Principle local equilibrium

When a two- or higher-phase system is used with two or more phases permeable to the solute of interest and when interactions between the phases is possible, it would be necessary to apply the principle of local mass equilibrium [427] in order to derive a single effective diffusion coefficient that will be used in a one-equation model for the transport. Extensive justification of the principle of local thermdl equilibrium has been presented by Whitaker [425,432]. If the transport is in series rather than in parallel, assuming local equilibrium with equilibrium partition coefficients equal to unity, the effective diffusion coefficient is... 

A distinction between solid/fluid and solid/solid boundaries is irrelevant from the point of view of transport theory. Solid/fluid boundaries in reacting systems are, for example, (A,B)/A, B, X (aq) or (A,B)/X2(g). More important is the distinction according to the number of components. In isothermal binary systems, the boundary is invariant if local equilibrium prevails. In higher than binary systems, the state of the a/fi interface is, in principle, variable and will be determined by the reaction kinetics, including the diffusion in the adjacent bulk phases. 

Various properties of crystals can be used to inspect c,( ,r), provided that appropriate detectors for the intensity of input and output signals are available. If the monitor response is sufficiently fast, one may determine the time dependence of solid state reactions. The monitoring of reactants and/or reaction products can serve this purpose, but the relation between signal intensity (property) and concentration Cj) must always be established first. Since functions of state are related to one another in a unique way, any equilibrium property can, in principle, be used to determine However, the necessary assumption of local equilibrium must still be... 

The reaction quotient may be measured, at least in principle, for the reacting system at any time. If Z is observed not to change, the system is at equilibrium, or trapped in a metastable state that serves as a local equilibrium. In informal work, a time-independent Z is identified direcdy with the equilibrium constant... 

Another method has been suggested to measure the activity of the surface oxygens on NiO catalysts as a quantity proportional to the ratio PcOj/Fco of the gas mixture of C02 and CO which, on contacting the surface of NiO, does not change in its composition [59]. The contact time allowed should be short so as not to oxidize or reduce the bulk oxygen, but long enough so that the local equilibrium between the gas phase and the surface is established. Thus the use of a differential reactor is most effective. In this way, the activities of the surface species alone can in principle, be determined. 

Inequalities (3.2) and (3.3) are generalizations of the principle of the minimal entropy production rate in the course of spontaneous evolution of its system to the stationary state. They are independent of any assump tions on the nature of interrelations of fluxes and forces under the condi tions of the local equilibrium. Expression (3.2), due to its very general nature, is referred to as the Qlansdorf-Prigogine universal criterion of evolution. The criterion implies that in any nonequilibrium system with the fixed boundary conditions, the spontaneous processes lead to a decrease in the rate of changes of the entropy production rate induced by spontaneous variations in thermodynamic forces due to processes inside the system (i.e., due to the changes in internal variables). The equals sign in expres sion (3.2) refers to the stationary state. 

In view of the results just given, we might guess that local average rates of chemical heat release in turbulent diffusion flames are calculable in some fashion from P(Z). However, this is not possible even in principle for equilibrium flows. As equilibrium is approached, the chemical production terms in the equations for species conservation become indeterminate, involving differences of large numbers that cancel (for example, see Section B.2.5.2). A more circuitous route is therefore needed to find the average rate of heat release [15], [20], [27], [28]. The necessary expressions will be developed here. 

In d.c. electrokinetics the time scale of the applied disturbance is usually much larger than that. The implication is that locally, in each segment of the double layer, the double layer may be considered at equilibrium, although over the particle as a whole, there is a tangential gradient in the ionic concentration. In sec. 3.13 we have referred to this feature as the local equilibrium principle. This principle starts to fail when the time scale of the applied disturbance is shorter than 10" s. For Instance, this may be the case in a.c. fields of frequencies 5 20 MHz. 

This result is sometimes called the principle of stress equilibrium, because it shows that the surface forces must be in local equilibrium for any arbitrarily small volume element centered at any point x in the fluid. This is true independent of the source or detailed form of the surface forces. 

In 3.1.2 we pointed out that thermodynamically based geochemical models can in principle only be used successfully for natural systems which exhibit areas of local equilibrium. We must now examine this idea more closely, and develop criteria for deciding whether or not the local equilibrium assumption (LEA) is a valid approximation in given systems. The conditions under which the LEA is applicable to groundwater and geological systems have been discussed extensively (Bahr, 1990 Bahr and Rubin,... 

Thus, the effect of reaction on diffusion is similar to the effect it has on infiltration. In principle it is possible to determine the relative abundance of the major isotopes in the fluid compared to the rock, though the separation of curves is much less pronounced (Baumgartner and Rumble 1988). Solutions to the local equilibrium reaction/diffusion Equations (42) and (43) and the non-reactive diffusion Equation (44) have the same form. A wealth of solutions are available in the classic reference of Crank (1975). 

Besides, small subsystems of the solution relax, i.e., reach equilibrium much sooner than the entire solution. As a result, chemical equilibrium in separate parts of the solution is reached at different times. Equilibrium, reached in a separate part of the solution, is called local chemical equilibrium. The local equilibrium principle maintains that each small (but macroscopic) element of volume in a nonequilibrium overall system at any moment in time is in the state of equilibrium. Special significance is attributed to local equilibrium at the boxmdary of different media, which determines the nature and rate of the mass exchange between them. 

