Models local equilibrium

The first transport model (local equilibrium sorption) is formulated by assuming that local equilibrium exists along the column. No immobile-water phase is present. The water velocity is... 

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. 

Local Thermodynamic Equilibrium (LTE). This LTE model is of historical importance only. The idea was that under ion bombardment a near-surface plasma is generated, in which the sputtered atoms are ionized [3.48]. The plasma should be under local equilibrium, so that the Saha-Eggert equation for determination of the ionization probability can be used. The important condition was the plasma temperature, and this could be determined from a knowledge of the concentration of one of the elements present. The theoretical background of the model is not applicable. The reason why it gives semi-quantitative results is that the exponential term of the Saha-Eggert equation also fits quantum-mechanical expressions. 

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. 

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... 

FIG. 16-35 Elution curves under trace conditions with a constant separation factor isotherm for different feed loadings and N = 80. Solid lines, rate model dashed line, local equilibrium theory for ZF = 0.4. 

The standard wall function is of limited applicability, being restricted to cases of near-wall turbulence in local equilibrium. Especially the constant shear stress and the local equilibrium assumptions restrict the universality of the standard wall functions. The local equilibrium assumption states that the turbulence kinetic energy production and dissipation are equal in the wall-bounded control volumes. In cases where there is a strong pressure gradient near the wall (increased shear stress) or the flow does not satisfy the local equilibrium condition an alternate model, the nonequilibrium model, is recommended (Kim and Choudhury, 1995). In the nonequilibrium wall function the heat transfer procedure remains exactly the same, but the mean velocity is made more sensitive to pressure gradient effects. 

In light of the small solubilities of many minerals, the extent of reaction predicted by this type of calculation may be smaller than expected. Considerable amounts of diagenetic cements are commonly observed, for example, in sedimentary rocks, and crystalline rocks can be highly altered by weathering or hydrothermal fluids. A titration model may predict that the proper cements or alteration products form, but explaining the quantities of these minerals observed in nature will probably require that the rock react repeatedly as its pore fluid is replaced. Local equilibrium models of this nature are described later in this section. 

Reaction between rocks and the groundwaters migrating through them is most appropriately conceptualized by using a model configuration based on the assumption of local equilibrium (Section 13.3). 

A final variant of local equilibrium models is the dump option (Wolery, 1979). Here, once the equilibrium state of the initial system is determined, the minerals... 

Beginning in the late 1980s, a number of groups have worked to develop reactive transport models of geochemical reaction in systems open to groundwater flow. As models of this class have become more sophisticated, reliable, and accessible, they have assumed increased importance in the geosciences (e.g., Steefel et al., 2005). The models are a natural marriage (Rubin, 1983 Bahr and Rubin, 1987) of the local equilibrium and kinetic models already discussed with the mass transport... 

The great value of kinetic theory is that it frees us from many of the constraints of the equilibrium model and its variants (partial equilibrium, local equilibrium, and so on see Chapter 2). In early studies (e.g., Lasaga, 1984), geochemists were openly optimistic that the results of laboratory experiments could be applied directly to the study of natural systems. Transferring the laboratory results to field situations, however, has proved to be much more challenging than many first imagined. 

Only for the intermediate cases - those with velocities in the range of about 100 m yr-1 to 1000 m yr-1 - does silica concentration and reaction rate vary greatly across the main part of the domain. Significantly, only these cases benefit from the extra effort of calculating a reactive transport model. For more rapid flows, the same result is given by a lumped parameter simulation, or box model, as we could construct in REACT. And for slower flow, a local equilibrium model suffices. 

Valocchi, A.J., 1985, Validity of the local equilibrium assumption for modeling sorbing solute transport through homogeneous soils. Water Resources Research 21, 808-820. 

However, we have to reflect on one of our model assumptions (Table 5.1). It is certainly not justified to assume a completely uniform oxide surface. The dissolution is favored at a few localized (active) sites where the reactions have lower activation energy. The overall reaction rate is the sum of the rates of the various types of sites. The reactions occurring at differently active sites are parallel reaction steps occurring at different rates (Table 5.1). In parallel reactions the fast reaction is rate determining. We can assume that the ratio (mol fraction, %a) of active sites to total (active plus less active) sites remains constant during the dissolution that is the active sites are continuously regenerated after AI(III) detachment and thus steady state conditions are maintained, i.e., a mean field rate law can generalize the dissolution rate. The reaction constant k in Eq. (5.9) includes %a, which is a function of the particular material used (see remark 4 in Table 5.1). In the activated complex theory the surface complex is the precursor of the activated complex (Fig. 5.4) and is in local equilibrium with it. The detachment corresponds to the desorption of the activated surface complex. 

Given the character of the water-water interaction, particularly its strength, directionality and saturability, it is tempting to formulate a lattice model, or a cell model, of the liquid. In such models, local structure is the most important of the factors determining equilibrium properties. This structure appears when the molecular motion is defined relative to the vertices of a virtual lattice that spans the volume occupied by the liquid. In general, the translational motion of a molecule is either suppressed completely (static lattice model), or confined to the interior of a small region defined by repulsive interactions with surrounding molecules (cell model). Clearly, the nature of these models is such that they describe best those properties which are structure determined, and describe poorly those properties which, in some sense, depend on the breakdown of positional and orientational correlations between molecules. 

