Heterogeneous media, diffusion

Most solutes in soils are to some extent adsorbed on the soil solid only a small fraction is in the solution in the pores. However some adsorbed solutes, particularly exchangeable cations, can have considerable mobility on soil surfaces (see below), so it is important to consider the solid phase pathway as well as the solution. Because the diffusing solute passes rapidly between the solid and solution, the two pathways partly act in series. In such a heterogeneous medium as soil it is not realistic to account for the mobilities and concentration gradients of solutes in all the constituent parts. But if the soil volumes and reaction times... 

The lognormal distribution of the zth component of a multifractal of dimension r, obtained shows that if 1-3 components are isolated in a real multifractal, then 1-3 lognormal modes corresponding to these characteristic scales can be isolated in an experiment connected with the diffusion of radiation in a heterogeneous medium. 

In diffusion in a heterogeneous medium the intermediate linear part in (he Z(0 plot is dominant, and this similarly applies to the plot of q vs. In / the range of applicability of the Elovich equation is very wide and often all the measured data are in that range (Fig. 1-3). We expect the slope of q/qgreater than 10 [the actual value cannot be estimated as q is not reached, (Aharoni et al., unpublished data)]. 

Immiscible blends. Immiscible polymer blends having good interphase bonding provide a continuous two-phase heterogenous medium to diffusion and sorption. A detailed study of these cases is beyond the scope of this chapter. Let us consider the simple case of a two-polymer immiscible blend in which one phase remains continuous across a composition range, and the other is a discontinuous phase dispersed as spherical particles. The effective permeability is described by the following equation ... 

As the synthesis of the affinity support is carried out in a heterogeneous medium, the amount of immobilized ligand on the solid support may turn out to be rather low and difficult to appreciate. Moreover, due to steric hindrance and diffusion limitations, the accessibility of the target molecule to these binding sites may be somewhat limited. The binding capacity of the affinity matrix as well as the conditions for the release of the target molecule... 

For heterogeneous media composed of solvent and fibers, it was proposed to treat the fiber array as an effective medium, where the hydrodynamic drag is characterized by only one parameter, i.e., Darcy s permeability. This hydrodynamic parameter can be experimentally determined or estimated based upon the structural details of the network [297]. Using Brinkman s equation [49] to compute the drag on a sphere, and combining it with Einstein s equation relating the diffusion and friction coefficients, the following expression was obtained ... 

The form of the effective mobility tensor remains unchanged as in Eq. (125), which imphes that the fluid flow does not affect the mobility terms. This is reasonable for an uncharged medium, where there is no interaction between the electric field and the convective flow field. However, the hydrodynamic term, Eq. (128), is affected by the electric field, since electroconvective flux at the boundary between the two phases causes solute to transport from one phase to the other, which can change the mean effective velocity through the system. One can also note that even if no electric field is applied, the mean velocity is affected by the diffusive transport into the stationary phase. Paine et al. [285] developed expressions to show that reversible adsorption and heterogeneous reaction affected the effective dispersion terms for flow in a capillary tube the present problem shows how partitioning, driven both by electrophoresis and diffusion, into the second phase will affect the overall dispersion and mean velocity terms. 

The development of the theory of solute diffusion in soils was largely due to the work of Nye and his coworkers in the late sixties and early seventies, culminating in their essential reference work (5). They adapted the Fickian diffusion equations to describe diffusion in a heterogeneous porous medium. Pick s law describes the relationship between the flux of a solute (mass per unit surface area per unit time, Ji) and the concentration gradient driving the flux. In vector terms. 

Chapter 14 - It was shown, that the conception of reactive medium heterogeneity is connected with free volume representations, that it was to be expected for diffusion-controlled solid phase reactions. If free volume microvoids were not connected with one... 

Compared to rivers and lakes, transport in porous media is generally slow, three-dimensional, and spatially variable due to heterogeneities in the medium. The velocity of transport differs by orders of magnitude among the phases of air, water, colloids, and solids. Due to the small size of the pores, transport is seldom turbulent. Molecular diffusion and dispersion along the flow are the main producers of randomness in the mass flux of chemical compounds. 

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. 

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