Nonlinear chemical kinetics models

S. Vajda, H. Rabitz, E. Walter, Y. Lecourtier, Qualitative and quantitative analysis of nonlinear chemical kinetic models, Chem. Eng. Comm., 83, 191-219 (1989)... 

Thus, from the point of view of modem phenomenological thermodynamics, the current outputs of classical equilibrium thermodynamics (e.g. the description of thermochemistry of mixtures) and the tasks of irreversible thermodynamics, like the description of linear transport phenomena and nonlinear chemical kinetics, are valid much more generally, e.g. even when all these processes mn simultaneously. As we noted above, these properties are not expected to be valid in any material models in some models the local equilibrium may not be valid, reaction rates may depend not only on concentrations and temperature, etc. 

Verneuil et al. (Verneuil, V.S., P. Yan, and F. Madron, Banish Bad Plant Data, Chemical Engineering Progress, October 1992, 45-51) emphasize the importance of proper model development. Systematic errors result not only from the measurements but also from the model used to analyze the measurements. Advanced methods of measurement processing will not substitute for accurate measurements. If highly nonlinear models (e.g., Cropley s kinetic model or typical distillation models) are used to analyze unit measurements and estimate parameters, the Hkelihood for arriving at erroneous models increases. Consequently, resultant models should be treated as approximations. 

Until now, we have dealt with kinetic models and rate constants as the nonlinear parameters to be fitted to spectrophotometric absorbance data. However, measurements can be of a different kind and particularly titrations (e.g. pH-titrations) are often used for quantitative chemical analyses. In such instances concentrations can also be parameters. In fact, any variable used to calculate the residuals is a potential parameter to be fitted. 

There are several control problems in chemical reactors. One of the most commonly studied is the temperature stabilization in exothermic monomolec-ular irreversible reaction A B in a cooled continuous-stirred tank reactor, CSTR. Main theoretical questions in control of chemical reactors address the design of control functions such that, for instance (i) feedback compensates the nonlinear nature of the chemical process to induce linear stable behavior (ii) stabilization is attained in spite of constrains in input control (e.g., bounded control or anti-reset windup) (iii) temperature is regulated in spite of uncertain kinetic model (parametric or kinetics type) or (iv) stabilization is achieved in presence of recycle streams. In addition, reactor stabilization should be achieved for set of physically realizable initial conditions, (i.e., global... 

If the behaviour of complex chemical (in our case catalytic) reactions is known, it will be clear in what way these reactions can be carried out under optimal conditions. The results of studying kinetic models must be used as a basis for the mathematical modelling of chemical reactors to perform processes with probable non trivial kinetic behaviour. It is real systems that can appear to show such behaviour first far from equilibrium, second nonlinear, and third multi dimensional. One can hardly believe that their associated difficulties will be overcome completely, but it is necessary to approach an effective theory accounting for several important problems and first of all provide fundamentals to interpret the dependence between the type of observed kinetic relationships and the mechanism structure. 

This comparison is performed on the basis of an optimality criterion, which allows one to adapt the model to the data by changing the values of the adjustable parameters. Thus, the optimality criteria and the objective functions of maximum likelihood and of weighted least squares are derived from the concept of conditioned probability. Then, optimization techniques are discussed in the cases of both linear and nonlinear explicit models and of nonlinear implicit models, which are very often encountered in chemical kinetics. Finally, a short account of the methods of statistical analysis of the results is given. 

For models in which the dependent variables are linear functions of the parameters, the solution to the above-mentioned optimization problems can be obtained in closed form when the least squares objective functions (3.22) and (3.24) are considered. However, in chemical kinetics, linear problems are encountered only in very simple cases, so that optimization techniques for nonlinear models must be considered. 

The relevance of the model is a matter of controversal discussions. Though the chemical reaction is a rather speculative one, one should realize that the special type of nonlinear reaction can be replaced by an other one. The important step is the combined existence of both, a special chemical kinetics and a related electric behaviour. Both terms can be modified, but they must be based on physical laws and the extraordinary dielectric properties of the material. 

So far we have shown how multivariate absorbance data can be fitted to Beer-Lamberf s law on the basis of an underlying kinetic model. The process of nonlinear parameter fitting is essentially the same for any kinetic model. The crucial step of the analysis is the translation of the chemical model into the kinetic rate law, i.e., the set of ODEs, and their subsequent integration to derive the corresponding concentration profiles. 

Model-based nonlinear least-squares fitting is not the only method for the analysis of multiwavelength kinetics. Such data sets can be analyzed by so-called model-free or soft-modeling methods. These methods do not rely on a chemical model, but only on simple physical restrictions such as positiveness for concentrations and molar absorptivities. Soft-modeling methods are discussed in detail in Chapter 11 of this book. They can be a powerful alternative to hard-modeling methods described in this chapter. In particular, this is the case where there is no functional relationship that can describe the data quantitatively. These methods can also be invaluable aids in the development of the correct kinetic model that should be used to analyze the data by hard-modeling techniques. 

Most attempts at describing CWA PK and PD have used classical kinetic models that often fit one set of animal experimental data, at lethal doses, with extrapolation to low-dose or repeated exposure scenarios having limited confidence. This is due to the inherent nonlinearity in high-dose to low-dose extrapolations. Also, the classical approach is less adept at addressing multidose and multiroute exposure scenarios, as occurs with agents like VX, where there is pulmonary absorption of agent, as well as dermal absorption. PBPK models of chemical warfare nerve agents (CWNAs) provide an analytical approach to address many of these limitations. 

Models linear in 6, with unconstrained parameters, can be fitted directly by solving Eq. (6.3-5). Efficient algorithms and software for such problems are available [Lawson and Hanson (1974, 1995) Dongarra. Bunch, Moler, and Stewart (1979) Anderson et ah. (1992)], and will not be elaborated here. We will focus on nonlinear models with bounded parameters, which are common in chemical kinetics and chemical reaction engineering. 

Dose to Dose PBPK modeling permits reasonable extrapolation from one dose to another, if adequate information on physiology, physicochemical properties, and biochemistry is available. If the dynamic processes modeled by the PBPK approach are all directly proportional to administered concentrations, then the extrapolation can be relatively straightforward. However, this is not often the case, especially at higher doses, where saturation of metabolic or clearance processes can occur [14,19]. Further causes of nonlinearity of chemical kinetics include the induction and inhibition of metabolic enzymes [14], Despite these difficulties successful applications of dose extrapolation using PBPK models for many chemicals have been published [20,21], and... 

Liu et al. (2004, 2005) examined a three-dimensional non-linear coupled auto-catalytic cure kinetic model and transient-heat-transfer model solved by finite-element methods to simulate the microwave cure process for underfill materials. Temperature and conversion inside the underfill during a microwave cure process were evaluated by solving the nonlinear anisotropic heat-conduction equation including internal heat generation produced by exothermic chemical reactions. 

Chemical kinetics equations are commonly nonlinear and may represent diverse phenomena of a catastrophe type. Theoretical studies in this area fall into two groups. Purely model considerations belong to the first group. A certain sequence of elementary reactions — the reaction mechanism, permitted from the chemical standpoint (see the Korzukhin theorem, Chapter 4) is postulated, the corresponding system of kinetic equations is found and its solutions are examined. Such a procedure allows us to predict a possible behaviour of chemical systems. The second approach involves the investigation of a mechanism of a specific chemical reaction, having interesting dynamics. 

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