Main simplification

The theory was very similar to that described earlier, but was simplified in view of the complexity of the problem. A number of reaction intermediates were considered explicitly, and the corresponding signals were calculated by molecular dynamics simulation. Kinetic equations governing the reaction sequence were established and were solved numerically. The main simplification of the theory is that, when calculating A5[r, r], the lower limit of the Fourier integral was shifted from 0 to a small value q. The authors wrote [59]... 

One approach is to roughly estimate how the degree of crystallization would vary with time by making main simplifications in treating solidification, leading to the Avrami equation. 

The present synthesis is an adaptation of a previously reported synthesis4 in a divided cell (i.e., separate anode and cathode compartments). The overriding consideration in making this modification has been to simplify the operations involved and render the synthesis more attractive to chemists not well acquainted with electrochemical procedures. The main simplification achieved is that the pH is controlled internally via the anodic generation of protons as noted above (in the reported procedure4 this is achieved by periodic addition of acetic acid to the cathode compartment). A further simplification has been to run the reaction with a constant current rather than at controlled cathode potential. After the electrolysis has been initiated, the reaction requires no special attention. A small price is paid for the simplicity of the present synthesis in that the yield is somewhat lower than that obtained previously.4 The major by-product formed is diethyl succinate, which results from a 2e reduction of diethyl fumarate or diethyl maleate ... 

The usual approach in the modeling of an extremely complicated process is the consideration of some extreme cases with further synthesis of the results obtained. The next three main simplifications are inherent in the citrrent state of emulsion dynamics modeling the neglect of the influence of the gravitational field, i.e., neglect of... 

The main simplification in all existing models for emulsion dynamics (23, 39) is the neglect of the coalescence time dependence on droplet dimensions. This simplification is not justified and decreases very much the value of the prediction, which can now be made with use of the PBE. For elimination of this unjustified simplification it is necessary to determine the coalescence time for emulsion films between droplets of different dimensions i and j, namely, Tpy, similar to flie existing analytical expressions for the doublet-fragmentation time, (12). The determination of a large set of values by means of a comparison of ex-... 

The finite element method is already estabhshed in modelling for eddy current sensors the same as analytical methods. For example, reference [1] solves for the absolute impedance of core-less coils, and [2], [3] solve for the induced voltage in the coil. The problem with these analytical models is the dependency on the crack size, shape and location, and the equations needs to be modified all the time. Consequently, the results may not necessarily be comparable. When the FEA is used instead, the results remain comparable for changed model parameters. Hence, the aim of this study is to build a 3D model to calculate the complex coil impedance for encircling coils that measure rods with any kind of cracks. In doing so, two main simplifications are applied as follows. 

This is the main simplification introduced by the tubeless siphon as opposed to the entry of a capillary. However, our discussion remains qualitatively valid for the latter case (provided that the solution is dilute). 

In the model of Roisman et al. [26], a main simplification concerns the flow in the lamella. The lamella is approximated by an axially symmetric flow of a liquid... 

As discussed in Sec. 4, the icomplex function of temperature, pressure, and equilibrium vapor- and hquid-phase compositions. However, for mixtures of compounds of similar molecular structure and size, the K value depends mainly on temperature and pressure. For example, several major graphical ilight-hydrocarbon systems. The easiest to use are the DePriester charts [Chem. Eng. Prog. Symp. Ser 7, 49, 1 (1953)], which cover 12 hydrocarbons (methane, ethylene, ethane, propylene, propane, isobutane, isobutylene, /i-butane, isopentane, /1-pentane, /i-hexane, and /i-heptane). These charts are a simplification of the Kellogg charts [Liquid-Vapor Equilibiia in Mixtures of Light Hydrocarbons, MWK Equilibnum Con.stants, Polyco Data, (1950)] and include additional experimental data. The Kellogg charts, and hence the DePriester charts, are based primarily on the Benedict-Webb-Rubin equation of state [Chem. Eng. Prog., 47,419 (1951) 47, 449 (1951)], which can represent both the liquid and the vapor phases and can predict K values quite accurately when the equation constants are available for the components in question. 

The simplification of the local anesthetic phaimacophore of cocaine to an aryl substituted ester of ethanolamine has been described previously. Atropine (S2) is a structurally closely related natural product whose main biologic action depends on inhibition of the parasympathetic nervous system. Among its many other actions, the compound exerts useful spasmolytic effects. 

The different studies are mainly reported on the OBLIN site (NASA). The main goal is to simplify and integrate all the functions to be achieved in the beam combiner by using integrated optics and optical fibre components. Consequently, the compactness, the stability and the reliability are significantly improved. Figure 15 illustrates the simplification of the experimental setup. 

The model presented in this paper will display the concentration gradients in a cylindrical, fully wax-filled pore of the catalyst. For simplification reasons, only the reactants CO and H2 as well as the reaction product H20 will be considered. C02 formation is disregarded in the model due to its comparatively low concentration in FT products and because compared to steam, C02 has no impact on the main reaction rate. Methane formation (usually not more than 5% of total CO conversion) was also neglected. 

The main disadvantage of (15) is that the number of directed bonds 2B may be quite large, (in the example discussed presently, 2B = 30 while the matrix A has dimension 7. However, the tree structure of most of the isospectral graphs under consideration here can be used to obtain a much simplified, and almost explicit form of the secular function. This simplification is the object of the present appendix. 

The assumption made is called the quasi-steady-state approximation (QSSA). It is valid here mainly because of the great difference in densities between the reacting species (gaseous A and solid B). For liquid-isolid systems, this simplification cannot be made. 

The model proposed by Brandt et al. is consistent with the experimental observations, reproduces the peculiar shape of the kinetic curves in the absence and presence of dioxygen reasonably well, and predicts the same trends in the concentration dependencies of t, p that were observed experimentally (80). It was concluded that there is no need to assume the participation of oxo-complexes in the mechanism as it has been proposed in the literature (88-90). However, the model provides only a semi-quantitative description of the reaction because it was developed at constant pH by neglecting the acid-base equilibria of the sulfite ion and the reactive intermediates, as well as the possible complex-formation equilibria between various iron(III) species. In spite of the obvious constraints introduced by the simplifications, the results shed light on the general mechanistic features of the reaction and could be used to identify the main tasks for further model development. 

The models of PSs considered above mainly concern the packings of particles, that is, from the texturological point of view they can be considered as development of a corpuscular model by Kiselev. The morphology of a porous space in corpuscular systems is usually more complex. For simplification, it is traditionally simulated by a group or network of nonintersecting cylindrical channels of varying sizes or flat slits, the models of goffered channels are less often used, etc. 

Catalysts have been bonded to insoluble polymers to allow, in principle, an appreciable simplification of PTC the catalyst represents a third insoluble phase which can be easily recovered at the end of the reaction by filtration, thus avoiding tedious processes of distillation, chromatographic separation and so on. This is of potential interest mainly from the industrial point of view, due to the possibility of carrying on both discontinuous processes with a dispersed catalyst and continuous processes with the catalyst on a fixed bed. This technique was named "triphase catalysis" by Regen (13,33,34). 

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