Examples of Models

The first example to be considered is a static function approximation. The MATLAB neural network toolbox (Mathworks, 2006) will be used. A two-layer feed-forward neural network, consisting of a hidden layer and output layer, will be trained to approximate the sum of two sine waves. The function can be described by  

Data are generated at equidistant points and some noise is added  

The vectors x (input) andy (output) contain 201 data points each. The relationship between x andy is shown in Fig. 27.7. 

The created net is stored in the object named net. The net can now be used to test and/or validate the model. For this purpose new points are created within the region that was selected for training. 

The results are shown in Fig. 27.8. Usually the model will perform well with data from within the training region. 

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The examples of modelling discussed in section C2.5.2 and section C2.5.3 are meant to illustrate tlie ideas behind tlie tlieoretical and computational approaches to protein folding. It should be borne in mind tliat we have discussed only a very limited aspect of tlie rich field of protein folding. The computations described in section C2.5.3 can be carried out easily on a desktop computer. Such an exercise is, perhaps, tlie best of way of appreciating tlie simple approach to get at tlie principles tliat govern tlie folding of proteins. 

Another example of model transformation to achieve linearity is the change from the nonlinear rate equation... 

The representation of the dispersion viscosity given above related to suspensions of particles of spherical form. Upon a transition to anisodiametrical particles a number of new effects arises. The effect of nonsphericity is often discussed on the example of model dispersions of particles of ellipsoidal form. An exact form of particles to a first approximation is not very significant, the degree of anisodiametricity is only important, or for ellipsoidal-particles, their eccentricity. 

Dusek (1), Shy (2 and Bauer (3) give examples of modelling the structure-property relation of several networks (Tg, gel point, etc.). Examples are described of models on solvent evaporation, calculation of functionalities, molecular weight of resins (4), etc. 

Examples of models of the environmental fate of chemicals, utilizing partition coefficients and fngacities, are given in the works cited earlier, and also in Chapter 3 in Walker et al. (2000, 2006). 

In this paper, we first briefly describe both the single-channel 1-D model and the more comprehensive 3-D model, with particular emphasis on the comparison of the features included and their capabilities/limitations. We then discuss some examples of model applications to illustrate how the monolith models can be used to provide guidance in emission control system design and implementation. This will be followed by brief discussion of future research needs and directions in catalytic converter modeling, including the development of elementary reaction step-based kinetic models. 

Recall Eq. (10-46) that Hj and H2 are defined entirely by the four plant functions Gy. This is another example of model-based control. With the definitions of Hj and H2 given in (10-46), the calculations are best performed with a computer. 

As an example of modeling a fluid in redox disequilibrium, we use an analysis, slightly simplified from Nordstrom et al. (1992), of a groundwater sampled near the Morro do Ferro ore district in Brazil (Table 7.2). There are three measures of oxidation state in the analysis the Eh value determined by platinum electrode, the dissolved oxygen content, and the distribution of iron between ferrous and ferric species. 

TABLE 6.7. Example of Model Parameters Used in the Sewer Process Model Outlined in Table 6.6. 

The comparative estimation of efficiency of zinc oxide and ZnCFO similar concentrations (3,0 5,0 7,0 phr) as the agents of metaloxide vulcanization system was carried out on example of modelling unfilled elastomeric compositions from chloroprene rubber of recipe, phr chloroprene rubber - 100,0 magnesium oxide - 7,0. Kinetic curves of rubber mixes curing process at 155°C are shown on fig. 8. The analysis of the submitted data testifies, that at increase of zinc oxide contents vulcanization kinetics is changed as follows the scorch time and optimum cure time are decreased, the cure rate is increase. Vulcanization... 

Saturated complex polyesters, particularly, poly (butylene terephthalate) (PBT) are used as engineering thermoplastics possesing good thermo - and wearstability, excellent moulding. These properties also allow to use them as matrix material for polymer composites [1], One of the perspective ways of search of effective catalysts for such systems is kinetic study of the reesterification model reaction, performed in the presence of various catalysts and comparison it with the results of the similar reaction without catalyst. Clarification on the example of model system of the most effective catalysts list allows to use them for obtaining both filled and nonfilled PBT and compare catalytic activity of various catalysts. The purpose of the... 

