Consistent model

A variation on the exact soiutions is the so-caiied seif-consistent modei that is explained in simpiest engineering terms by Whitney and Riiey [3-12]. Their modei has a singie hollow fiber embedded in a concentric cylinder of matrix material as in Figure 3-26. That is, only one inclusion is considered. The volume fraction of the inclusion in the composite cylinder is the same as that of the entire body of fibers in the composite material. Such an assumption is not entirely valid because the matrix material might tend to coat the fibers imperfectiy and hence ieave voids. Note that there is no association of this model with any particular array of fibers. Also recognize the similarity between this model and the concentric-cylinder model of Hashin and Rosen [3-8]. Other more complex self-consistent models include those by Hill [3-13] and Hermans [3-14] which are discussed by Chamis and Sendeckyj [3-5]. Whitney extended his model to transversely isotropic fibers [3-15] and to twisted fibers [3-16]. 

Theessence of the procedure is that Halpin and Tsai [3-17] showed that Hermans solution [3-14] generalizing Hill s self-consistent model [3-13] can be reduced to the approximate form... 

R. Hill, Theory of Mechanical Properties of Fibre-Strengthened Materials - III. Self-Consistent Model, Journal of the Mechanics and Physics of Solids, August 1965, pp. 189-198. 

In order to solve the system of the above-described equations, and which are derived by applying the self-consistent model, applied for composites by Budiansky 7), it is necessary to evaluate experimentally the moduli of elasticity (tension, shear, bulk) and Poisson s ratios of the constituent phases and the composite. Thus, the only unknown are the radius r of the mesophase layer and its mechanical properties and thermal expansion coefficient, which are then derived. 

A three-layer model for fiber composites may be developed, based on the theory of self-consistent models and adapting this theory to a three-layered cylinder, delineating the representative volume element for the fiber composite. 

Indeed, the self-consistent model averages the stresses and strains in either phase of a two-phase material, and it determines them, by solving separate problems, whose superposition yields the final configuration of the model 7). 

In the self-consistent model the matrix material outside the inclusion is assumed as possessing the effective macroscopic properties of the composite. Moreover, two consecutive problems were solved by assuming either phase of the composite as occupying its position and surrounded by this average material. In both cases the average values of the composite are determined from the values of the characteristic... 

An important variation of the self-consistent model is the three-phase model, introduced by Kerner 20), according to which the inclusion is enveloped by a matrix annulus, which in turn is embedded in an infinite medium with the unknown macroscopic properties of the composite. 

To achieve these consistencies, MODEL.LA. provides a series of semantic relationships among its modeling elements, which are defined at different levels of abstraction. For example, the semantic relationship (see 21 1), is-disaggregated-in, triggers the generation of a series of relationships between the abstract entity (e.g., overall plant) and the entities (e.g., process sections) that it was decomposed to. The relationships establish the requisite consistency in the (1) topological structure and (2) the state (variables, terms, constraints) of the systems. For more detailed discussion on how MODEL.LA. maintains consistency among the various hierarchical descriptions of a plant, the reader should consult 21 1. 

Newns DM. 1969. Self consistent model of hydrogen chemisorption. Phys Rev 178 ... 

Although the extended ab initio atomistic thermodynamics approach provides an exact expression for the interfacial stability, the formalism requires self-consistent modeling of the entire electrochemical system, or electrode/electrolyte interface, exceeding presently available computational capabilities. Therefore, certain assumptions had to be made that reduce the effort to the calculation of the electrode surface only. Even with this simplified approach, which has been applied to the two examples discussed in this chapter, the qualitative behavior can be reproduced. 

The main reason for using a 1D model is the reduction of the computational effort compared to the higher-dimensional models. This reduction to one dimension is acceptable because it is possible to study the sustaining mechanisms and the chemistry in the discharge with a ID self-consistent model. 

A comparison of several of the PY and IM types of catalysts in intramolecular reactions of allylic diazoacetates led to a consistent model for the enantioselectivity. The highest e.e. values are observed for ds-substituted allylic esters. Both R and R1 are directed toward the catalyst and introduce steric interactions that detract from enantioselectivity.208... 

The oxidation of a-tocopherol (1) to dimers29,50 and trimers15,51 has been reported already in the early days of vitamin E chemistry, including standard procedures for near-quantitative preparation of these compounds. The formation generally proceeds via orf/zo-quinone methide 3 as the key intermediate. The dimerization of 3 into spiro dimer 9 is one of the most frequently occurring reactions in tocopherol chemistry, being almost ubiquitous as side reaction as soon as the o-QM 3 occurs as reaction intermediate. Early accounts proposed numerous incorrect structures,52 which found entry into review articles and thus survived in the literature until today.22 Also several different proposals as to the formation mechanisms of these compounds existed. Only recently, a consistent model of their formation pathways and interconversions as well as a complete NMR assignment of the different diastereomers was achieved.28... 

