Non-uniform diffusivity

Because a vacuum is applied for the removal of the solutes on the membrane downstream face, this side of the membrane is ideally dry in comparison to the more swollen (if polymeric membranes are employed) and hence more flexible membrane upstream face resulting from the solute uptake. This anisotropy of the membrane in the direction of the diffusion of the solute always exists for polymeric membranes and results in a non-uniform diffusivity of solute within the membrane. In other words, the diffusion coefficient of solute i in the membrane, will be position-dependent and not constant across the membrane. 

In a eompressor with a vaned diffuser followed by a typieal easing, the non-uniform, eireumferential flow resistanee aeross the diffuser walls induees an asymmetrie gas pressure around the wheel. Non-uniform peripheral gas pressure results in unbalaneed loading on the wheel and, henee, a radial bearing load. 

The distribution of tracer molecule residence times in the reactor is the result of molecular diffusion and turbulent mixing if tlie Reynolds number exceeds a critical value. Additionally, a non-uniform velocity profile causes different portions of the tracer to move at different rates, and this results in a spreading of the measured response at the reactor outlet. The dispersion coefficient D (m /sec) represents this result in the tracer cloud. Therefore, a large D indicates a rapid spreading of the tracer curve, a small D indicates slow spreading, and D = 0 means no spreading (hence, plug flow). 

Fiek s diffusion law is used to deseribe dispersion. In a tubular reaetor, either empty or paeked, the depletion of the reaetant and non-uniform flow veloeity profiles result in eoneentration gradients, and thus dispersion in both axial and radial direetions. Fiek s law for moleeular diffusion in the x-direetion is defined by... 

Trinh, S Locke, BR Arce, P, Diffusive-Convective and Diffusive-Elechoconvective Transport in Non-Uniform Channels with Application to Macromolecular Separations, Separation and Purification Technology 15, 255, 1999. 

Chapman, S., Cowling, T, The Mathematical Theory of Non-uniform Gases An account of the Kinetic Theory of Viscosity, Thermal Gonduction and Diffusion in Gases, Cambridge University Press, Cambridge (1970). 

Structure and mobility of adsorbed molecules may exhibit a wide spectrum of features. As a general view it is commonly accepted that layers are formed on increasing abundance. The mobility of molecules in the first layer depends strongly on interactions present in a given case. Local domains of adsorbed molecules may be formed in the case of non-uniform surface. Analysis of deuteron spectra for a series of molecules adsorbed on neutral alumina led to the conclusion that while the exchange of molecules between layers in a given domain is fast, the diffusion rate between domains is slow [5],... 

Whatever the typology of immobilized biophase, kinetics assessment and modeling studies should not neglect the relevance of the profiles reported in Fig. 4. In agreement with Bailey and Ollis [51], the non uniform profile of the concentrations of azo-dye and of the products may be expressed in terms of the effectiveness factor of the immobilized biophase the ratio of actual reaction rate to the reaction rate without diffusion limitation. 

The circulation conditions obtained in the small laboratory cell cannot be attained in a full-scale cell. The effects of Na+ diffusion to the membrane and non-uniformity in its intracell concentration cannot be entirely eliminated, and a greater decrease in current efficiency will tend to occur at high current densities. 

The polydispersity of melt-phase samples is generally lower than that of solid-state samples. Gel permeation chromatography (GPC) analysis of samples prepared from the solid-state showed polydispersity values in the range of 2.57 to 2.84 compared to 2.27 to 2.49 for melt samples [107], The higher polydispersity of solid-state samples can largely be explained by the non-uniformity in the average molecular weight across the pellet radius caused by a SSP reaction rate that is diffusion controlled [11],... 

Overall, the rib effects are important when examining the water and local current distributions in a fuel cell. They also clearly show that diffusion media are necessary from a transport perspective. The effect of flooding of the gas-diffusion layer and water transport is more dominant than the oxygen and electron transport. These effects all result in non-uniform reaction-rate distributions with higher current densities across from the channels. Such analysis can lead to optimized flow fields as well as... 

Also, mechanical data on the influence of low volume fractions (0.03-0.05) of rigid filler particles provide evidence of a localized plastic deformation which would not seem understandable by reference to a uniformly crosslinked network. A non-uniformly crosslinked matrix might also be invoked to account for insensitivity of the rate of diffusion of water on the apparent degree of crosslinking. However, an observed increase in the uptake of water with apparent degree of crosslinking remains unexplained. 

It will be demonstrated in this section that a narrow pore structure limits the reaction rate to an extent which casues the reaction rate to be either proportional to the square root of the specific surface area (per unit mass) or independent of it, depending on the mode of diffusion within the pore structure. Lest this departure of the reaction rate from direct proportionality with specific surface area might be thought to be accounted for in terms of a non-uniform distribution of surface energy over the catalyst surface, it should be pointed out that such in situ heterogeneity is usually only a small fraction of the total chemically active surface and cannot therefore explain the observed effects. 

First, we ask whether it is possible that the diffusion of the intermediate A and the conduction of heat along the box might destabilize a stable uniform state. An important condition for this is that the diffusion and conduction rates should proceed at different rates (i.e. be characterized by different timescales). Secondly, if the well-stirred system is unstable, can diffusion stabilize the system into a time-independent spatially non-uniform state Here we find a qualified yes , although the resulting steady patterns may be particularly fragile to some disturbances. 

Thus, when the stirring stops, the uniform state remains a stationary solution of the system. Diffusion does not affect the existence of the uniform state, but it may influence its stability. In particular we are interested in determining whether this state can become unstable to spatially non-uniform perturbations. 

If the mass diffusion coefficient is sufficiently large compared with the thermal diffusivity, so P > / c, the range between n2- and n+ will be non-zero. There is another consideration n can only have discrete integer values, the lowest of which for a non-uniform state is n = 1. Thus for observable patterns we must make sure that at least n exceeds unity. Equation (10.49) shows that this last requirement puts a lower bound on the size parameter y we need y > yc, where... 

An alternative way of portraying the pattern formation behaviour in systems of the sort under consideration here is to delineate the regions in chemical parameter space (the h k plane) over which the uniform state is unstable to non-uniform perturbations. We have already seen in chapter 4, and in Fig. 10.3, that we can locate the boundary of Hopf instability (where the uniform state is unstable to a uniform perturbation and at which spatially uniform time-dependent oscillations set in). We can use the equations derived in 10.3.2 to draw similar loci for instability to spatial pattern formation. For this, we can choose a value for the ratio of the diffusivities / and then find the conditions where eqn (10.48), regarded as a quadratic in either y or n, has two real positive solutions. The latter requires that... 

In the previous section we have taken care to keep well away from parameter values /i and k for which the uniform stationary state is unstable to Hopf bifurcations. Thus, instabilities have been induced solely by the inequality of the diffusivities. We now wish to look at a different problem and ask whether diffusion processes can have a stabilizing effect. We will be interested in conditions where the uniform state shows time-dependent periodic oscillations, i.e. for which /i and k lie inside the Hopf locus. We wish to see whether, as an alternative to uniform oscillations, the system can move on to a time-independent, stable, but spatially non-uniform, pattern. In fact the... 

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