Local equilibrium postulate

Within the proposed context the quasi-steady assumption is imposed by neglecting the local spatial variability of the concentration c°, bulk phase pressure and streaming potential tp° in the clusters by assuming these quantities equal to their corresponding counterparts in the macro-pore system. Therefore al local equilibrium we postulate c (x, y, t) = C(j x.l) ip°(x,y,t) =... 

Engineering systems mainly involve a single-phase fluid mixture with n components, subject to fluid friction, heat transfer, mass transfer, and a number of / chemical reactions. A local thermodynamic state of the fluid is specified by two intensive parameters, for example, velocity of the fluid and the chemical composition in terms of component mass fractions wr For a unique description of the system, balance equations must be derived for the mass, momentum, energy, and entropy. The balance equations, considered on a per unit volume basis, can be written in terms of the partial time derivative with an observer at rest, and in terms of the substantial derivative with an observer moving along with the fluid. Later, the balance equations are used in the Gibbs relation to determine the rate of entropy production. The balance equations allow us to clearly identify the importance of the local thermodynamic equilibrium postulate in deriving the relationships for entropy production. 

The purpose of the study of irreversible thermodynamics is to extend classical thermodynamics to include systems in which irreversible processes (e.g., diffusion and heat transfer) are taking place. Such an extension is made possible by assuming that for systems not too far from equilibrium the postulate of local equilibrium applies Departures from local equilibrium are sufficiently small that all thermodynamic state quantities may be defined locally by the same relations as for systems at equilibrium. ... 

The first postulate of irreversible thermodynamics is that the fluxes (or dependent variables) are directly proportional to the driving forces (or independent variables). [Actually, it may be shown that the assumption of local equilibrium follows from the assumption of a linear relation between the fluxes and driving forces (Truesdell, 1969).] If we take the di as dependent variables and the (m, — Wy) as independent variables we may, therefore, write... 

This theory, developed by Lewis19 and Whitman, a> supposes that motion in the two phases dies out near the interface and the entire resistance to transfer is considered as heing contained in two fictitious films on either side of the interface, in which transfer occurs by purely molecular diffusion. It is postulated that local equilibrium prevails at the interface and that the cimeeniralion gradiants are established so rapidly in the films compared to the total time of contact thel steady-state diffusion may be assumed,... 

One of the important postulates of irreversible thermodynamics is the postulate of local equilibrium discussed in Chapter 10. Accordingly, the local rate of change of the entropy density is... 

In equilibrium thermodynamics model A and in model B not far from equilibrium (and with no memory to temperature) the entropy may be calculated up to a constant. Namely, in both cases S = S(V, T) (2.6)2, (2.25) and we can use the equilibrium processes (2.28) in B or arbitrary processes in A for classical calculation of entropy change by integration of dS/dT or dS/dV expressible by Gibbs equations (2.18), (2.19), (2.38) through measurable heat capacity dU/dT or state Eqs.(2.6>, (2.33) (with equilibrium pressure P° in model B). This seems to accord with such a property as in (1.11), (1.40) in Sects. 1.3, 1.4. As we noted above, here the Gibbs equations used were proved to be valid not only in classical equilibrium thermodynamics (2.18), (2.19) but also in the nonequilibrium model B (2.38) and this expresses the local equilibrium hypothesis in model B (it will be proved also in nonuniform models in Chaps.3 (Sect. 3.6), 4, while in classical theories of irreversible processes [12, 16] it must be taken as a postulate). 

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... 

The quasi-equilibrium postulate states that systems are not far from equilibrium the gradients, or the thermodynamic forces are not too large. Within the system, local thermodynamic equilihrium holds. 

Kinetic Theory of Transport and Postulate of Local Equilibrium... 

