States global equilibrium

Systems involving an interface are often metastable, that is, essentially in equilibrium in some aspects although in principle evolving slowly to a final state of global equilibrium. The solid-vapor interface is a good example of this. We can have adsorption equilibrium and calculate various thermodynamic quantities for the adsorption process yet the particles of a solid are unstable toward a drift to the final equilibrium condition of a single, perfect crystal. Much of Chapters IX and XVII are thus thermodynamic in content. 

The term balance means that all forces generated by, or acting on, the rotating element of a machine-train are in a state of equilibrium. Any change in this state of equilibrium creates an imbalance. In the global sense, imbalance is one of the most common abnormal vibration profiles exhibited by all process machinery. 

In a similar vein, Riemann s formalism finds useful application in expressing the global thermodynamic behavior of a system S. The metric geometry governed by M( ) represents thermodynamic responses (as before), while labels distinct states of equilibrium, each exhibiting its own local geometry of responses. The state-specifier manifold may actually be chosen rather freely, for example, as any/independent intensive variables (such as gi = T, 2 = P 3 = Mr, > = l c-p)- For our purposes, it is particularly convenient to... 

In many cases, the study of kinetics concerns itself with the paths and rates adopted by systems approaching equilibrium. Thermodynamics provides invaluable information about the final state of a system, thus providing a basic reference state for any kinetic theory. Kinetic processes in a large system are typically rapid over short length scales, so that equilibrium is nearly satisfied locally at the same time, longer-length-scale kinetic processes result in a slower approach to global equilibrium. Therefore, much of the machinery of thermodynamics can be applied locally under an assumption of local equilibrium. It is clear, therefore, that the subject of thermodynamics is closely intertwined with kinetics. 

In the global equilibrium state (see the Euler Eq. 4) one obtains the local hardness equalization at the global hardness level ... 

It can be easily verified that iu[Pg.56]

The electrolytic permeability is a property of any solid electrolyte, since a local equilibrium involving ions and electrons is required by - thermodynamics for any conditions close to steady-state or global equilibrium. However, it is possible to optimize the level of permeability, depending on particular applications. In many cases, the permeability is a parasitic phenomenon leading to power losses in - fuel cells and - batteries, lower efficiency of solid-state electrolyzers and -> electrochemi-... 

States away from global equilibrium are called the thermodynamic branch (Figure 2.2). Systems not far from global equilibrium may be extrapolated around equilibrium state. For systems near equilibrium, linear phenomenological equations may represent the transport and rate processes. The linear nonequilibrium thermodynamics theory determines the dissipation function or the rate of entropy production to describe such systems in the vicinity of equilibrium. This theory is particularly useful to describe coupled phenomena, and quantify the level of coupling in physical, chemical, and biological systems without detailed process mechanisms. 

Far from global equilibrium, the reaction velocity is not only related to affinity but also depends on the concentration of species. If we expand Eq. (9.123) and consider the near global equilibrium state (, 4 / RT . l), then we have a linear relationship between the reaction velocity and the chemical affinity for an elementary reaction... 

The second law for isolated systems shows that the excess entropy, A.V S SKI < 0, increases monotonically in time, d(AS)/dt > 0. Therefore, it plays the role of a Lyapunov function, and defines a global stability. So, dfi/dt is a Lyapunov function that guarantees the global stability of stationary states that are close to global equilibrium. 

In the linear nonequilibrium thermodynamics theory, the stability of stationary states is associated with Prigogine s principle of minimum entropy production. Prigogine s principle is restricted to stationary states close to global thermodynamic equilibrium where the entropy production serves as a Lyapunov function. The principle is not applicable to the stability of continuous reaction systems involving stable and unstable steady states far from global equilibrium. 

Irreversible processes may promote disorder at near equilibrium, and promote order at far from equilibrium known as the nonlinear region. For systems at far from global equilibrium, flows are no longer linear functions of the forces, and there are no general extremum principles to predict the final state. Chemical reactions may reach the nonlinear region easily, since the affinities of such systems are in the range of 10-100 kJ/mol. However, transport processes mainly take place in the linear region of the thermodynamic branch. 

The stability of transport and rate systems is studied either by nonequilibrium thermodynamics or by conventional rate theory. In the latter, the analysis is based on Poincare s variational equations and Lyapunov functions. We may investigate the stability of a steady state by analyzing the response of a reaction system to small disturbances around the stationary state variables. The disturbed quantities are replaced by linear combinations of their undisturbed stationary values. In nonequilibrium thermodynamics theory, the stability of stationary states is associated with Progogine s principle of minimum entropy production. Stable states are characterized by the lowest value of the entropy production in irreversible processes. The applicability of Prigogine s principle of minimum entropy production is restricted to stationary states close to global thermodynamic equilibrium. It is not applicable to the stability of continuous reaction systems involving stable and unstable steady states far from global equilibrium. The steady-state deviation of entropy production serves as a Lyapunov function. 

To solve highly nonlinear differential equations for systems far from global equilibrium, the method of cellular automata may be used (Ross and Vlad, 1999). For example, for nonlinear chemical reactions, the reaction space is divided into discrete cells where the time is measured, and local and state variables are attached to these cells. By introducing a set of interaction rules consistent with the macroscopic law of diffusion and with the mass action law, semimicroscopic to macroscopic rate processes or reaction-diffusion systems can be described. 

Furthermore, for non-isothermal situations we need to be able to calculate the thermodynamics of fluid dynamics. However, thermodynamics deals with relatively permanent states, called equilibrium states, within uniform fields of matter [7] [145] [42] [54]. Any changes are assumed to be extremely slow. On the other hand, the fluid motions of interest in fluid mechanics are not necessary slow. Nevertheless, it has been assumed that the classical thermodynamics can be directly applied to any flow system provided that an instantaneous local thermodynamic state is considered and that the rates of change are not too large [168]. A more common statement is that the thermodynamics require that the fluids are close to local equilibrium, but may not be in global equilibrium. However, all systems are supposed to be relaxing towards a state of global thermodynamic equilibrium. 

Normally, the point defects are expected to be in a local or global equilibrium state when the thermodynamic approach can be used. Within the framework of this approach, the defects and their simplest associates are treated as chemical species [9,15-19]. Therefore, the chemical potential of each structural element (p ), which may correspond to atoms (ions) in their regular positions or defects, and the Gibbs energy change for any process involving the i-type species (AG), can be written as... 

The AIM chemical potentials defined by the partial functional derivatives of equation (35), calculated for the fixed external potential and the frozen embedding densities pp a r) of the remaining subsystems, are equalized only when the subsystems are mutually open [4,5], This is the case in the global equilibrium state considered in the preceding section. In what follows we shall denote such open subsystem condition by the vertical broken fines in the symbolic representation of the molecular system as a whole, Mg = (a fi y. ..), in the global (g, intersubsystem) equilibrium of the ground-state of an externally open system ... 

Following the global equilibrium development of the preceding section one defines the thermodynamic potentials corresponding to the four alternative sets of the constrained-equilibrium states, for the alternative sets of the state parameters,... 

In the global equilibrium state, when all constituent subsystems are mutually open, the transformations of infinitesimal displacements (perturbations) of the global and local parameters of state, which determine the Legendre transformed representation under consideration, into differentials of the respective conjugate state-variables (responses) can be summarized in terms of the following matrix integral equations [4,5,8,12,13] ... 

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