Mathematical models classification

Another mathematical model classification is proposed by Froment and Bischoff (1990) and is based on the level of complexity, and consequently of accuracy. 

Classification Process simulation refers to the activity in which mathematical models of chemical processes and refineries are modeled with equations, usually on the computer. The usual distinction must be made between steady-state models and transient models, following the ideas presented in the introduction to this sec tion. In a chemical process, of course, the process is nearly always in a transient mode, at some level of precision, but when the time-dependent fluctuations are below some value, a steady-state model can be formulated. This subsection presents briefly the ideas behind steady-state process simulation (also called flowsheeting), which are embodied in commercial codes. The transient simulations are important for designing startup of plants and are especially useful for the operating of chemical plants. 

Focus For the purposes of this discussion, a model is a mathematical representation of the unit. The purpose of the model is to tie operating specifications and unit input to the products. A model can be used for troubleshooting, fault detection, control, and design. Development and refinement of the unit model is one of the principal results of analysis of plant performance. There are two broad model classifications. 

C.G. Vayenas, S. Brosda, and C. Pliangos, Rules and Mathematical Modeling of Electrochemical and Chemical Promotion 1. Reaction Classification and Promotional Rules,/. Catal., in press (2001). 

They cannot be part of a mathematical model whose purpose would be to turn the classification into a continuous quantitative variable. In particular, the example of physical factors illustrates this. Whereas for the highest degree criteria are the same as those of the NFPA code, the simple fact of wanting to add in physical factors to these calculation models forced the originators of this technique to forget about the NFPA code. 

The primary classification that is employed throughout this book is algebraic versus differential equation models. Namely, the mathematical model is comprised of a set of algebraic equations or by a set of ordinary (ODE) or partial differential equations (PDE). The majority of mathematical models for physical or engineered systems can be classified in one of these two categories. 

When describing mathematical modeling in general (not just for classification of bacteria), it is important to point out the mathematical meaning of pattern recognition the mapping of an n-dimensional function to describe a set of... 

This appendix consists of two parts, (a) a conceptual classification and review of the conversion system and (b) a review of conceptual models applied to the heat and mass transport of the thermochemical conversion of biomass. Both these parts are analysed in the context of PBC and the three-step model. Mathematical modelling is outside the scope of this survey. 

Coal combustion processes can be classified based on process type (see Table 9.1), even though classification based on the particle size, the flame type, the reactor flow type, or the mathematical model complexity is also possible [7]. 

The systematization of biological data through mathematical modeling is not a trivial task and requires the intensive use of material and human resources. Moreover, the models often incorporate experimental uncertainties that do not always allow the identification of the trends of the culture, with the desired precision. Figure 8.1 summarizes the main types of models applied to the description of cellular metabolism, according to the classification proposed by Tsuchiya et al., (1966). 

Another important feature of mathematical modeling techniques is the nature of the response data that they are capable of handling. Some methods are designed to work with data that are measured on a nominal or ordinal scale this means the results are divided into two or more classes that may bear some relation to one another. Male and female, dead and alive, and aromatic and nonaromatic, are all classifications (dichotomous in this case) based on a nominal scale. Toxic, slightly toxic, and non-toxic are classifications based on an ordinal scale since they can be written as toxic > slightly toxic > non-toxic. The rest of this section is divided into three parts methods that deal with classified responses, methods that handle continuous data, and artificial neural networks that can be used for both. 

There are a large number of mediods for supervised pattern recognition, mostly aimed at classification. Multivariate statisticians have developed many discriminant functions, some of direct relevance to chemists. A classical application is the detection of forgery of banknotes. Can physical measurements such as width and height of a series of banknotes be used to identify forgeries Often one measurement is not enough, so several parameters are required before an adequate mathematical model is available. 

Such discrepancy is observed in many other cases [12, 13] as well as in cases of tumors growth considered above. Obviously, mathematical model of growth (8) is a very rough approximation. It may be used for rough estimation, for example for classification of population development [16, 17] and also for description of experimental data on separate sections of growth curves. 

Level 2 Classification of Toxicity Samples identified as dissimilar to matched control samples can be fitted to a series of mathematical models that define the multivariate boundaries for known classes of toxicity (Figures 5.4 and 5.5) [6,7,12,63]. Therefore, biofluid or tissue samples from experimental animals treated with novel drugs can be tested to ascertain if the drug induces biochemical effects that would infer a particular site or mechanism of toxicity. 

The physical context concept in the conceptual model is extended to describe the behavior of plastics in the form of pellets through the class solid state condition which encapsulates properties such as pellet type. This part of the implementation model concerns the mathematical modeling of some of the properties of polymers, which correspond to their djmamic or flow behavior. A class for a concrete mathematical model not only holds declarative information such as the list of parameters, but also provides a method for calculating the value of the property modeled. This method requires an implementation which is usually different from the one for another mathematical model. Therefore, mathematical models are organized in this application through further classification. 

In order to form a bridge between the laboratory (chemical) experiments and the theoretical (mathematical) models we refer to Table I. In a traditional approach, experimental chemists are concerned with Column I of Table I. As this table implies there are various types of research areas thus research interests. Chemists interested in the characteristics of reactants and products resemble mathematicians who are interested in characteristics of variables, e.g. number theorists, real and complex variables theorists, etc. Chemists who. are interested in reaction mechanism thus in chemical kinetics may be compared to mathematicians interested in dynamics. Finally, chemists interested in findings resulting from the study of reactions are like mathematicians interested in critical solutions and their classifications. In chemical reactions, the equilibrium state which corresponds to the stable steady states is the expected result. However, it is recently that all interesting solutions both stationary and oscillatory, have been recognized as worthwhile to consider. 

Classification of oscillations in chemical systems will be made in terms of three groupings chemical reactions, chemical elements, and oscillatory mathematical solutions. In the first grouping an enumeration of different reactions , are considered. In the second, a listing of those chemical elements which take part in oscillatory reactions are considered. Finally, the oscillatory solutions of the mathematical model represented by the rate equations are classified on the basis of known examples. 

Chapter 1 introduces the reader to system theory and the classification of systems. It also covers the main principles for the development of mathematical models in general and for fixed bed catalytic reactors in particular. 

According to Ennis (1988), the application of the various multivariate analysis techniques (factor, cluster, discriminant analysis, multidimensional scaling) to classification in sensory analysis has been very valuable but is of little help for understanding the modes of perception. Mathematical models are proposed for predicting human sensory responses and the author concludes that they need development before they are able to improve the understanding of the complex perceptions associated with foods and beverages . 

---