Local equilibrium definition

The terms in (2.5) with n 2 decay rapidly they describe how the local equilibrium in either well is established. The identification (2.6) is, of course, valid only with the margin of uncertainty of the definition of the escape times, i.e., disregarding terms of relative order X1/X2. 

Within the small volume of material shown at r in Fig. 1.5, a certain quantity of species i is expected. This specifies a concentration for that particular small box this concentration will be in local equilibrium with some diffusion potential. However, materials are comprised of discrete atoms (molecules), which complicates the definition of local concentration when the volume sampled becomes comparable to the mean distance between atoms being counted. In Fig. 1.5, for the physical... 

Definition of Critical and Rate-Limiting Bottlenecks" The hypothesis of local equilibrium within the reservoirs means that the set of transitions from reservoir to reservoir can be described as a Markov process without memory, with the transition probabilities given by eq. 4. Assuming the canonical ensemble and microscopic reversibility, the rate constant Wji, for transitions from reservoir i to reservoir j can be written... 

The kinetic theory leads to the definitions of the temperature, pressure, internal energy, heat flow density, diffusion flows, entropy flow, and entropy source in terms of definite integrals of the distribution function with respect to the molecular velocities. The classical phenomenological expressions for the entropy flow and entropy source (the product of flows and forces) follow from the approximate solution of the Boltzmann kinetic equation. This corresponds to the linear nonequilibrium thermodynamics approach of irreversible processes, and to Onsager s symmetry relations with the assumption of local equilibrium. 

The application of the z-transform and of the coherence theory to the study of displacement chromatography were initially presented by Helfferich [35] and later described in detail by Helfferich and Klein [9]. These methods were used by Frenz and Horvath [14]. The coherence theory assumes local equilibrium between the mobile and the stationary phase gleets the influence of the mass transfer resistances and of axial dispersion (i.e., it uses the ideal model) and assumes also that the separation factors for all successive pairs of components of the system are constant. With these assumptions and using a nonlinear transform of the variables, the so-called li-transform, it is possible to derive a simple set of algebraic equations through which the displacement process can be described. In these critical publications, Helfferich [9,35] and Frenz and Horvath [14] used a convention that is opposite to ours regarding the definition of the elution order of the feed components. In this section as in the corresponding subsection of Chapter 4, we will assume with them that the most retained solute (i.e., the displacer) is component 1 and that component n is the least retained feed component, so that... 

The formation-dissolution mechanism assumes a total dissolution of a micelle in order to reestablish the local equilibrium monomer concentration. This model is based on an idealised distribution of only monomers and micelles with a definite aggregation number. Mechanism 2 is based on the existence of micelles of different size and therefore, a broad micelle size... 

The temperature-dependent physical constants in the mass balance (i.e., the kinetic rate constant and the equilibrium constant) are expressed in terms of nonequilibrium conversion x using the linear relation (3-42). The concept of local equilibrium allows one to rationalize the definition of temperature and calculate an equilibrium constant when the system is influenced strongly by kinetic changes. In this manner, the mass balance is written with nonequilibrium conversion of CO as the only dependent variable, and the problem can be solved by integrating only one ordinary differential equation for x as a function of reactor volume. 

Here T is the local-equilibrium temperature. In extended irreversible thermodynamics fluxes are independent variables. The kinetic temperature associated to the three spatial directions of along the flow, along the velocity gradient, and perpendicular to the previous to directions may be different from each other. To define temperature from the entropy is the most fundamental definition, and the nonequilibrium temperature may come from the derivative of a nonequilibrium entropy du/dS) -p. Effective nonequilibrium temperature may be defined from the fluctuation-dissipation theorem relating response function and correlation function. 

A Xi is the reaction layer thickness in the case where it is formed in a reaction starting with the pure reactants. A Xj, on the other hand, is the reaction layer thickness in the case of a reaction where the reaction layer under consideration is formed from the saturated adjacent phases. These definitions hold for a one-phase reaction product as well as for a multiphase reaction product. Since the average diffusion coefficients in a certain reaction layer (phase k) cannot depend upon the starting material if local equilibrium prevails, one may return to eqs. (7-35) and (7-36) in order to obtain a relation between the reaction rate constants of the first and second kind. By taking into account the definitions, the relation between the two rate constants for a reaction product with phases of very narrow ranges of homogeneity can be shown to be ... 

In the introductory chapter we clearly distinguished between so-called global and local equilibrium states. As we know, in global equilibrium states there are no spatial or temporal variations in the system density, temperature, pressure, entropy, and the like. How are such states arrived at when we know that matter is composed of molecules in constant random or thermal motion with continuous intermolecu-lar interactions and dynamics Our goal in this chapter is to see how such states theoretically come about and to introduce formal, molecular-based definitions of temperature, pressure, entropy, and internal... 

In the next chapter, we will consider the nonequilibrium behavior of matter in the most general way by deriving the spatial and temporal variations in density, average velocity, internal and kinetic energy, and entropy. We will use the formal definitions of these quantities introduced in this chapter, including the possibility of their spatial and temporal variations via the probability density function described by the full Liouville equation. In the next chapter, we will also formally define local equilibrium behavior and look at some specific, well-known examples of such behavior in science and engineering. 

The justification of variational transition state theory is rigorous only in a classical mechanical world because, when the local equilibrium assumption is valid, VTST provides an upper bound on the classical mechanical rate constant. One optimizes the definition of the transition state to minimize recrossing, and the calculated rate constant converges to the exact rate constant from above. 

E] Use s BoUes Fair (Ref. 75) data base to determine new effective area to use with Onda et al. (Ref. 126) correlation. Same definitions as 5-28-D. P = total pressure, atm Mq = gas, molecular weight m = local slope of equilibrium curve Lf /Gf = slope operating line Z = height of packing in feet. 

The subscripts m, L, S, and G will represent the local two-phase mixture, liquid phase, solid phase and gas phase, respectively. The definitions below are given in terms of solid-liquid (S-L) mixtures, where the solid is the more dense distributed phase and the liquid the less dense continuous phase. The same definitions can be applied to gas-liquid (G-L) flows if the subscript S is replaced by L (the more dense phase) and the L by G (the less dense phase). The symbol

volume fraction of the more dense phase, and s is the volume fraction of the less dense phase (obviously (p = 1 — e). An important distinction is made between ([Pg.444]

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