In addition one member of group D was consistently modeled as a member of group C. 

It must be noted that there are a number of more or less arbitrary assumptions made in this work30,31 which need justification, as well as parameters whose values should be calculated rather than assumed. For instance, the importance of the distance dl9 taken as equal to Rc, has been mentioned. In principle, the value of this distance is a consequence of the forces between components of the metal and molecules of solvent, and would be calculated in a consistent model of the complete interface. This was pointed out by Yeager,18 who noted that the electron density tail of the metal determines the distance of closest approach of solvent in the interface, as well as the behavior of the solvent dipoles on the surface. Since changing qM will move the electron density tail in and out, dx should depend on the state of charge of the interface. In fact, it turns out31 that if dx varies linearly with surface charge according to... 

The calculations involved in a fully consistent model of the interface are difficult, but, as may be seen in the work discussed above, quite practicable. It may well be that when a complete model, calculating all terms properly and giving properties in agreement with experiment, is available, it will appear that some of the simplifications and assumptions used in the less complete models of the past, such as separation into diffuse and compact layers, will still be valid. However, it will no longer be possible to neglect the structure of the metal and reduce it to an uninteresting ideal conductor. 

Recent research now concentrates on the more physical models involving theories of rotation. The long-term aim of these attempts are to provide fully self-consistent models which include stellar evolution, rotation, transport of angular momentum and of chemical species. The key players in this field are P. Denis-senkov [11,12] and C. Charbonnel and coworkers (for their approach, see the contributions by Charbonnel and Palacios in this volume). Both groups employ the theoretical description of rotation by Zahn and Maeder [27,19]. 

Rotation remains the best candidate for extra-mixing in RGB stars. The present modelling of the rotational mixing is however still incomplete, and agreement between self-consistent models and observations might be achieved by improving the description of the hydrodynamics related to rotation. 

E. Bakker, R.K. Meruva, E. Pretsch, and M.E. Meyerhoff, Selectivity of polymer membrane-based ion-selective electrodes — self-consistent model describing the potentiometric response in mixed ion solutions of different charge. Anal. Chem. 66, 3021—3030 (1994). 

The LSR model must be applied in conjunction with a consistent Lagrangian mixing model for For example, if the Lagrangian FP model is used, one consistent model has the form... 

Consistent modeling of scalars in turbulent flows. The Physics of Fluids 26, 404 108. 

The approximate kinetic formats discussed above face inherent difficulties to account for fundamental physicochemical properties of biochemical reactions, such as the Haldane relation discussed in Section III.C.4 a major drawback when aiming to formulate thermodynamically consistent models. 

Such ambiguity and also the low structural resolution of the method require that the spectroscopic properties of protein fluorophores and their reactions in electronic excited states be thoroughly studied and characterized in simple model systems. Furthermore, the reliability of the results should increase with the inclusion of this additional information into the analysis and with the comparison of the complementary data. Recently, there has been a tendency not only to study certain fluorescence parameters and to establish their correlation with protein dynamics but also to analyze them jointly, to treat the spectroscopic data multiparametrically, and to construct self-consistent models of the dynamic process which take into account these data as a whole. Fluorescence spectroscopy gives a researcher ample opportunities to combine different parameters determined experimentally and to study their interrelationships (Figure 2.1). This opportunity should be exploited to the fullest. 

These studies were with polycrystalline copper, but more detailed results (Fig. 9) were obtained subsequently (44) with Cu(100), Cu(l 11), and Cu(l 10) surfaces XPS, UPS, LEED, and mass spectrometric data being combined to provide a self-consistent model. The surface species formed with their associated N(ls) and 0(ls) binding energies (eV) on exposing copper surfaces to NO(g) are listed in Table I. 

M. Otta, H.H. Fischer, M. Maiwald, K. Albert, and H. Hasse, Kinetics of oligomerization reactions in formaldehyde solutions NMR experiments up to 373K and thermodynamically consistent model, Chem. Eng. Process., 44, 653-660 (2005). 

The use of an ideal-solution model meant that there were a number of instances where calculated and experimental results were quantitatively at variance. However, the approach very successfully predicted the general form of most of the phase diagrams, for example whether they were peritectic or eutectic, and accounted for the appearance of intermediate phases in systems such as Cr-Rh. That the approach could do this using such simple and internally self-consistent models is a demonstration of the inherent power of CALPHAD methods. The importance of this first step therefore cannot be overestimated, although its significance was not... 

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