Explicit model

In spite of the importance of reaction prediction, only a few systems have been developed to tackle this problem, largely due to its complexity it demands a huge amount of work before a system is obtained that can make predictions of sufficient quality to be useful to a chemist. The most difficult task in the development of a system for the simulation of chemical reactions is the prediction of the course of chemical reactions. This can be achieved by using knowledge automatically extracted from reaction databases (see Section 10.3.1.2). Alternatively, explicit models of chemical reactivity will have to be included in a reaction simulation system. The modeling of chemical reactivity is a very complex task because so many factors can influence the course of a reaction (see Section 3.4). 

Five percent random error was added to the error-free dataset to make the simulation more realistic. Data for kinetic analysis are presented in Table 6.4.3 (Berty 1989), and were given to the participants to develop a kinetic model for design purposes. For a more practical comparison, participants were asked to simulate the performance of a well defined shell and tube reactor of industrial size at well defined process conditions. Participants came from 8 countries and a total of 19 working groups. Some submitted more than one model. The explicit models are listed in loc.cit. and here only those results that can be graphically presented are given. 

Here the effects of any one fractionating step can be expressed in a change in isotopic composition in a wider range of body tissue components, including the product as well as the precursor of a (reversible) reaction. The details depend on the explicit model, for example how rates depend on metabolite concentrations. Therefore, where a metabolic pathway is, or becomes, reversible, the effect on isofractionation on measured body components can be more widespread. 

These disadvantages are overcome by the methodology we will describe in the subsequent paragraph developed by Bakshi and Stephanopoulos. Effects of the curse of dimensionality may be decreased by using the hierarchical representation of process data, described in Section III. Such a multiscale representation of process data permits hierarchical development of the empirical model, by increasing the amount of input information in a stepwise and controlled manner. An explicit model between the features in the process trends, and the process conditions may be learned... 

The complementary roles of trans, entering, and leaving ligands when contrasted with the much smaller role of variable cis ligands suggest an explicit model for... 

Both the frequency of the well and its depth cancel, so that the free energy of activation is determined by the height of the maximum in the potential of mean force. The height of this maximum varies with the applied overpotential (see Fig. 13). To a first approximation this dependence is linear, and a Butler-Volmer type relation should hold over a limited range of potentials. Explicit model calculation gives transfer coefficients between zero and unity there is no reason why they should be close to 1/2. For large overpotentials the barrier disappears, and the rate will then be determined by ion transport. 

These considerations are borne out by explicit model calculations (see Fig. 16), which give results close to those predicted by simple application of the Verwey-Niessen... 

It isn t a case of extreme difficulty . It is a situation where, in one case you use a factor which happens to be based upon an explicit model (i.e. linearity) which is correct... 

In this fashion, we extend our deterministic model with a prediction horizon of H = 2 to a multi-stage model. The multi-stage tree of the possible outcomes of the demand within this horizon (starting from period i = 1) with four scenarios is shown in Figure 9.5. Each scenario represents the combination fc out of the set of all combinations of the demand outcomes within the horizon. The production decision x has to be taken under uncertainty in all future demands. The decision xj can react to each of the two outcomes of d i, but has to be taken under uncertainty in the demand di. The corrective decisions are explicitly modeled by replacing xj by two variables 2,1 and 2.2 ... 

Ghezell-Ayagh et al. (2001) An explicit model for direct reforming carbonate Fuel Cell stack, IEEE Trans. Energy Conversion, Vol. 16, No. 3. 

In a classical regression approach, the measurements of the independent variables are assumed to be free of error (i.e., for explicit models), while the observations of the dependent variables, the responses of the system, are subject to errors. However, in some engineering problems, observations of the independent variables also contain errors (i.e., for implicit models). In this case, the distinction between independent and dependent variables is no longer clear. 

A single experiment consists of the measurement of each of the g observed variables for a given set of state variables (dependent, independent). Now if the independent state variables are error-free (explicit models), the optimization need only be performed in the parameter space, which is usually small. 

In the present version of the SR model, the fractions y, and yn are assumed to be time-independent functions of Rei and Sc. Likewise, the scalar-variance source term Va is closed with a gradient-diffusion model. The SR model could thus be further refined (with increased computational expense) by including an explicit model for the scalar-flux spectrum. 

Note that, unlike with the continuous representation (A.2), the discrete representation used in the SR model requires an explicit model for the scalar dissipation rate ea. 

The explicit model can be used to determine the behavior of the TCC kiln when the initial temperature T is varied. A good index of the kiln performance is the distance down the kiln at which 99% of the carbon has been removed. This distance Zb is an important indicator of the extent to... 

It was found that for a large spectrum of commercial operations, the kiln behaved as if Ty was approximately 920°F (494°C) regardless of reactor outlet temperature. This observation allowed the explicit model to be extensively used to explore the effects of some of the variables such as catalyst diSiisivity and air inlet location on kiln performance. However, the absence of fast coke in the model limited its usefulness, here leading us to develop a model which included the plume burner and fast coke. This next stage of advance was not possible without numerical integration. 

Computed results from this model are compared to actual kiln performance in Table VI and the operating conditions taken from kiln samples are given in Table VII. There are no unit factors or adjustable parameters in this model. As with the explicit model, all kinetic data are determined from laboratory experiments. Values of the frequency factors and activation energies are given in Table VIII. Diffusivity values are also included. The amount of fast coke was determined from Eq. (49). With the exception of the T-B (5/12) survey, the agreement between observed and computed values of CO, CO2, and O2 is very good considering that there are no adjustable parameters used to fit the model to each kiln. In the kiln survey T-212/10, the CO conversion activity of the catalyst has been considerably deactivated and a different frequency factor was used in this simulation. 

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