Modeling necessity

The necessity to calculate the electrostatic contribution to both the ion-electrode attraction and the ion-ion repulsion energies, bearing in mind that there are at least two dielectric ftmction discontinuities hr the simple double-layer model above. 

The ordinary BO approximate equations failed to predict the proper symmetry allowed transitions in the quasi-JT model whereas the extended BO equation either by including a vector potential in the system Hamiltonian or by multiplying a phase factor onto the basis set can reproduce the so-called exact results obtained by the two-surface diabatic calculation. Thus, the calculated hansition probabilities in the quasi-JT model using the extended BO equations clearly demonshate the GP effect. The multiplication of a phase factor with the adiabatic nuclear wave function is an approximate treatment when the position of the conical intersection does not coincide with the origin of the coordinate axis, as shown by the results of [60]. Moreover, even if the total energy of the system is far below the conical intersection point, transition probabilities in the JT model clearly indicate the importance of the extended BO equation and its necessity. 

As already explained the necessity to satisfy the BB stability condition restricts the types of available elements in the modelling of incompressible flow problems by the U-V P method. To eliminate this restriction the continuity equation representing the incompressible flow is replaced by an equation corresponding to slightly compressible fluids, given as... 

Conformational analysis is far simpler m cyclopropane than m any other cycloalkane Cyclopropane s three carbon atoms are of geometric necessity coplanar and rotation about Its carbon-carbon bonds is impossible You saw m Section 3 4 how angle strain m cyclopropane leads to an abnormally large heat of combustion Let s now look at cyclopropane m more detail to see how our orbital hybridization bonding model may be adapted to molecules of unusual geometry... 

When considering mathematical models of plates and shells, the authors clearly perceived the necessity for a reasonable compromise so that, on the one hand, the used models should describe the principle of a physical phenomenon and, on the other, they should be quite simple in order that the mathematical tool could be usefully employed. 

Implementation of advanced performance degradation models, necessitate the inclusion of advanced instrumentation and sensors such as pyrometers for monitoring hot section components, dynamic pressure transducers for detection of surge and other flow instabilities such as combustion especially in the new dry low NO combustors. To fully round out a condition monitoring system the use of expert systems in determining fault and life cycle of various components is a necessity. 

Though a powerfiil technique, Neutron Reflectivity has a number of drawbacks. Two are experimental the necessity to go to a neutron source and, because of the extreme grazing angles, a requirement that the sample be optically flat over at least a 5-cm diameter. Two drawbacks are concerned with data interpretation the reflec-tivity-versus-angle data does not directly give a a depth profile this must be obtained by calculation for an assumed model where layer thickness and interface width are parameters (cf., XRF and VASE determination of film thicknesses. Chapters 6 and 7). The second problem is that roughness at an interface produces the same effect on specular reflection as true interdiffiision. 

It is clear that a large number of parameters influence the formation of the sheets in a tubule and that their relative importance is still unknown, as is also the cause of the occurrence of the defects responsible for the eventual polygonization of the sheets. Although the model presented here highlights the necessity of including, as one of the parameters, the uncertainty 8r or dd on the separation of successive cylindrical sheets, it is impossible to predict with absolute certainty the final characteristics of any of these sheets, symmetric or not, on the basis of the characteristics of the previous one. Nevertheless, a number of features of their structure, such as the presence or absence of helicity, and the presence of groups of sheets with nearly the same angle of pitch, can be explained and quantified. 

Since this text is not solely devoted to HRA, the above process is by necessity simplified. A more defined interaction is shown in Figure 4.5-2. Here HRA interacts with the development of the system models and feeds into not only the accident sequences but also the physical analysis of the inplant and explant accident progress,... 

However, in most cases the AW(D) dependencies are distinctly nonlinear (Fig. 9), which gives impulse to further speculations. Clearly, dependencies of this type can result only from mutual suppression of the hydrogel particles because of their nonuniform distribution over the pores as well as from the presence of a distribution with respect to pore size which does not coincide with the size distribution of the SAH swollen particles. A considerable loss in swelling followed from the W(D) dependencies, as shown in Fig. 9, need a serious analysis which most probably would lead to the necessity of correlating the hydrogel particle sizes with those of the soil pores as well as choice of the technique of the SAH mixing with the soil. Attempts to create the appropriate mathematical model have failed, for they do not give adequate results. 

The various copolymerization models that appear in the literature (terminal, penultimate, complex dissociation, complex participation, etc.) should not be considered as alternative descriptions. They are approximations made through necessity to reduce complexity. They should, at best, be considered as a subset of some overall scheme for copolymerization. Any unified theory, if such is possible, would have to take into account all of the factors mentioned above. The models used to describe copolymerization reaction mechanisms arc normally chosen to be the simplest possible model capable of explaining a given set of experimental data. They do not necessarily provide, nor are they meant to be, a complete description of the mechanism. Much of the impetus for model development and drive for understanding of the mechanism of copolymerization conies from the need to predict composition and rates. Developments in models have followed the development and application of analytical techniques that demonstrate the inadequacy of an earlier model. 

