Spatial nonuniformity

A gas is not in equilibrium when its distribution function differs from the Maxwell-Boltzman distribution. On the other hand, it can also be shown that if a system possesses a slight spatial nonuniformity and is not in equilibrium, then the distribution function will monotonically relax in velocity space to a local Maxwell-Boltzman distribution, or to a distribution where p = N/V, v and temperature T all show a spatial dependence [bal75]. 

Other less definite yet important effects such as profile changes due to nonlinear refractive index alteration in spatially nonuniform high power beams must be carefully considered. As example, the use of nonidentical liquids and optical paths prior to and in, say, EFISH cells and the usual quartz calibration cells could cause potentially inaccurate x determinations. Obviously these types of considerations are important when precise experimentation to test fine models of molecular behavior are intended, but have not stood as obstacle to uncovering the important general trends in molecular nonlinearity enhancement. 

Situations that depart from thermodynamic equilibrium in general do so in two ways the relative concentrations of different species that can interconvert are not equilibrated at a given position in space, and the various chemical potentials are spatially nonuniform. In this section we shall consider the first type of nonequilibrium by itself, and examine how the rates of the various possible reactions depend on the various concentrations and the lattice temperature. 

In this illustration, the spatially nonuniform body forces cause an internal source of vortic-ity. Vorticity is also generated due to shearing behavior at the walls. 

From an operational point of view, the choice of an appropriate polymerization reactor depends on six requirements temperature control mixing product accumulation and reactor foul-up follow-up separation processes the desired form of the product and safety. Heats of polymerization are typically high, so that maintaining the reactor at a desired temperature level is not always a simple task. Temperature can become spatially nonuniform and globally out of control (causing inconsistency of the reaction medium). Nonuniformity in temperature can lead to localized zones of poor mixing or even dead zones. In a polymerization reactor, temperature, mixing, viscosity,... 

Second, the use of meshed particles versus a pressed wafer will typically lead to nonuniformity of X-ray absorption thickness. This can be directly observed by placing an X-ray sensitive camera behind the sample a sample of a powder pressed into a wafer is spectroscopically more uniform than a catalyst bed of meshed particles. Naturally the contrast becomes more extreme as the meshed particles become larger. Moreover, if the sample is spatially nonuniform then severe constraints are placed on the positional stability of the X-ray beam. Any motion of the position of the X-ray beam will then probe different thicknesses of the sample, with direct consequences on the measured S/N. From the perspective of XAFS spectroscopy, any nonuniformity of the sample thickness could directly affect the accuracy of the measurement of the amplitude of the X-ray absorption coefficient. It is the amplitude that contains information about the coordination number and site disorder. As has been discussed elsewhere (Koningsberger and Prins, 1988), these amplitude distorting effects are given the general heading of "thickness effects." In brief, a thickness effect occurs when part of the incident X-ray beam is not attenuated by the sample. In the case of meshed particles this would be in the form of pinholes in the sample. 

Real systems are typically spatially nonuniform even if they are inhomoge neous. Such systems are obviously nonequibbrium, since the nonuniform in respect to temperature, pressure, concentrations, and so on generates the matter and/or heat fluxes. This section considers some mathematical tools for calculating thermodynamic forces and fluxes in such systems, as well as the relation of these quantities with conventional thermo dynamic parameters. 

Calculating the Thermodynamic Forces in Spatially Nonuniform Systems... 

To calculate the thermodynamic forces in spatially nonuniform systems, mathematical notion of divergence is essential. The divergence of a vecto rial field a(M) in point (x,y,z) is determined as a scalar quantity... 

In real systems, especially in heterogeneous catalytic and biological sys terns, the reactants are often arranged irregularly in space. Therefore, an arising instability may cause simultaneous diffusion of substances from one point to another inside the system to make the reactant concentration oscillations arranged in a certain manner in space during the occurrence of nonlinear chemical transformations. As a result, a new dissipative structure arises with a spatially nonuniform distribution of certain reac tants. This is a consequence of the interaction between the process of diffusion, which tends to create uniformity of the system composition, and local processes of the concentration variations in the course of nonlinear... 

