walls (see Eq. 5-236). The tortuosity T that expresses this hindrance has been estimated from geometric arguments. Unfortunately, measured values are often an order of magnitude greater than those estimates. Thus, the effective diffusivity D f (and hence t) is normally determined by comparing a diffusion model to experimental measurements. The normal range of tortuosities for sihca gel, alumina, and other porous solids is 2 < T < 6, but for activated carbon, 5 < T < 65. 

The ratio of the overall rate of reaction to that which would be achieved in the absence of a mass transfer resistance is referred to as the effectiveness factor rj. SCOTT and Dullion(29) describe an apparatus incorporating a diffusion cell in which the effective diffusivity De of a gas in a porous medium may be measured. This approach allows for the combined effects of molecular and Knudsen diffusion, and takes into account the effect of the complex structure of the porous solid, and the influence of tortuosity which affects the path length to be traversed by the molecules. 

Porous solid as pseudo-homogeneous medium 635 effective diffusivity 635 Portable mixers 306 Positive displacement pumps. 315... 

The presence of particles in the fluid medium complicates diffusion in a sediment due to the effects of porosity, represented by n, and tortuosity. Since tortuosity of natural sediments is seldom known it is more convenient to use the term "formation factor" or "lithological factor," denoted L, which takes into account everything but porosity. Tick s diffusion constant D is replaced by the whole sediment diffusion constant Ds, where < D. 

Solutions of surfactant-stabilized nanogels share both the advantage of gels (drastic reduction of molecular diffusion and of internal dynamics of solubilizates entrapped in the micellar aggregates) and of nonviscous liquids (nanogel-containing reversed micelles diffuse and are dispersed in a macroscopicaUy nonviscous medium). Effects on the lifetime of excited species and on the catalytic activity and stability of immobilized enzymes can be expected. 

The aqueous diffusivities of charged permeants are equivalent to those of uncharged species in a medium of sufficiently high ionic strength. The product DF(r/R) is the effective diffusion coefficient for the pore. It is implicit in k that adsorption of the cations does not occur, so that the fixed surface charges on the wall of the pore are not neutralized. Adsorption is more likely to occur with multivalent cations than with univalent ones. 

The Effectiveness Factor Analysis in Terms of Effective Diffusivities First-Order Reactions on Spherical Pellets. Useful expressions for catalyst effectiveness factors may also be developed in terms of the concept of effective diffusivities. This approach permits one to write an expression for the mass transfer within the pellet in terms of a form of Fick s first law based on the superficial cross-sectional area of a porous medium. We thereby circumvent the necessity of developing a detailed mathematical model of the pore geometry and size distribution. This subsection is devoted to an analysis of simultaneous mass transfer and chemical reaction in porous catalyst pellets in terms of the effective diffusivity. In order to use the analysis with confidence, the effective diffusivity should be determined experimentally, since it is difficult to obtain accurate estimates of this parameter on an a priori basis. 

In terms of an effective diffusivity De and a mean concentration gradient across a porous medium of thickness L, the flux through the medium may be written as ... 

J. H. Nam and M. Kaviany. Effective diffusivity and water-saturation distribution in single- and two-layer PEMEC diffusion medium. International Journal of Heat Mass Transfer 46 (2003) 4595-4611. 

To calculate the release through diffusion of an entrapped residue, Barraclough et al. (2005) considered the size of organic matter particles (effective radius 10" to 10 cm) and the effective diffusion coefficient of small organic molecules in a sorbing medium (D 10 cm s )- The time for 50% of the material in a sphere to diffuse out is given by... 

Regardless of the transport equation considered, the major effect of sorption on contaminant breakthrough curves is to delay the entire curve on the time axis, relative to a passive (nonsorbing) contaminant. Because of the longer residence time in the porous medium, advective-diffusive-dispersive interactions also are affected, so that longer (non-Fickian) tailing in the breakthrough curves is often observed. 

The collapse of the networks was studied experimentally mainly for the case of mixed solvents. From the practical point of view, it would be interesting to generate the collapse in a purely aqueous medium. In this case, by changing the interactions between subchains of the network in aqueous solution it is possible, for example, to influence effectively diffusion in gels that are used as carriers of enzymes or pharmacologically active substances. One of the new directions in the research of network polymers is the study of their interaction with linear macromolecules. The results of the theoretical analysis of the behavior of polymer networks swollen in polymer solutions have been discussed in Ref. [34,35] (see Sect. 2.4.2). 