Again, the following examples of modelling are textbook cases that cannot be used for design purposes. 

One of the most interesting and unique approaches to the design of sampled-data controllers is called minimal-prototype design. It is one of the earliest examples of model-based controllers. 

Examples of models with two stereogenic units, two-dimensional enantiomers, and familiar examples of diastereomers are illustrated by examples 5-7. 

Practical identifiability is not the only problem that can be adressed by principal component analysis of the sensitivity matrix. In (refs. 29-30) several examples of model reduction based on this technique are discussed. 

When the rate equation is complex, the values predicted by the two models are not necessarily limiting. Complexities can arise from multiple reactions, variation of density or pressure or temperature, incomplete mixing of feed streams, minimax rate behavior as in autocatalytic processes, and possibly other behaviors. Sensitivity of the reaction to the mixing pattern can be established in such cases, but the nature of the conversion limits will not be ascertained. Some other, possibly more realistic models will have to be devised to represent the reaction behavior. The literature has many examples of models but not really any correlations (Naumann and Buffham, 1983 Wen and Fan Westerterp et al., 1984). 

Some typical examples of modeling with 0-1 variables exclusively include ... 

As we shall discuss below, it is also more straightforward to calculate the relative intensity of vibrational modes observed by inelastic neutron scattering than in electron-energy-loss and optical spectroscopies. The relative intensity of the modes, as well as their frequency, can then be used to identify the atomic displacement pattern or eigenvector of the mode. We shall also see through examples of model calculations how the relative intensity of surface vibratory modes is sensitive to the orientation of the adsorbed molecule and the strength and location of its bond to the surface. 

The above example gives us an idea of the difficulties in stating a rigorous kinetic model for the free-radical polymerization of formulations containing polyfunctional monomers. An example of efforts to introduce a mechanistic analysis for this kind of reaction, is the case of (meth)acrylate polymerizations, where Bowman and Peppas (1991) coupled free-volume derived expressions for diffusion-controlled kp and kt values to expressions describing the time-dependent evolution of the free volume. Further work expanded this initial analysis to take into account different possible elemental steps of the kinetic scheme (Anseth and Bowman, 1992/93 Kurdikar and Peppas, 1994 Scott and Peppas, 1999). The analysis of these mechanistic models is beyond our scope. Instead, one example of models that capture the main concepts of a rigorous description, but include phenomenological equations to account for the variation of specific rate constants with conversion, will be discussed. 

As a final example of modeling thermoset processing, the process called the shell molding or Croning process will be analyzed. This is just one of many processes found in a variety of industries (automobile, electronics, sports goods, furniture, etc.) that involve the cure of thermosetting polymers. The selected example is a typical process used in foundries. 

The third example of model outputs deals with gas hydrates (Table A.4), which was part of our snowball Earth simulations (Fig. 5.6). The Pco2, in this case, was set equal to 0.12 bars and was independent of total pressure this served to prevent Pco2 from becoming too high and beyond the validity... 

As shown below, the basic principles of peroxidase-mimetic sensor appliance operation are developed using the example of model peroxidase reaction of ethyl alcohol electrochemical oxidation to aldehyde. 

In this example of model reactive polymer processing of two immiscible blend components, as with Example 11.1, we have three characteristic process times tD,, and the time to increase the interfacial area, all affecting the RME results. This example of stacked miscible layers is appealing because of the simple and direct connection between the interfacial layer and the stress required to stretch the multilayer sample. In Example 11.1 the initially segregated samples do create with time at 270°C an interfacial layer around each PET particulate, but the torsional dynamic steady deformation torques can not be simply related to the thickness of the interfacial layer, <5/. However, the initially segregated morphology of the powder samples of Example 11.1 are more representative of real particulate blend reaction systems. 

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