A global representation of the P cycle, by necessity, will be general. It will combine a wide variety of P-containing components into relatively few reservoirs and will parameterize intricate processes and feedback mechanisms into simple first-order transfers. To appreciate the rationale behind the construction of such a model and to understand its limitations, the transfers of P within a hypothetical terrestrial ecosystem and in a generalized ocean system will be discussed first. 

This calculation has shown the importance of the basis set and in particular the polarization functions necessary in such computations. We have studied this problem through the calculation of the static polarizability and even hyperpolarizability. The very good results of the hyperpolarizabilities obtained for various systems give proof of the ability of our approach based on suitable polarization functions derived from an hydrogenic model. Field—induced polarization functions have been constructed from the first- and second-order perturbed hydrogenic wavefunctions in which the exponent is determined by optimization with the maximum polarizability criterion. We have demonstrated the necessity of describing the wavefunction the best we can, so that the polarization functions participate solely in the calculation of polarizabilities or hyperpolarizabilities. 

Consequences of the Snyder and Soczewinski model are manifold, and their praetieal importance is very signifieant. The most speetaeular conclusions of this model are (1) a possibility to quantify adsorbents ehromatographic activity and (2) a possibility to dehne and quantify chromatographic polarity of solvents (known as the solvents elution strength). These two conclusions could only be drawn on the assumption as to the displacement mechanism of solute retention. An obvious necessity was to quantify the effect of displacement, which resulted in the following relationship for the thermodynamic equilibrium constant of adsorption, K,, in the case of an active chromatographic adsorbent and of the monocomponent eluent ... 

From the theoretical standpoint the above issues are addressed by quantum chemistry. On the basis of calculations of various cluster models [191] the properties of surfaces of solid body are being studied as well as issues dealing with interaction of gas with the surface of adsorbent. However, fairly good results have been obtained in this area only to calculate adsorption on metals. The necessity to account for more complex structure of the adsorption value as well as availability of various functional groups on the surface of adsorbent in case of adsorption on semiconductors geometrically complicates such calculations. 

Another kind of situation arises when it is necessary to take into account the long-range effects. Here, as a rule, attempts to obtain analytical results have not met with success. Unlike the case of the ideal model the equations for statistical moments of distribution of polymers for size and composition as well as for the fractions of the fragments of macromolecules turn out normally to be unclosed. Consequently, to determine the above statistical characteristics, the necessity arises for a numerical solution to the material balance equations for the concentration of molecules with a fixed number of monomeric units and reactive centers. The difficulties in solving the infinite set of ordinary differential equations emerging here can be obviated by switching from discrete variables, characterizing macromolecule size and composition, to continuous ones. In this case the mathematical problem may be reduced to the solution of one or several partial differential equations. 

Monomers employed in a polycondensation process in respect to its kinetics can be subdivided into two types. To the first of them belong monomers in which the reactivity of any functional group does not depend on whether or not the remaining groups of the monomer have reacted. Most aliphatic monomers meet this condition with the accuracy needed for practical purposes. On the other hand, aromatic monomers more often have dependent functional groups and, thus, pertain to the second type. Obviously, when selecting a kinetic model for the description of polycondensation of such monomers, the necessity arises to take account of the substitution effects whereas the polycondensation of the majority of monomers of the first type can be fairly described by the ideal kinetic model. The latter, due to its simplicity and experimental verification for many systems, is currently the most commonly accepted in macromolecular chemistry of polycondensation processes. 

Figure 3.5.2 shows the results obtained using M-5 and TS-500 samples with S/V values of 3.03 x 107 and 3.28 x 107 m 1, respectively, and porosities of 0.936 and 0.938, respectively. Note the significant deviation of the relaxation behavior from that ofbulk CF4 gas (dotted lines in Figure 3.5.2). The experimental data were first fitted to the model described above, assuming an increase in collision frequency due purely to the inclusion of gas-wall collisions, assuming normal bulk gas density. However, this model merely shifts the T) versus pressure curve to the left, whereas the data also have a steeper slope than bulk gas data. This pressure dependence can be empirically accounted for in the model via the inclusion of an additional fit parameter. Two possible physical mechanisms can explain the necessity of this parameter.

Orion Model 95-64). In practice, one simply determines E ntot by calibration with a standard solution without the necessity of knowing the various constants mentioned. The S02 electrode allows the determination of concentrations down to 10 8 Af with a response time of a few minutes. From the above it appears that the gas-sensing electrodes show Nemstian behaviour provided that the concentrations to be measured are not high there is little or no interference by other components in the sample solution. 

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