The Integral Method of Kinetic Analysis

C The integral Method of Kinetic Analysis The integration of the rate equation leads to 

The regression is garmaUy nonlinear and in the second case the computations are even more complicated because the equation is implicit in x. Peterson and Lajndus [42] used the integral method with nonlinear regression on Franckaerts and Froment s data and found excellent agreement, as shown by Table 2.3.C-1. A further illustration of such agreement is based on Hosten and Froment s data on the isomerization of n-pentane [38] as analyzed by Froment and Mezaki [43], 

In this case the expression W/Fm versus/(x) was linear in two groups containing the parameters, so that linear regression was possible when the sum of squares on WjF Q was minimized. When the objective function was based on the conversion itself, an implicit equation had to solved and the regression was nonlinear. Only approximate confidence intervals can then be calculated from a linearization of the model equation in the vicinity of the minimum of the objective function. 

Again the agreement between the linear and nonlinear regression is excellent, which is probably due to the precision of the data. Poor data may give differences, but they probably do not deserve such a refined treatment, in any event. 

The problem of estimation in algebraic equations that are nonlinear in the parameters was recently reviewed by Seinfeld [44] and by Froment [45] and [46], who give extensive lists of references. Standard textbooks dealing with this topic are by Wilde and Beightler [47], Beveridge and Schechter [48], Hoffmann and Hofmann [49], Himmelblau [50], and Rosenbrock and Storey [51]. 

For a simple order, the rate expression can be integrated and special plots utilized to determine the rate coefficient. A plot of MCa versus t or xjj -x versus t is used similarly for a second-order irreversible reaction. For reversible reactions with first order in both directions a plot of ln(Ci - Qeq)/(Qo - Qeq) or ln(l - XaIxa versus f yields ( i + ki) from the slope of the straight line. Using the thermodynamic equilibrium constant K = kxlkz, both k and kz are obtained. Certain more complicated reaction rate forms can be rearranged into such linear forms. These plots are useful for an estimate of the quality of the fit to the experimental data and can also provide initial estimates to formal regression techniques that will be systematically discussed in Chapter 2. 

INTEGRATED FORMS OF SIMPLE KINETIC EXPRESSIONS FOR REACTIONS WITH CONSTANT DENSITY 

Graphical representation of various integrated kinetic equations. [Caddell and Hurt, 1951]. 

The integrated forms of several other simple-order kinetic expressions, obtained under the assumption of constant density, are listed in Table 1.2.4.2-1. Fig. 1.2.4.2-1 graphically represents the various integrated kinetic equations of Table 1.2.4.2-1 [Caddell Hurt, 1951). Note that for a second order reaction with a large ratio of feed components, the reaction order degenerates into pseudo-first-order. 

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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... 

Discrimination Using the Integral Method of Kinetic Analysis... 

The integral method of kinetic analysis can be conveniently used when the expression for can be analytically integrated. When the differential method is applied, N, i4 is obtained as the slope of a curve giving (px)hi (Pa)i>m a function of p, VIF, arrived at by measuring the amount of A abmrbed at different gas flow rates. 

L2.4.1 The Differential Method of Kinetic Analysis L2.4.2 The Integral Method of Kinetic Analysis Coupled Reactions... 

When the integral method of kinetic analysis is applied, numerical integration of the continuity equations containing the rate equations is generally necessary for the comparison of the predicted and experimental responses for each experiment in each iteration cycle of the parameter estimation. Examples can be found in the work of De Pauw and Froment [1975] on n-pentane reforming in the presence of coke formation, in the work of Emig, Hofmann, and Friedrich [1972] on methanol oxidation, and in Example 2.6.4.A on benzothiophene hydrogenolysis. 

ETHANOL DEHYDROGENATION SEQUENTIAL DISCRIMINATION USING THE INTEGRAL METHOD OF KINETIC ANALYSIS... 

The data can also be obtained in an integral fixed reactor, of course. Information on the coke content profile in a tubular reactor may yield valuable information as to the mechanism of coking—parallel or consecutive—and, therefore, as to the form of Pq, as will be shown in the next section. If the integral method of kinetic analysis is applied to the data, as was done by De Pauw and Froment [1975], the conversion 4 replaces the rate Pa in the objective function, requiring integration of the rate equation. 

Apply the differential and integral methods of kinetic analysis (see Chapter 2) to determine the rate coefficients and order at the different temperatures. To work out the integral method of kinetic analysis, it is necessary to express pA as a function of x. A rigorous expression would only be possible if all reactions taking place were exactly known. Therefore, undertake an empirical fit of this function. 

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 data given below are provided by J. H. Raley, F. E. Rust, and W. E. Vaughn J.A.ChS., 70,98 (1948)]. They were obtained at 154.6°C under a 4.2-mmHg partial pressure of nitrogen, which was used to feed the peroxide to the reactor. Determine t he rate coefficient by means of the differential and integral method of kinetic analysis. 

The approach to be followed in the determination of rates or detailed kinetics of the reaction in a liquid phase between a component of a gas and a component of the liquid is, in principle, the same as that outlined in Chapter 2 for gas-phase reactions on a solid catalyst. In general the experiments are carried out in flow reactors of the integral type. The data may be analyzed by the integral or the differential method of kinetic analysis. The continuity equations for the components, which contain the rate equations, of course depend on the type of reactor used in the experimental study. These continuity equations will be discussed in detail in the appropriate chapters, in particular Chapter 14 on multiphase flow reactors. Consider for the time being, by way of example, a tubular type of reactor with the gas and liquid in a perfectly ordered flow, called plug flow. The steady-state continuity equation for the component A of the gas, written in terms of partial pressure over a volume element dV and neglecting any variation in the total molar flow rate of the gas is as follows ... 

Determine a suitable kinetic model by means of both the differential and integral method of kinetic analysis. 

Integral Method of Kinetic Analysis Substituting the rate equation in Eq. 9.1-2 leads to ... 

Figure 3.2 Test of the fit of a reaction rate expression to kinetic data by the integral method of data analysis.

The experimental results may be analyzed in two ways, as mentioned already in Chapter 1—by the differential method of kinetic analysis or by the integral method, which uses the x versus W/Faq data. The results obtained in an integral reactor may be analyzed by the differential method provided the a versus W/Fao curves are differentiated to get the rate, as illustrated by Fig. 2.5-2. Both methods are discussed in the following section. 

From (9.1-1) it follows that the slope of the tangent of the curve Xa versus V/Fao is the rate of reaction of A at the conversion Xa. The rates are shown in Table 9.2.1-2. The kare calculated by both the integral and the differential method of kinetic analysis. 

Use the integral method of data analysis to test whether the reaction is zero, first, or second order in B12. If one of these kinetic models fits the data, determine the value of the rate... 

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... 

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 ... 

Methods of kinetic analysis that involve fitting of experimental data to assumed forms of the reaction model (first-order, second order, etc.) normally result in highly uncertain Arrhenius parameters. This is because errors in the form of the assumed reaction model can be masked by compensating errors in the values of E and A. The isoconversional technique eliminates the shortcomings associated with model-fitting methods. It assumes the unknown integrated form of the reaction model, g(a), as shown in Eq. (4), to be the same for all experiments. 

Tdjle 2 Thermal cracking of propane. Rate versus conversion, k-vaiues from the integral and differential method of kinetic analysis... 

Table I Thermal cracking of acetone. Rate coefficients and order by the integral and differential methods of kinetic analysis...

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