Single-component model

Here q is determined from the lAST model based on the single component model in eq. (7). 

Isotherm models have to be divided into single-component and multicomponent models. Because most of the multicomponent models are directly derived from single-component models, the latter will be presented first. 

Heise, M.S. and J.A. Schwarz, Preparation of metal distributions within catalyst supports III. Single component modeling of pH, ionic strength and concentration effects. J. Colloid. Int. Sci, 123 pp. 51-58, 1988. 

The characters of single-component model are as follow.23i (l) a key component is selected to be used for the estabhshment of mass balance for diffusions and reaction processes in catalyst particle, and the variation of molecular fraction for other reaction components is used for finding the relation with key component according to stoichiometric equations ... 

These assumptions are violated at high pressures when the vapor phase becomes nonideal and in the cases where the multicomponent nature of the mixture cannot be neglected. The extreme case of non-Kelvin behavior is the behavior of hydrocarbon mixtures in od-gas-condensate reservoirs. Although the Kelvin equation is apphed to tests of the porous media of the reservoirs [74,75], it cannot be used for modeling of equilibria in such reservoirs, not only because of the high pressure and the multicomponent composition of the mixture but also because this mixture exhibits retrograde behavior when the liquid phase precipitates as the pressure decreases. Such a behavior cannot, in principle, be described in terms of a single-component model. 

The study of mixed films has become of considerable interest. From the theoretical side, there are pleasing extensions of the various models for single-component films and from the more empirical side, one moves closer to modeling biological membranes. Following Gershfeld [200], we categorize systems as follows ... 

Ideal Adsorbed Solution Theory. Perhaps the most successful approach to the prediction of multicomponent equiUbria from single-component isotherm data is ideal adsorbed solution theory (14). In essence, the theory is based on the assumption that the adsorbed phase is thermodynamically ideal in the sense that the equiUbrium pressure for each component is simply the product of its mole fraction in the adsorbed phase and the equihbrium pressure for the pure component at the same spreadingpressure. The theoretical basis for this assumption and the details of the calculations required to predict the mixture isotherm are given in standard texts on adsorption (7) as well as in the original paper (14). Whereas the theory has been shown to work well for several systems, notably for mixtures of hydrocarbons on carbon adsorbents, there are a number of systems which do not obey this model. Azeotrope formation and selectivity reversal, which are observed quite commonly in real systems, ate not consistent with an ideal adsorbed... 

Eeavittloading Katio Correlation (LRCJ Method. The LRC model (17) for a single component i parallels Sips model (35) ... 

Adsorption. In the design of the adsorption step of gas-phase processes, two phenomena must be considered, equiUbrium and mass transfer. Sometimes adsorption equiUbrium can be regarded as that of a single component, but mote often several components and their interactions must be accounted for. Design techniques for each phenomenon exist as well as some combined models for dynamic performance. 

The MSF model (NUREG/CR-3837) is used principally to determine the level of dependence between safety systems introduced by maintenance, testing, and calibration activities. It is a mathematical model which modifies the independent failure probability of any single component by considering that a component with which it is redundant has already failed. This allows the conditional failure probabilities of redundant components to be calculated to determine the overall system failure probability. Documentation requirements are given in Table 4.5-6. 

The study of the behavior of reactions involving a single species has attracted theoretical interest. In fact, the models are quite simple and often exhibit IPT. In contrast to standard reversible transitions, IPTs are also observed in one-dimensional systems. The study of models in ID is very attractive because, in some cases, one can obtain exact analytical results [100-104]. There are many single-component nonequilibrium stochastic lattice reaction processes of interacting particle systems [100,101]. The common feature of these stochastic models is that particles are created autocatalytically and annihilated spontaneously (eventually particle diffusion is also considered). Furthermore, since there is no spontaneous creation of particles, the zero-particle... 

Presently it is not possible to relax the Cu lattice at the SCF level, since from a computational point of view it is composed of two different kinds of Cu atoms (those with and without the ECP). Also questions of wetting, i.e. whether the chemisorbed Be4 would prefer to remain as a tetrahedron (or distorted tetrahedron) or to spread out to a single layer are still not amenable to ab initio study. These questions have not yet been investigated using the parameterized model approach, because of the problems associated with modeling Be2 and Beg as accurately as larger Be clusters. Nonetheless, these preliminary results show that the parameterized and ab initio calculations can be used to complement each other in a multicomponent system, just as for single component systems. 

The Fowler-Guggenheim-Jovanovic model [3] assumes (as it was the earlier case also) the occurrence of intermolecular interactions among the molecules adsorbed as a monolayer but is based on the Jovanovic isotherm. The single-component isotherm is represented by the equation ... 

Surface Polarization in TFF The simplified model of polarization shown in Fig. 20-47 is used as a basis for analyzing more complex systems. Consider a single component with no reaction in a thin, two-dimensional boundary layer near the membrane surface. Axial diffusion is negligible along the membrane surface compared to convection. 

The classical multivariate calibration represents the transition of common single component analysis from one dependent variable y (measured value) to m dependent variables (e.g., wavelengths or sensors) which can be simultaneously included in the calibration model. The classical linear calibration (Danzer and Currie [1998] Danzer et al. [2004]) is therefore represented by the generalized matrix relation... 

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