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Simplified statistical model

Chung, R M. (1969). A simplified statistical model for turbulent, chemically reacting shear flows. AIAA Journal 7, 1982-1991. [Pg.410]

Relative reaction rates of hydrolysis, condensation, reesterification, and dissolution must be understood and controlled to dictate structural evolution. However, accurate values for rate constants are difficult to obtain because of the enormous number of distinguishable reactions as next nearest neighbors are considered, and to the concurrency of these reactions. Assink and Kay [45] use a simplified statistical model assuming that the local silicon environment does not affect reaction rates, and the reactions for a particular silicon species are the product of a statistical factor and rate constant. These assumptions ignore steric and inductive effects. For example, this model predicts that the relative rate constants for the four sequential hydrolysis steps leading from TMOS to Si(OH)4 would be 4 3 2 1. This model was applied to acid-catalyzed TMOS sols with W values ranging from 0.5 to 2.0. Si NMR spectra on the temporal evolution of various silicon species show the model is in excellent agreement with experimental results. A lower limit for fen was calculated as 0.2 L/mol-min. Values for few and feA are 0.006 and 0.001 L/mol-min, respectively. [Pg.642]

FIGURE 4.18. Comparison of experimental equilibrium data for the system C,ll4-C3H,-5/ sieve at 273 K, 8 torr, with the predictions of the ideal adsorbed solution theory (—). For this system the simplified statistical model leads to essentially the same predictions as the ideal adsorbed solution theory and the formation of an azeotrope is not predicted by cither approach. (Reprinted with permission from ref, 42. Copyright 1977 American Chemical Society.)... [Pg.111]

The first approximation of oil spill cost without considering damage to environment could be done by simplified statistical models. One of such model of cleanup cost reveal that unit cost is not dependant of spill size (Fig. 4). The model can be written as (Goryczko Gucma 2006, Etkin D.S., 2000) ... [Pg.116]

Another simplified model is the freely jointed or random flight chain model. It assumes all bond and conformation angles can have any value with no energy penalty, and gives a simplified statistical description of elasticity and average end-to-end distance. [Pg.308]

Classical surface and colloid chemistry generally treats systems experimentally in a statistical fashion, with phenomenological theories that are applicable only to building simplified microstructural models. In recent years scientists have learned not only to observe individual atoms or molecules but also to manipulate them with subangstrom precision. The characterization of surfaces and interfaces on nanoscopic and mesoscopic length scales is important both for a basic understanding of colloidal phenomena and for the creation and mastery of a multitude of industrial applications. [Pg.688]

For the interbipolycondensation the condition of quasiideality is the independence of the functional groups either in the intercomponent or in both comonomers. In the first case the sequence distribution in macromolecules will be described by the Bernoulli statistics [64] whereas, in the second case, the distribution will be characterized by a Markov chain. The latter result, as well as the parameters of the above mentioned chain, were firstly obtained within the framework of the simplified kinetic model [64] and later for its complete version [59]. If all three monomers involved in interbipolycondensation have dependent groups then, under a nonequilibrium regime, non-Markovian copolymers are known to form. [Pg.191]

A much more detailed and time-dependent study of complex hydrocarbon and carbon cluster formation has been prepared by Bettens and Herbst,83 84 who considered the detailed growth of unsaturated hydrocarbons and clusters via ion-molecule and neutral-neutral processes under the conditions of both dense and diffuse interstellar clouds. In order to include molecules up to 64 carbon atoms in size, these authors increased the size of their gas-phase model to include approximately 10,000reactions. The products of many of the unstudied reactions have been estimated via simplified statistical (RRKM) calculations coupled with ab initio and semiempirical energy calculations. The simplified RRKM approach posits a transition state between complex and products even when no obvious potential barrier... [Pg.33]

The inequality (2) considerably simplifies further treatment and is the starting point for a simple statistical model of the plasma43. ... [Pg.147]

The purpose of the recent work just reviewed was to develop and verify a reasonably simplified theoretical approach to heterogeneous reactions in a nonisothermal low pressure plasma. With this purpose in mind, we first considered a simple statistical model of the plasma which has brought about a better understanding of the dependence of the chemical composition of the plasma on energy. Comparison of this model with several real systems which had been experimentally investigated illustrated the applicability of the theoretical ideas to such systems as well as their limitations. [Pg.156]

The productivity for the same % yield calculated by the SQP technique was higher than that calculated using the RSM. The RSM calculates the ranges of the optimization variables and not exact values, so it is difficult to determine the exact global maximum point. In addition, the RSM uses statistical models for productivity and % yield and these are simplified models, while the SQP technique uses a rigorous process model. [Pg.490]

Another point to be discussed is the use of the rigorous model compared with the use of simplified models as the statistical models determined by factorial design and used with RSM. Although rigorous models are more realistic and probably lead to better results, for many processes they can be excessively complex, which hinders determination of the optimal result. [Pg.491]

The statistical models determined by factorial design can be used as simplified models with the SQP technique. In this work the results obtained through this approach are compared with the results obtained using a rigorous model of the process. Costa et al. (5) determined quadratic models for productivity and % yield as functions of the significant input variables. These equations evaluated productivity and % yield and the SQP technique to determine the optimal values for S0, tr, R, and r. The optimization problem is postulated as follows ... [Pg.491]

An alternative simulation procedure is to replace the explicit solvent molecules with a continuous medium having the bulk dielectric constant. - " Once the solvent has been simplified, it is much easier to employ quantum mechanical techniques for the ENP relaxation of electronic and molecular structure in solution thus this approach is complementary to simulation insofar as it typically focuses on the response of the solute to the solvent. Since the properties of the continuum solvent must represent an average over solvent configurations, such approaches are most accurately described as quantum statistical models. [Pg.7]

The classical theory of the Gibbs adsorption isotherm is based on the use of an equation of state for the adsorbed phase hence it assumes that this adsorbed phase is a mobile fluid layer covering the adsorbent surface. By contrast, in the statistical thermod)mamic theory of adsorption, developed mainly by Hill [15] and by Fowler and Guggenheim [12], the adsorbed molecules are supposed to be localized and are represented in terms of simplified physical models for which the appropriate partition function may be derived. The classical thermodynamic fimctions are then derived from these partition fimctions, using the usual relationships of statistical thermodynamics. [Pg.76]

Lau and Dill have also investigated the statistical mechanics of folding for simplified protein models on two-dimensional square lattices. They explored both conformational space (the set of all possible conformations) and sequence space (the set of all possible sequences) and concluded that many sequences have stable, compact, native-like structures. Another conclusion of these studies was that sequences tended to form a single, unique structure, even with only two types of residues (hydrophobic and polar). This tendency increased with chain length. Moreover, one or two mutations in these sequences did not greatly destabilize most folded states. [Pg.68]


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See also in sourсe #XX -- [ Pg.8 ]




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