Homogeneity, spatial

Consider a spatially homogeneous reacting mixture where concentration gradients are removed by stirring or rapid... 

A homogeneity index or significance coefficienf has been proposed to describe area or spatial homogeneity characteristics of solids based on data evaluation using chemometrical tools, such as analysis of variance, regression models, statistics of stochastic processes (time series analysis) and multivariate data analysis (Singer and... 

The airflow equations presented above are based on the assumption that the soil is a spatially homogeneous porous medium with constant intrinsic permeability. However, in most sites, the vadose zone is heterogeneous. For this reason, design calculations are rarely based on previous hydraulic conductivity measurements. One of the objectives of preliminary field testing is to collect data for the reliable estimation of permeability in the contaminated zone. The field tests include measurements of air flow rates at the extraction well, which are combined with the vacuum monitoring data at several distances to obtain a more accurate estimation of air permeability at the particular site. 

It is thus entirely expressed in terms of the zero wave number Fourier coefficient pQ(p t). Similarly, the pair correlation function in a spatially homogeneous system is defined by17... 

In particular, if we start with a spatially homogeneous system, which is such that ... 

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 stationary, spatially-homogeneous mean held solution of (3) (without taking into account of fluctuations) is... 

Reaction-diffusion systems have been studied for about 100 years, mostly in solutions of reactants, intermediates, and products of chemical reactions [1-3]. Such systems, if initially spatially homogeneous, may develop spatial structures, called Turing structures [4-7]. Chemical waves of various types, which are traveling concentrations profiles, may also exist in such systems [2, 3, 8]. There are biological examples of chemical waves, such as in parts of glycolysis, heart... 

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. 

The monodisperse materials described hereafter were obtained with the Couette type cell designed by Bibette et al. [ 150,159]. It consists of two concentric cylinders (rotor and stator) separated by a very narrow gap (100 pm), allowing application of spatially homogeneous shear rates over a very wide range (from 0 to 14280 s ), with shearing durations of the order of 10 s. 

A second limiting physical/hydrodynamic case is the soil as a porous bed. Often others simulate undisturbed soils in the lab with soil columns, however we have chosen to use a slice of such a column a differential volume reactor (DVR)-as the experimental design (22). This approach offers advantages in the ability to develop a more spatially homogeneous system and also contributes to the perturbation/response analysis needed for systems identification. 

A phenomenogical expression for the hydrodynamic force F may be constructed by assuming that this force is linear in the flux velocities and in the strength of any applied flow field. We consider a system that is subjected to a macroscopic flow field v(r) characterized by a spatially homogeneous macroscopic velocity gradient Vv. We assume that Fa vanishes for all a = 1in the equilibrium state, where the flux velocities and the macroscopic... 

The simplest and often most suitable modeling tool is the one-box model. One-box models describe the system as a single spatially homogeneous entity. Homogeneous means that no further spatial variation is considered. However, one-box models can have one or several state variables, for instance, the mean concentration of one or several compounds i which are influenced both by external forces (or inputs) and by internal processes (removal or transformation). A particular example, the model of the well-mixed reactor with one state variable, has been discussed in Section 12.4 (see Fig. 12.7). The mathematical solution of the model has been given for the special case that the model equation is linear (Box 12.1). It will be the starting point for our discussion on box models. 

Note The problem with diurnal input variation is underestimated by the one-box approach in which spatial homogeneity is assumed. In fact, the limiting factor is spatial mixing. 

The production of species i (number of moles per unit volume and time) is the velocity of reaction,. In the same sense, one understands the molar flux, jh of particles / per unit cross section and unit time. In a linear theory, the rate and the deviation from equilibrium are proportional to each other. The factors of proportionality are called reaction rate constants and transport coefficients respectively. They are state properties and thus depend only on the (local) thermodynamic state variables and not on their derivatives. They can be rationalized by crystal dynamics and atomic kinetics with the help of statistical theories. Irreversible thermodynamics is the theory of the rates of chemical processes in both spatially homogeneous systems (homogeneous reactions) and inhomogeneous systems (transport processes). If transport processes occur in multiphase systems, one is dealing with heterogeneous reactions. Heterogeneous systems stop reacting once one or more of the reactants are consumed and the systems became nonvariant. 

Fig. 1.27. Self-organization in spatially homogeneous and inhomogeneous media.

Spatial homogeneity of a system (needed for making use of the formal chemical kinetics) is secured, first of all, by complete particle mixing. On... 

The Hanusse theorem [23] discussed in Section 2.1.1 was later generalized for the case of diffusion by Tyson and Light [32], Therefore, the mono- and bimolecular reactions with one or two intermediate products are expected to strive asymptotically, as t —> oo, for the stationary spatially-homogeneous solution Ci(r, oo) = nt(oo) corresponding to equations (2.1.2) for a system with the complete particle mixing. 

Due to the spatial homogeneity it is independent of fj allowing us to introduce the simplest spatial characteristics - macroscopic densities of particles (concentrations)... 

Deviation from standard chemical kinetics described in (Section 2.1.1) can happen only if the reaction rate K (t) reveals its own non-monotonous time dependence. Since K(t) is a functional of the correlation functions, it means that these functions have to possess their own motion, practically independent on the time development of concentrations. The correlation functions characterize the intermediate order in the particle distribution in a spatially-homogeneous system. Change of such an intermediate order could be interpreted as a series of structural transitions. 

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