Inhomogeneity, spatial

This new design is sought to overcome the limits of conventional porous fixed-bed reactors using an electrode phase flowing through the pores [65]. The latter systems suffer from the low conductivity of the electrolyte phase. This generates electrical resistance and leads to accumulation of the electrical current in certain reactor zones and hence results in a spatially inhomogeneous reaction. This means poor exploitation of the catalyst and possible reductions in selectivity. 

OS 92] [R 32] [P 72/The iodate-arsenous acid reachon proceeds to one of two stationary states in different parts of the capillary when an electrical field of specific strength is applied [68]. Accordingly, a spatially inhomogeneous distribution of reaction products is generated along the capillary. 

Illustration Short-time behavior in well mixed systems. Consider the initial evolution of the size distribution of an aggregation process for small deviations from monodisperse initial conditions. Assume, as well, that the system is well-mixed so that spatial inhomogeneities may be ignored. Of particular interest is the growth rate of the average cluster size and how the polydispersity scales with the average cluster size. 

Autocorrelation and time series analysis have been successfully applied in testing spatial inhomogeneities (Ehrlich and Kluge [1989], Do-erffel et al. [1990]). This techniques are generalized in the theory of stochastic processes (Bohacek [1977a, b]) which is widely used in chemical process analysis and about them. 

If we had taken a spatially inhomogeneous field Eq, the connection between the conductivity and the external electric field would be much more complicated than Eq. (113), due to the polarization of the medium.18 However, for q strictly equal to zero, the system remains spatially homogeneous and Eq. (113) holds. 

The Gaussian expressions are not expected to be valid descriptions of turbulent diffusion close to the surface because of spatial inhomogeneities in the mean wind and the turbulence. To deal with diffusion in layers near the surface, recourse is generally had to the atmospheric diffusion equation, in which, as we have noted, the key problem is proper specification of the spatial dependence of the mean velocity and eddy difiusivities. Under steady-state conditions, turbulent diffusion in the direction of the mean wind is usually neglected (the slender-plume approximation), and if the wind direction coincides with the x axis, then = 0. Thus, it is necessary to specify only the lateral (Kyy) and vertical coefficients. It is generally assumed that horizontal homogeneity exists so that u, Kyy, and Ka are independent of y. Hence, Eq. (2.19) becomes... 

MSN. 115. I. Prigogine, A new microscopic level of irreversibility Meeting Spatial Inhomogeneities and Transient Behavior in Chemical Kinetics, Bruxelles, 1987. 

When the steady state becomes unstable, the system moves away from it and often undergoes sustained oscillations around the unstable steady state. In the phase space defined by the system s variables, sustained oscillations generally correspond to the evolution toward a limit cycle (Fig. 1). Evolution toward a limit cycle is not the only possible behavior when a steady state becomes unstable in a spatially homogeneous system. The system may evolve toward another stable steady state— when such a state exists. The most common case of multiple steady states, referred to as bistability, is of two stable steady states separated by an unstable one. This phenomenon is thought to play a role in differentiation [30]. When spatial inhomogeneities develop, instabilities may lead to the emergence of spatial or spatiotemporal dissipative stmctures [15]. These can take the form of propagating concentration waves, which are closely related to oscillations. 

Jurczek, E., Model Study of the Semiconductor-Metal Transition in BaBij xPbxOs by Use of a Spatially Inhomogeneous Order-Parameter Approximation. Phys. Rev. B 35(13) 6997 (1987). 

Finally, some simple estimates will be presented for the three-dimensional electrolyte concentration and electric potential fields resulting from concentration polarization in a diffusion layer adjacent to a spatially inhomogeneous ion-selective interface (membrane). It will be shown that the appropriate fields are incompatible with mechanical equilibrium in an ionic fluid, so that a related (nongravitational) convection is expected to arise at an inhomogeneous ion-exchange membrane upon the passage of an electric current. 

Well-stirred systems are particularly convenient for the theoretician but are often less easy to realize in practice. Indeed, spatial inhomogeneities, with consequent molecular diffusion or thermal conduction processes, arise in many important situations—as varied as a single biological cell and a haystack. In the next three chapters we turn to unstirred systems, again seeking to determine bifurcation phenomena driven by non-linear kinetics. 

We can think of the reactant concentration and some initial spatial distribution of the intermediate concentration and temperature profiles specifying a point on Fig. 10.9. If we choose a point above the neutral stability curve, then the first response of the system will be for spatial inhomogeneity to disappear. If the value of /r lies outside the range given by (10.79), then the system adjusts to a stable spatially uniform stationary state. If ji lies between H and n, we may find uniform oscillations. 

Ertl, G. (1989). The oscillatory catalytic oxidation of carbon monoxide on platinum surfaces. In Spatial inhomogeneities and transient behaviour in chemical kinetics, (ed. P. Gray, G. Nicolis, F. Baras, P. Borckmans, and S. K. Scott), ch. 37, pp. 563—76. Manchester University Press. 

If we neglect spatial inhomogeneities, the elastic free energy Fel, Eq. (2.19), becomes... 

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