The foundation of these models is the assumption of the principle of local equilibrium proposed by Ilia Romanovich Prigozhin (1917-2003). This concept assumes that if the study object is broken into blocks, each of them may be in total equilibrium even in the absence of such between the blocks. This concept is based on the idea that small volumes, due to small distances, reach equilibrium much faster than large ones. It enables breaking of a large non-equilibrium study object into numerous smaller size blocks, within bounds of which is reached total equilibrium. The greater the number of such blocks, the smaller they are, the lower the error... 

Morral" has criticized the Wagner analysis and argued that concentration profiles such as that shown in Figure B1 violate principles of local equilibrium and that solute enrichment in the zone of internal oxidation is not possible for the case of zero solubility product. Analysis of this issue is beyond the scope of this book but it should be remarked that many experimental observations are consistent with conclusions based on Wagner s model. 

The assertion that the results (3.171) with properties (3.174), (3.175) (in fact the same as in classical thermodynamics and proved in this model of nonsimple fluid) are valid even at nonequilibrium process (at nonzero o in (3.178)) is known as local equilibrium. This was taken as a starting principle in the classical theories of nonequilibrium processes [36, 80]. But in more complicated models local equilibrium need not be valid, cf. Sect. 2.2. 

If we assume that the ideal gas studied Mfils the local equihbrium (and this is the usual case ideal gas may be from the linear fluid models discussed here, but it may be also from some nonlinear models fulfilhng this principle, e.g. those in [78]), then property (3.213) follows from state equation (3.212). Indeed, the local equilibrium means the validity of Gibbs equations (3.200)i, (3.198)i, from... 

Therefore, the classical relations of thermochemistry were obtained. Especially, the Gibbs equations (4.201)-(4.206) are valid in arbitrary process in this chemically reacting mixture of fluids with linear transport properties, i.e. the principle of local equilibrium is valid in this mixture. But we show in the following relations that this accord with classical thermochemistry (e.g. [138]) is not quite identical indeed, if we differentiate (4.211) and use (4.22), (4.23) we obtain... 

These and all previous results of thermodynamic mixture which also fulfil Gibbs-Duhem equations (4.263) show the complete agreement with the classical thermodynamic of mixtures but moreover all these relations are valid much more generally. Namely, they are valid in this material model—linear fluid mixture—in all processes whether equilibrium or not. Linear irreversible thermodynamics [1-4], which studies the same model, postulates this agreement as the principle of local equilibrium. Here in rational thermodynamics, this property is proved in this special model and it cannot be expected to be valid in a more general model. We stress the difference in the cases when (4.184) is not valid—e.g. in a chemically reacting mixture out of equilibrium—the thermodynamic pressures P, Pa need not be the same as the measured pressure (as e.g. X =i Pa) therefore applications of these thermodynamic... 

Chapter 3 adds also the description of spatial distribution (gradients). Only single fluid is considered for the sake of simplicity and preparation of the basics for the subsequent treatment of mixtures. Mathematics necessary for the spatial description is introduced in Sect. 3.1. Section 3.2 in the same chapter stresses the importance of the referential frame (coordinate system) and its change in the mathematical description. Sections 3.3—3.6 shows the development of final material model (of a fluid) within our thermodynamic framework, consistent with general laws (balances) as well as with thermodynamic principles (the First and Second Laws and the principles of rational thermodynamics). The results of this development are simplified in Sect. 3.7 to the model of (single) fluid with linear transport properties. Sections 3.6 and 3.7 also show that the local equilibrium hypothesis is proved for fluid models. The linear fluid model is used in Sect. 3.8 to demonstrate how the stability of equilibrium is analysed in our approach. 

To illustrate the combination of rate and equilibrium principles, we next consider a widely used separation method, which is inherently unsteady packed bed adsorption. We imagine a packed bed of finely granulated (porous) solid (e.g., charcoal) contacting a binary mixture, one component of which selectively adsorbs (physisorption) onto and within the solid material. The physical process of adsorption is so fast relative to other slow steps (diffusion within the solid particle), that in and near the solid particles, local equilibrium exists... 

We shall illustrate the principles of the perturbation chromatography on the adsorption model for pore diffusion with local equilibrium. 

Local equilibrium is spoken of when the local values of thermodynamic and optical properties relate to each other as if the substance were in the state of general thermodynamic equilibrium. This principle of local equilibrium underlies hydrodynamics (Landau and Lifshitz, 1988) and unequilibrium thermodynamics (Glansdorff and Prigogine, 1971). 

On the hypothesis of local equilibrium, a description analogous to that proposed by Kroger can be used to determine the local situation in the crystal. The basic principles and assumptions of this description are briefly recalled in what follows. Here we will use as an independent variable the electron chemical potential Pe. It is the most appropriate parameter to characterize the degree of trap filling. Within a constant, it equals the distance between the Fermi level and the valence band edge. 

The fundamental hypothesis of CIT is the existence of a local-equilibrium condition. A series of finite volume cells is considered in a material body, in which local variables such as temperature and entropy are uniform and in equilibrium, but time-dependent. The variables can take different values from cell to cell. The majority of textbooks are written using this formulation (see, e.g., Kestin 1979, which refers to this as the principle of local state). The most important result of CIT under the local-equilibrium hypothesis is that, as a natural result of the Second Law of Thermodynamics in the course of a mechano-thermal process, we have the following entropy inequality ... 

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