Similar equations can be written for all relevant compartments. If parameters are chosen, the resulting set of nonlinear ordinary differential equations can be solved numerically to yield predictions of the concentration of the drug and metabolite(s) in each of the compartments as a function of time. Of course, the simplifying assumptions here can be relaxed to include much more detail concerning plasma and tissue binding, transport at the level of the blood capillary and cell membrane, and spatial nonuniformity — but at the cost of increasing complexity and the requirement for more parameters. 

Kaul et at (58) also used the elementary steps of Eq. (19) to model their results for CO oxidation over Pt/Si02, for which they used the experimental techniques of transient Fourier-transform infrared (FTIR), temperature-programmed reaction, and concentration-programmed reaction (59). They later applied the same methods to the CO oxidation over Rh/Si02. In the numerical calculations many parameters were taken from surface science results, and the agreement between experiment and simulation is good 60) when spatial nonuniformities are not present. 

Rossi R. C. and Lewis N. S. (2001), Investigation of the size-scaling behavior of spatially nonuniform barrier height contacts to semiconductor surfaces using ordered nanometer-scale nickel arrays on silicon electrodes , J. Phys. Chem. B 105, 12303-12318. 

Figure 8.3 A layer of fluid-filled permeable material that is subjected to a spatially nonuniform load. The layer is seen edge-on at the bottom of each picture and is imagined to extend indefinitely perpendicular to the page. In (a) we imagine the load to be imposed by heavy rods of unequal length in (b), we imagine a permeable material overlying the layer of interest, and a second fluid occupying the pore space of this material to nonuniform depths.

ON THE THEORY OF THE ORIGIN OF SPATIALLY NONUNIFORM STATIONARY STATES (DISSIPATIVE STRUCTURES) IN HETEROGENEOUS CATALYTIC SYSTEMS... 

A. Existence of Spatially Nonuniform Self-Oscillatory Modes... 

In this work, the numerical experiment was carried out for y = 1, i.e., in the region where uniform oscillations in system (3,4) are impossible. Can the distributedness of the system act, under such conditions, as a factor capable of inducing a spatially nonuniform self-oscillatory process There is no strict criterion at the present time for the existence of this new type of self-oscillatory instability. Numerical experiments, however, provide a positive answer to this question. Figure 36 shows a time display of a self-oscillatory process discovered in the course of DS transformation as the catalytic element length / was varied. Other parameters were ... 

Volpert, A. I. and Ivanova, A. N., On the Spatially Nonuniform Solutions of Nonlinear Diffusion Equations, Preprint (in Russian), Chemogolovka Branch of Inst. Chem. Phys., USSR Acad. Sci., Chemogolovka, 1981. 

On The Theory Of The Origin Of Spatially Nonuniform Stationary States (Dissipative Structures) In Heterogeneous Catalytic Systems 551... 

In Section 1.1 we briefly described the velocity field for some cases of gradient flows with nonuniform structure. For particles whose size is much less than the characteristic scale of spatial nonuniformity of the flow, the velocity distribution (1.1.15) can be used as the velocity distribution remote from the particle in problems about mass transfer to a particle in a linear shear flow. 

J. Kadaksham, P. Singh and N. Aubry, Dynamics of electrorheological suspensions subjected to spatially nonuniform electric fields. Journal of Fluids Engineering, 126(2), 170-179 (2004). 

A spiral wave is created if some special initial conditions are used or if a single propagating wave front is broken down. Although the spatial nonuniformity of the medium can simplify the emergence of spiral waves, they exist in completely homogeneous media as well. The influence of the boundary conditions decreases very fast with the distance from the boundary and practically vanishes if this distance exceeds the spiral wavelength. Thus, once created spiral waves represent very robust sources of wave activity and oscillations in excitable media. 

This observation constitutes the basic idea of the local equilibrium model of Prigogine, Nicolis, and Misguich (hereafter referred to as PNM). One considers the case of a spatially nonuniform system and deduces from (3) an integral equation for the pair correlation function that is linear in the gradients. This equation is then approximated in a simple way that enables one to derive explicit expressions for all thermal transport coefficients (viscosities, thermal conductivity), both in simple liquids and in binary mixtures, excluding of course the diffusion coefficient. The latter is a purely kinetic quantity, which cannot be obtained from a local equilibrium hypothesis. 

The agreement is remarkably good, considering that the computer codes take into account the changes in cross sections and reactor parameters that occur as fuel composition changes, and also follow spatial nonuniformities in flux and fuel composition. All these factors have been neglected in this section. 

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