Thus, the sorption of chemicals on the surface of the solid matrix may become important even for substances with medium or even small solid-fluid equilibrium distribution coefficients. For the case of strongly sorbing chemicals only a tiny fraction of the chemical actually remains in the fluid. As diffusion on solids is so small that it usually can be neglected, only the chemical in the fluid phase is available for diffusive transport. Thus, the diffusivity of the total (fluid and sorbed) chemical, the effective diffusivity DieS, may be several orders of magnitude smaller than diffusivity of a nonsorbing chemical. We expect that the fraction which is not directly available for diffusion increases with the chemical s affinity to the sorbed phase. Therefore, the effective diffusivity must be inversely related to the solid-fluid distribution coefficient of the chemical and to the concentration of surface sites per fluid volume. 

Both Knudsen and molecular diffusion can be described adequately for homogeneous media. However, a porous mass of solid usually contains pores of non-uniform cross-section which pursue a very tortuous path through the particle and which may intersect with many other pores. Thus the flux predicted by an equation for normal bulk diffusion (or for Knudsen diffusion) should be multiplied by a geometric factor which takes into account the tortuosity and the fact that the flow will be impeded by that fraction of the total pellet volume which is solid. It is therefore expedient to define an effective diffusivity De in such a way that the flux of material may be thought of as flowing through an equivalent homogeneous medium. We may then write ... 

To calculate the effective diffusivity in the region of molecular flow, the estimated value of D must be multiplied by the geometric factor e/x which is descriptive of the heterogeneous nature of the porous medium through which diffusion occurs. 

One particular model of a biological system is a set of barriers in an otherwise homogeneous medium. If diffusion is observed over a short enough time, very little of the substance experiences the effect of the barriers, and the observed motion is characteristic of the medium alone. As the time of observation is extended, more of the substance is reflected at barriers and thus, its total displacement is less than would have been the case without the barrier bringing about less attenuation of the PG spin-echo [48, 49]. From an analysis of such data one can, in principle, obtain information on the geometry of the domains in which restricted diffusion occurs information on cell size, droplet size distribution [50] etc. 

Here, the length L in (7.38) has been replaced by porous layer thickness d and the surface area Aeff. The effective diffusion coefficient D0,eff characterizes the transport through porous medium and includes both regular diffusion and the Knudsen diffusion coefficient >o,Kn, which has a different temperature dependence from diffusion in bulk. 

For reactions which are first order in the dissolving reactant, the rate will be proportional to the solubility of that species, regardless of whether the reaction occurs in the bulk solution or in the film (Equations 5.4 and 5.7). Inorganic salts usually depress the solubility of organic solutes in water [5], which can lead to unusual effects. Fig. 5.6 shows howthe rate ofreaction of n-butyl formate with potassium hydroxide changes with hydroxide concentration [6]. There are three reasons for the decrease in the rate as the concentration of hydroxide is increased (i) salting out of the ester, (ii) a decrease in the diffusion coefficient of the n-butyl formate in water and (iii) a sharp decrease in the second-order rate constant presumably due to medium effects on lc2-... 

In the porous medium, diffusion is affected by the porosity and tortuosity of the medium itself therefore Knudsen diffusion is computed as well as the ordinary diffusion. Eventually, an effective diffusion coefficient is calculated that depends on the ordinary and Knudsen diffusion coefficients and on the ratio between porosity and tortuosity of the medium (Equation (3.58)). 

The second explanation for the solvent isotope effect arises from the dynamic medium effect . At 25 °C the rotational and translational diffusion of DjO molecules in D20 is some 20% slower than H20 molecules in H20 (Albery, 1975a) the viscosity of D20 is also 20% greater than H20. Hence any reaction which is diffusion controlled will be 20% slower in D20 than in H20. This effect would certainly apply to transition state D in Fig. 3 where in the transition state the leaving group is diffusing away. A similar effect may also apply to the classical SN1 and SN2 transition states, if the rotational diffusion of water molecules to form the solvation shell is part of the motion along the reaction co-ordinate in the transition state. Robertson (Laughton and Robertson, 1959 Heppolette and Robertson, 1961) has indeed correlated solvent isotope effects for both SN1 and SN2 reactions with the relative fluidities of H20 and D20. However, while the correlation shows that this is a possible explanation, it may also be that the temperature variation of the solvent isotope effect and of the relative fluidities just happen to be very similar (see below). 

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