Mass and energy transport occur throughout all of the various sandwich layers. These processes, along with electrochemical kinetics, are key in describing how fuel cells function. In this section, thermal transport is not considered, and all of the models discussed are isothermal and at steady state. Some other assumptions include local equilibrium, well-mixed gas channels, and ideal-gas behavior. The section is outlined as follows. First, the general fundamental equations are presented. This is followed by an examination of the various models for the fuel-cell sandwich in terms of the layers shown in Figure 5. Finally, the interplay between the various layers and the results of sandwich models are discussed. 

A critical part of the calculations is to calculate the tie-line at the interface corresponding to local equilibrium, and Enomoto (1992) used the central atoms model to predict the thermodynamic properties of a and 7. Some assumptions were made concerning the growth mode and the calculation of this tie-line is dependent on whether growth occurred under the following alternative conditions ... 

A simple model that illustrates the connection between faceting and step attraction is provided by the following hypothetical linear function F(n) giving the free energy of a bundle of n steps per unit length in local equilibrium on the vicinal surface... 

To quantify this treatment of migration as influenced by kinetics, a model has been developed in which instantaneous or local equilibrium is not assumed. The model is called the Argonne Dispersion Code (ARDISC) ( ). In the model, adsorption and desorption are treated independently and the rates for adsorption and desorption are taken into account. The model treats one dimensional flow and assumes a constant velocity of solution through a uniform homogeneous media. 

The work that was performed in this set of experiments was an extension of work performed by Inoue and Kaufman (7). In the previous work, the migration of strontium in glauconite was modeled using conditions of local equilibrium for flows up to 6.3 kilometers per year (72 cm/hr). The differences between the predicted and experimental results in the experiments performed by Inoue and Kaufman may be due to the existence of non-equilibrium behavior. 

In other cases, however, and in particular when sublattices are occupied by rather immobile components, the point defect concentrations may not be in local equilibrium during transport and reaction. For example, in ternary oxide solutions, component transport (at high temperatures) occurs almost exclusively in the cation sublattices. It is mediated by the predominant point defects, which are cation vacancies. The nearly perfect oxygen sublattice, by contrast, serves as a rigid matrix. These oxides can thus be regarded as models for closed or partially closed systems. These characteristic features make an AO-BO (or rather A, O-B, a 0) interdiffusion experiment a critical test for possible deviations from local point defect equilibrium. We therefore develop the concept and quantitative analysis using this inhomogeneous model solid solution. 

In many non-equilibrium situations, this local equilibrium assumption holds for the crystal bulk. However, its verification at the phase boundaries and interfaces (internal and external surfaces) is often difficult. This urges us to pay particular attention to the appropriate kinetic modeling of interfaces, an endeavour which is still in its infancy. 

The smoothing of a rough isotropic surface such as illustrated in Fig. 3.7 due to vacancy flow follows from Eq. 3.69 and the boundary conditions imposed on the vacancy concentration at the surface.12 In general, the surface acts as an efficient source or sink for vacancies and the equilibrium vacancy concentration will be maintained in its vicinity. The boundary condition on cy at the surface will therefore correspond to the local equilibrium concentration. Alternatively, if cy, and therefore Xy, do not vary significantly throughout the crystal, smoothing can be modeled using the diffusion potential and Eq. 3.72 subject to the boundary conditions on a at the surface and in the bulk.13... 

The geometry of a B-rich /9-phase platelet growing edgewise in an a-phase matrix is shown in Fig. 20.7. The growing edge is modeled as a cylindrical interface of radius R where local equilibrium between precipitate and matrix is maintained. Adapting Eq. 15.4 to this cylindrical interface, the concentration in the a phase... 

A cell model is presented for the description of the separation of two-component gas mixtures by pressure swing adsorption processes. Local equilibrium is assumed with linear, independent isotherms. The model is used to determine the light gas enrichment and recovery performance of a single-column recovery process and a two-column recovery and purification process. The results are discussed in general terms and with reference to the separation of helium and methane. 

The single-column process (Figure 1) is similar to that of Jones et al. (1). This process is useful for bulk separations. It produces a high pressure product enriched in light components. Local equilibrium models of this process have been described by Turnock and Kadlec (2), Flores Fernandez and Kenney (3), and Hill (4). Various approaches were used including direct numerical solution of partial differential equations, use of a cell model, and use of the method of characteristics. Flores Fernandez and Kenney s work was reported to employ a cell model but no details were given. Equilibrium models predict... 

The two-column process (Figure 2) is the heatless adsorption process of Skarstrom (5 ) This process can perform both recovery and purification of a light component Local equilibrium models of this process have been presented by Shendalman and Mitchell (6), Chan et al (7), and Knaebel and Hill (8) In each case the method of characteristics was used. Models with finite mass transfer rates have been published by Kawazoe and Kawai (9 ), Mitchell and Shendalman (10), Chihara and Suzuki (11), and Richter et al (12). In these models the method of characteristics and direct integration of partial differential equations were employed ... 

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