Kinetic methods, integral

Integral methods Kinetic methods based on an integrated form of the rate law. 

When even second-order reactions are included in a group to be analyzed, individual integration methods maybe needed. Three cases of coupled first- and second-order reactions will be touched on. All of them are amenable only with difficulty to the evaluation of specific rates from kinetic data. Numerical integrations are often necessary. 

The basic problem of design was solved mathematically before any reliable kinetic model was available. As mentioned at start, the existence of solutions—that is, the integration method for reactor performance calculation—gave the first motivation to generate better experimental kinetic results and the models derived from them. 

A General Integral Method for the Analysis of Kinetic Data—Graphical Procedure. 

Illustrations 3.2 and 3.3 are examples of the use of the graphical integral method for the analysis of kinetic data. 

Integral Methods for the Analysis of Kinetic Data—Numerical Procedures. While the graphical procedures discussed in the previous section are perhaps the most practical and useful of the simple methods for determining rate constants, a number of simple numerical procedures exist for accomplishing this task. 

It is also possible to use integral methods to determine the concentration dependence of the reaction rate expression and the kinetic parameters involved. In using such approaches one again requires a knowledge of the equilibrium constant for use with one of the integrated forms developed in Section 5.1.1. 

There are two procedures for analyzing kinetic data, the integral and the differential methods. In the integral method of analysis we guess a particular form of rate equation and, after appropriate integration and mathematical manipulation, predict that the plot of a certain concentration function versus time... 

Schoenemann, E., Hahn, H., and Bracht, A. (1991), Determination of kinetic parameters from non-isothermal conductivity measurements by an integral method, Thermochim. Acta, 185(1), 171-176. 

The rate expressions Rj — Rj(T,ck,6m x) typically contain functional dependencies on reaction conditions (temperature, gas-phase and surface concentrations of reactants and products) as well as on adaptive parameters x (i.e., selected pre-exponential factors k0j, activation energies Ej, inhibition constants K, effective storage capacities i//ec and adsorption capacities T03 1 and Q). Such rate parameters are estimated by multiresponse non-linear regression according to the integral method of kinetic analysis based on classical least-squares principles (Froment and Bischoff, 1979). The objective function to be minimized in the weighted least squares method is... 

The rates of liquid-phase reactions can generally be obtained by measuring the time-dependent concentrations of reactants and/or products in a constant-volume batch reactor. From experimental data, the reaction kinetics can be analyzed either by the integration method or by the differential method ... 

This is not by itself a kinetic method. It must be combined with either the differential or the integration method and involves keeping all the reactants but one in large excess so that their concentration does not vary through the reaction under these conditions, the observed reaction order is that of the limiting reagent. For example, the simple second-order reaction of Equation 3.18,... 

Product Distribution at the Riser Exit With X or calculated, YQ, YE, Y and Y at the riser exit must be now be computed. This can be done by integrating the kinetic equations by numerical methods with the following boundary conditions YA = YAo and YQ = Yg = YQ = Y = 0. 

Lateral interactions between adsorbed species may make the kinetic parameters a function of coverage. In this case, it would be incorrect to rely on integral methods that depend on the properties of the entire TPD curve. Miller et al. [33] and Nieskens et al. [36] showed that doing so may induce artificial compensation effects in the results. Differential methods, which analyze a part of a trace are - in... 

The first, called the integral method of data analysis, consists of hypothesizing rate expressions and then testing the data to see if the hypothesized rate expression fits the experimental data. These types of graphing approaches are well covered in most textbooks on kinetics or reactor design. 

The second approach is a fractional-step method we call asymptotic timestep-splitting. It is developed by consideration of the specific physics of the problem being solved. Stiffness in the governing equations can be handled "asymptotically" as well as implicitly. The individual terms, including those which lead to the stiff behavior, are solved as independently and accurately as possible. Examples of such methods include the Selected Asymptotic Integration Method (4,5) for kinetics problems and the asymptotic slow flow algorithm for hydrodynamic problems where the sound speed is so fast that the pressure is essentially constant (6, 2). ... 

The text reviews the methodology of kinetic analysis for simple as well as complex reactions. Attention is focused on the differential and integral methods of kinetic modelling. The statistical testing of the model and the parameter estimates required by the stochastic character of experimental data is described in detail and illustrated by several practical examples. Sequential experimental design procedures for discrimination between rival models and for obtaining parameter estimates with the greatest attainable precision are developed and applied to real cases. 

Such a method of kinetic analysis is termed the differential method since the residual sum of squares is based on rates. The required differentiation of XA versus W/FA0 data can be a source of errors, however. To avoid this, the same set of data can be analyzed by the so-called integral method, which consists of minimizing a residual sum of squares based on the directly observed conversions ... 

The random degradation and reaction kinetics of high-molecular weight polymers can be determined by an approximate integral method (Ozawa 1965, Hirose and Hatakeyama 1986). Generally, the fractional weight, W, of a reacting material can be expressed as a function of the fraction of a structural quantity which is represented by x, i.e.,... 

In general the interpretation of the data is somewhat more complicated than for the differential method. Especially for an unknown complicated kinetic functions, the derivation of the correct reaction rate expression RA from experimental results using Equations 5.41 and 5.33 is more cumbersome than fitting Equations 5.30 and 5.33. This is especially true for complex reaction networks, as in the isomerization and cracking reactions of crude oil fractions, where the integral method is very laborious with which to derive individual rate constants. 

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