In the above approach, an implicit assumption is that the thermodynamic quantities do not depend on the gradients in the system, i.e. it is postulated that entropy 5 is a function of the temperature T and the mole numbers density Hk, but not their gradients. However, flows represent a level of organization. This implies that the local entropy in a nonequilibiium system may be expected to be smaller than the equilibrium entropy. In the recently developed formalism of extended thermodynamics, gradients are included in the basic formalism and there appears a small correction to the local entropy due to the flows. We shall not be discussing this more advanced formalism. For a detailed exposition of extended thermodynamics, we refer the reader to some recent books [8-11]. Extended thermodynamics finds application in systems where there are large gradients, such as in shock waves. For almost all systems that we encounter, thermodynamics based on local equilibrium has excellent validity. 

It is postulated that the given relations are valid for a system at local equilibrium, the differential d may thus be substituted by the V-operator in the forthcoming... 

There are three different approaches to a thermodynamic theory of continuum that can be distinguished. These approaches differ from each other by the fundamental postulates on which the theory is based. All of them are characterized by the same fundamental requirement that the results should be obtained without having recourse to statistical or kinetic theories. None of these approaches is concerned with the atomic structure of the material. Therefore, they represent a pure phenomenological approach. The principal postulates of the first approach, usually called the classical thermodynamics of irreversible processes, are documented. The principle of local state is assumed to be valid. The equation of entropy balance is assumed to involve a term expressing the entropy production which can be represented as a sum of products of fluxes and forces. This term is zero for a state of equilibrium and positive for an irreversible process. The fluxes are function of forces, not necessarily linear. However, the reciprocity relations concern only coefficients of the linear terms of the series expansions. Using methods of this approach, a thermodynamic description of elastic, rheologic and plastic materials was obtained. 

The third approach is called the thermodynamic theory of passive systems. It is based on the following postulates (1) The introduction of the notion of entropy is avoided for nonequilibrium states and the principle of local state is not assumed, (2) The inequality is replaced by an inequality expressing the fundamental property of passivity. This inequality follows from the second law of thermodynamics and the condition of thermodynamic stability. Further the inequality is known to have sense only for states of equilibrium, (3) The temperature is assumed to exist for non-equilibrium states, (4) As a consequence of the fundamental inequality the class of processes under consideration is limited to processes in which deviations from the equilibrium conditions are small. This enables full linearization of the constitutive equations. An important feature of this approach is the clear physical interpretation of all the quantities introduced. 

Microtubules in the cytoskeleton and mitotic apparatus are also in a state of dynamic equilibrium and flux with unpolymerized tubulin, and tubulin appears to be an excellent example of the proteins which Pauling (1953) postulated to exist as globular protomers or as insoluble, fibrous, supramolecular structures akin to unpolymerized and polymeric hemoglobin S. The current view of the microtubule cytoskeleton in nondividing celb comes from the development of tubulin-specific antibodies for indirect immunofluorescent localization of microtubules (Fuller et al., 1975 Weber et al., 1975). The general structural features of such cyto-... 

The basic postulate of irreversible thermodynamics is that, near equilibrium, the local entropy production is nonnegative ... 

The postulate of local thermodynamic equilibrium in a discontinuous system is replaced by the requirement that the intensive properties change very slowly in each part, so that the parts are in thermodynamic equilibrium at every instant. The intensive properties are a function of time only, and they are discontinuous at the interface and may change by jumps. In the following sections, thermomechanical effects and thermoelectricity are summarized. 

Some essential discoveries concerning the organization of the adsorbed layer derive from the various spectroscopic measurements [38-46]. Here considerable experimental evidence is consistent with the postulate that ionic surfactants form localized aggregates on the solid surface. Microscopic properties like polarity and viscosity as well as aggregation number of such adsorbate microstructures for different regions in the adsorption isotherm of the sodium dedecyl sulfate/water/alumina system were determined by fluorescence decay (FDS) and electron spin resonance (ESR) spectroscopic methods. Two types of molecular probes incorporated in the solid-liquid interface under in situ equilibrium conditions... 

This relation is recognized from introductory subjects on thermodynamics. Recall that in equilibrium thermodynamics a local formulation is generally not needed, since the intensive state variables are independent of the space coordinates. This fundamental formulation of the total energy balance is known as the first law of thermodynamics for a closed system, which expresses the fundamental physical principle that the total energy of the system, Etotab is conserved (a postulate). 

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