Basic Kinetic Models

Three basic kinetic laws have been used to describe the oxidation rates of pure metals. It is important to bear in mind that these laws are based on relatively simple oxidation models. Practical oxidation problems usually involve alloys and considerably more complicated oxidation mechanisms and scale properties than considered in these simple models. 

If the oxide film or scale cracks or is porous, that is, if the corrosive gas can continue to penetrate readily and react with the base metal in a catastrophic manner, no protection will be afforded and attack will proceed at a rate determined essentially by the availability of the corrosive gas. In this case, the rate will not sensibly change with time, and, as is apparent from Fig. 15.12, the weight change or depth of penetration from oxidation is a straight line or linear function of time and may be expressed as 

The logarithmic rate behavior follows an empirical relationship, which has no fundamental underlying mechanism. This behavior, also shown in Fig. 15.12, is mainly applicable to thin oxide films 

on the other hand, the scale formed is continuous, adherent, and prevents easy access of the corrosive gas to the underl5dng base metal, a considerable measure of protection may occur, and the extent of protection will increase as the scale thickens. In this case, the availability of the corrosive gas will not determine the reaction rate. 

After an initial period of time, the rate of scale growth will decrease to a low level, that is, a considerable measure of oxidation resistance will be obtained and, provided the scale remains intact, will be maintained for very long times. 

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Torkelson and coworkers [274,275] have developed kinetic models to describe the formation of gels in free-radical pol5nnerization. They have incorporated diffusion limitations into the kinetic coefficient for radical termination and have compared their simulations to experimental results on methyl methacrylate polymerization. A basic kinetic model with initiation, propagation, and termination steps, including the diffusion hmitations, was found to describe the gelation effect, or time for gel formation, of several samples sets of experimental data. 

The basic kinetic model for enzyme catalysed conversions in water and in w/o-microemulsions is based on the theory of MichaeHs and Menten [83]. Although the Michaelis-Menten-model is often sufficient to describe the kinetics, the bi-bi-models (e. g. random bi-bi, orderedbi-bi or ping-pongbi-bi), which describe the sequences of substrate bindings to the enzyme are the more accurate kinetic models [84]. 

A Basic Kinetic Model to Begin the Study of Polymer Aging... 

A proper fit of the time-courses of some batch reactor experiments at different starting concentrations represents an appropriate test of the rate equation. This implies that numerical integration of the rate equation (e. g. by the Runge Kutta method11121), yielding a simulated time-course, has to fit the data of the measured time-course over the whole range of conversion (compare to Fig. 7-17 B). Examples of these methods will be given after the presentation of the basic kinetic models. 

In this section the basic kinetic model for enzyme-catalyzed bioconversions is presented. Understanding this model is the foundation for deriving more complex models. In their theory of enzyme catalysis, Michaelis and Menten 113 postulated the existence of an enzyme substrate complex (ES), which is built up in a reversible... 

Eq. (37) is the basic kinetic model for isomerizations and racemizations. For example, the kinetic constants of an alanine racemase of Bacillus stearothermophilus are found to be... 

Basic Kinetic Models of Catalytic Heterogenous Reactions... 

This appendix contains a series of computer simulations that are thought to represent the most significant basic kinetic models in bioprocessing. The models are summarized in Table 11.1. The simulations in the figures also contain the values of model parameters chosen for demonstration. Mainly two different kinds of plots are presented, the first showing concentration/time curves and the second the corresponding time curves of specific rates of bioprocesses. The models are as follows ... 

In contrast, the value analysis is more focused on the construction of a basic kinetic model by excluding the excessively low-value steps from the expanded primary reaction model. 

David M-O, Gref R, Nguyen T Q and Neel J (1991a), Pervaporation-esterification coupling part I. Basic kinetic model , Trans Inst Chem Eng, 69,335-340. 

David, M.-O., Gref, R., Nguyen, T. Q., Neel, J. (1991). Pervaporation—esterification coupling. Part I. Basic kinetic model. Chemical Engineering Research and Design, 69, 335—340. 

As our understanding of the elementary catalytic reaction steps and our analytical capabilities improved, the basic kinetic models were expanded in order to reflect this finer level of detail. 

To include the details of colorants in the numerical modeling is difficult, if not impossible. Tactically, in a practical computation, the basic kinetic model for materials with and without colorants can remain the same, while the model parameters should be carefully determined for each colored polymer such that the nucleation and growth trends are correlated with the colorant effects. 

Table 3.10 Properties of the function Y( ) for the basic kinetic model...

In spite of being the first reaction ever studied [95], esterification has been under investigation ever since, and much knowledge has accumulated, even if some points are stUl less clear. The basic kinetic model for polyesterification was established by Flory and is summarized in his classic book [5j. Esterification was shown to be acid-catalyzed, it is first-order with respect to hydroxyls and, with respect to carboxyls, its order is either one in the presence of foreign strong protic acids, or two in their absence [Eq. (52)]. 

We have shown how the development of processes in the chemical industry is supported by means of mathematical models. Recognizing that the basic kinetic model still has several weak points, we are convinced that these have hardly any effect on the trends disclosed by the optimization, and also that the results of our study will contribute towards improved operation of the reactor section in existing plants as well as towards more exact design of new installations. 

To find expressions for X and z, we need a slightly more elaborate version of the kinetic model of gases. The basic kinetic model supposes that the molecules are effectively pointlike however, to obtain collisions, we need to assume that two points score a hit whenever they come within a certain range d of each other, where d can be thought of as the diameter of the molecules (Fig. 7.20). The collision cross-section, a (sigma), the target area presented by one molecule to another, is therefore the area of a circle of radius d, so O = nd. When this quantity is built into the kinetic model, we find that... 

In equation (1) the term f(a) represents the mathematical expression of the kinetic model. The most frequently cited basic kinetic models are summarised in Table 2. 

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