Non-linear Mathematical Model

The actual behaviour of the electrorefining system can be more complicated owing to non-linear effects, which may manifest themselves in the dependencies of the parameters on the variables of Eqs. (5.13) and (5.14). The influence of the positive feedback between the amount of deposit and its accumulation rate is considered below. 

Let us suppose that the parameter J depends on the variable Y because of the temperature dependence of the part of the ionic conductivity of the film, which can be expressed in the form 

The temperature dependence of the electronic conductivity is neglected, and the dependence of the ionic conductivity is assumed to be exponential  

The dimensionless temperature 6 = AT/T can be expressed as a function of the variable Y from the heat balance with respect to the heat exchange taking place under stationary conditions  

The heat transfer into the bulk is proportional to the temperature difference 

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A non-linear mathematical model, which is a set of ordinary differential equations, for the process in the SPBER was developed.19 The model accounts for the heterogeneous electrochemical reaction and homogeneous reaction in the bulk solution. The lateral distributions of potential, current density and concentration in the reactor were studied. The potential distribution in the lateral dimension, x, of the packed bed was described by a one dimensional Poisson equation as ... 

Thus, we have derived the non-linear mathematic model of the electrochemical process with low-conductive surface film that can decompose with an exponential temperature-dependent rate. Essentially, this is an example of a thermokinetic model. 

Sobol, I.M. 1993. Sensitivity analysis for non-linear mathematical models. Mathematical Modelling and Computational Experiment 1 407-414 Translated from Russian I.M. SoboT. 1990. Sensitivity estimates for nonlinear mathematical models, Matematicheskoe Modelirovanie 2 112-118. 

Sobol , I.M. Sensitivity analysis for non-linear mathematical models. Math. Model. Comp. Exp. 

The program of catastrophe theory has been formulated in the Thom s book Structural Stability and Morphogenesis. The philosophical, mathematical and experimental incentives to the program of catastrophe theory may be found there. On the position of catastrophe theory among methods of investigation of stability of solutions of non-linear equations (models) one may learn from a book by Thompson. Stewart s and Zeeman s papers on the perspectives of the development of generalized catastrophe theory are also worth study. 

It should be emphasized that there have been exceptions to this attitude. In 1910 and 1920 Lotka published his theory of chemical reactions in which the oscillations of reagent concentrations could appear. An essential feature of the Lotka models was nonlinearity. In mathematics and physics a trend has long persisted to examine linear systems and phenomena and to replace non-linear models by (approximate) linear models. The trend, originating from insufficient mathematical means, has turned into specific philosophy. The non-linear Lotka models thus constituted a deviation from a canon. Hence, general arguments of thermodynamic nature, lack of interest in non-linear models and commonness of observations of a monotonic attainment of the equilibrium in chemical reactions were the reasons for skepticism and disbelief which the results of Belousov have met with. 

Computer programming, as FORTRAN or Visual Basic, may be used to formulate mathematical functions of any complexity. This is a powerful feature of simulators often underestimated. An example is the study of steady state controllability directly from non-linear plant models (see Chapter 12). Different controllability measures can be calculated directly by programming, or by exporting data to other packages. 

Finite-element techniques can cope with large, highly non-linear deformations, making it possible to model soft tissues such as skin. When relatively large areas of skin are replaced during plastic surgery, there is a problem that excessive distortion of the apphed skin will prevent adequate adhesion. Finite-element models can be used to determine, either by rapid trial-and-error modelhng or by mathematical optimisation, the best way of... 

See also the theoretical description of a micro reactor for optical photocatalytic dissociation of non-linear molecules in [140]. Here, a mathematical model for a novel type of micro reactor is given. Rotating non-linear molecules at excitation of valent vibrations are considered, having a magnetic moment. Resonance decay of molecules can be utilized with comparatively weak external energy sources only. 

In the last decades not only thousands of chemical descriptors but also many advanced, powerful modeling algorithms have been made available, The older QSAR models were linear equations with one or a few parameters. Then, other tools have been introduced, such as artificial neural network, fuzzy logic, and data mining algorithms, making possible non linear models and automatic generation of mathematical solutions. 

More advanced mathematical aspects of the graph-theoretical models for aromaticity are given in other references [36, 48, 49]. Some alternative methods, beyond the scope of this chapter, for the study of aromaticity in deltahedral molecules include tensor surface harmonic theory [51-53] and the topological solutions of non-linear field theory related to the Skyrmions of nuclear physics [54]. 

IVIVCs can be divided into non-quantitative and quantitative. In non-quantitative correlations the two variables are not related to each other via a mathematical relationship. A characteristic example is the rank-order correlation that was popular in the 1970s (10-15). In quantitative correlations the in vitro variable correlates with the in vivo variable via a linear or a non-linear equation. A quantitative IVIVC can be established, with or without the framework of a model, by using estimated values of characteristic parameters of the in vitro dissolution process and estimated values of the characteristic parameters of the in vivo arrival-in-bloodstream... 

This exponential temperature dependenee represents one of the most severe non-linearities in chemical engineering systems. Keep in mind that the apparent temperature dependence of a reaction may not be exponential if the reaction is mass-transfer limited, not chemical-rate limited. If both zones are eneountered in the operation of the reactor, the mathematical model must obviously include both reaction-rate and mass-transfer effeets. 

For non-threshold mechanisms of genotoxic carcinogenicity, the dose-response relationship is considered to be linear. The observed dose-response curve in some cases represents a single ratedetermining step however, in many cases it may be more complex and represent a superposition of a number of dose-response curves for the various steps involved in the tumor formation (EC 2003). Because of the small number of doses tested experimentally, i.e., usually only two or three, almost all data sets fit equally well various mathematical functions, and it is generally not possible to determine valid dose-response curves on the basis of mathematical modeling. This issue is addressed in further detail in Chapter 6. 

The present text is aimed partly at chemists, to encourage more of us to be less afraid of the mathematics, but also with the hope of catching the attention of mathematicians, engineers, etc., who should find that the chemical world offers exemplary non-linearities which can be realized in practice as well as in numero. The book is loosely divided into two unequal parts. The main thrust is to introduce a variety of the techniques used by the non-linear kineticist, with special attention to particularly simple model schemes. This part encompasses chapters 2 to 13. Other readers may prefer to begin with real experimental results, and these are presented in chapters 14 and 15. 

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