Equilibrium approach, local

MSN.47. I. Prigogine, G. Nicolis, and J. Misguich, Local equilibrium approach to transport processes in dense media, J. Chem. Phys. 43, 4516-4521 (1965). 

When considering heat transfer in a multiphase system, one can either treat each phase separately or the phases can be assumed to be in local equilibrium. In general, it is more complicated to treat each phase separately hence, we will use the local equilibrium approach that assumes all the phases have the same local average temperature. By performing an energy balance over a differential resin element one obtains,... 

According to the previous discussion, a PEVD process relies on mass and charge transport in two solid state ionic materials of a PEVD system, i.e., the solid electrolyte (E) and the product (D). Since mass and charge transport occur in solid state ionic materials, the conductivity mechanism imposes some restrictions, and fundamental considerations in a PEVD system can be obtained through the local equilibrium approach. In the following, mass and charge transport in both phases will be discussed. 

Let us examine the process of diffusion in clays at the first stage of their consolidation, before the moment when m = 0. We will suppose that the diffusion solution contains cations of the same type, as cations of the exchanging complex of clay meaning that there is no ion exchange reaction. We will also suppose that in every point of environment the equilibrium between the solution in transport pores and the solution between clay particles is established immediately (its parameters we will be marked by the overline). Like this we will successively build the model of diffusion in clays in a local equilibrium approach. The conditions of equilibrium of cations (index 1) and anions (index 2) of two solutions are the equality of the chemical potentials, =. Pi= where... 

Energy minimization is a powerful tool for establishing global deformation characteristics of a strained film-substrate system on the basis of a particular set of assumptions and approximations, as demonstrated above. Essentially, the same assumptions and approximations can be used as a basis for a local equilibrium approach to arrive at the same results. Indeed, by exploiting the features of translational invariance and symmetry, it is revealed that some of the assumptions on which the energy analysis is based are known a priori to be true. These include the vanishing of the normal stress component Uzz, the shear stress component <7re and the shear strain components e z and egz- This approach was adopted by Freund (1993) and the main features are sketched out here for future reference. 

Benson, S.W. The induction period in chain reactions. J. Chtan. Phys. 20, 1605-1612 (1952) Beretta, G.P., Keck, J.C., Janhozorgi, M., Metghalchi, H. The rate-controlled constrained-equilibrium approach to far-from-local-equilibrium thermodynamics. Entropy 14, 92-130 (2012) Bhattacharjee, B., Schwer, D.A., Barton, P.I., Great, W.H. Optimally-reduced kinetic models reaction elimination in large-scale kinetic mechanisms. Cranbust. Flame 135,191-208 (2003)... 

Surface elasticity in the sense under consideration cannot exist in a system of pure liquid phases. In a system containing surfactant molecules, gradients in interfacial tension can arise as a result of the formation of new area, as mentioned above, or because of the loss of interfacial area. In the former case, the time lag between the formation of new interface and the diffusion of surfactant to that interface will produce an interfacial tension that is higher than equilibrium. The local value of the surface excess T, will fall and the value of a, will approach that of the pure system. The net effect will be a tendency for the interface to contract, providing a healing effect to reduce the chance of droplet coalescence. In the case of loss of interfacial area, there will be a time lag from the point of compression of the interfacial film until the excess surfactant molecules can desorb and diffuse away from the interface. 

In a recent paper [11] this approach has been generalized to deal with reactions at surfaces, notably dissociation of molecules. A lattice gas model is employed for homonuclear molecules with both atoms and molecules present on the surface, also accounting for lateral interactions between all species. In a series of model calculations equilibrium properties, such as heats of adsorption, are discussed, and the role of dissociation disequilibrium on the time evolution of an adsorbate during temperature-programmed desorption is examined. This approach is adaptable to more complicated systems, provided the individual species remain in local equilibrium, allowing of course for dissociation and reaction disequilibria. 

If it cannot be guaranteed that the adsorbate remains in local equilibrium during its time evolution, then a set of macroscopic variables is not sufficient and an approach based on nonequihbrium statistical mechanics involving time-dependent distribution functions must be invoked. The kinetic lattice gas model is an example of such a theory [56]. It is derived from a Markovian master equation, but is not totally microscopic in that it is based on a phenomenological Hamiltonian. We demonstrate this approach... 

For adsorbates out of local equilibrium, an analytic approach to the kinetic lattice gas model is a powerful theoretical tool by which, in addition to numerical results, explicit formulas can be obtained to elucidate the underlying physics. This allows one to extract simplified pictures of and approximations to complicated processes, as shown above with precursor-mediated adsorption as an example. This task of theory is increasingly overlooked with the trend to using cheaper computer power for numerical simulations. Unfortunately, many of the simulations of adsorbate kinetics are based on unnecessarily oversimplified assumptions (for example, constant sticking coefficients, constant prefactors etc.) which rarely are spelled out because the physics has been introduced in terms of a set of computational instructions rather than formulating the theory rigorously, e.g., based on a master equation. 

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. 

The possible advantages of this system over the equilibrium-limited reactor system are smaller catalyst beds, lower gas recycle requirements, and lower capital requirements. The possible disadvantages of this system are (a) practically no turn-down since any turn-down would be equivalent to decreased space velocities, closer approach to equilibrium, and higher temperature rises (b) maldistribution of gases across the bed would give rise to excessive temperature rises in zones of low flow and (c) considerably shortened catalyst life because of possible high local or zonal temperature and, concurrently, greater chances for carbon laydown. 

Figure 5.10 Representation of the formation of the lone pair in the PF3 molecule, (a) An isolated P3 + ion consisting of a P5+ core surrounded by two nonbonding electrons in a spherical distribution, (b) Three approaching F ions distort the distribution of the two valence shell electrons pushing them to one side of the P5+ core, (c) When the F ligands reach their equilibrium positions, the two nonbonding electrons are localized into a lone pair, which acts as a pseudo-ligand giving the PF3 molecule its pyramidal geometry.

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