The Integral Method of Data Analysis

3 THE INTEGRAL METHOD OF DATA ANALYSIS 6.3.1 Usii the Integral Method 

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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. 

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

Integral Method Alternatively, the integral method of data analysis maybe used with a trial procedxire, in which a particular rate equation is assumed and integrated to yield a relationship between space time (W/Pao) concentration (or conversion, in the case of more than one reactant) of the following form ... 

The Integral Method of Data Analysis 173 Table 6-6 Values of A, E, and ko for AIBN Decomposition in Supercritical Carbon Dioxide... 

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... 

Demonstrate how you would use the integral method of data analysis to test (graphically) this rate equation against the experimental data. Carry out any mathematical operations that are required. Sketch the graph that you would make, showing what you would plot against whaL... 

Use an integral method of data analysis and report the value of the rate constant for the rate expression that provides a suitable fit of the data. The reaction was carried out isothermaUy at 25° C with an initial HCl concentration of 4.50 mol/kg solution. 

It has been suggested that this hquid phase reaction is second-order in HCl and first-order in A. Use an integral method of data analysis to determine if the data below are consistent with the proposed rate expression. If the data are consistent, what is the third-order rate constant at 25°C The chemical shift is a property that is an additive function of the contributions of the various species. AO . 0 mol/kg solution and — 4. 4 mol/kg solution. 

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. 

Two important points have emerged from the discussions of the differential and integral methods of data analysis ... 

Test of reaction rate data by the integral method of analysis. 

Further investigation should involve the integration of lithological and grain-size data as well as other methods of data analysis. 

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... 

Batch reactors are used primarily to determine rate law parameters for homo, geneous reactions. This determination Ls usually achieved by measuring coa centration as a function of time and then using either the differential, integral, or least squares method of data analysis to determine the reaction order, a, and specific reaction rate, k. If some reaction parameter other than concentration i s monitored, such as pressure, the mole bMance must be rewritten in terms of the measured variable (e.g., pressure). 

The parameters K and V , can readily be determined from batch reactor data by using the integral method of analysis, Dividing both sides of Equation (7-95) by tKJV and rearranging yields... 

In the integral method of analysis of rate data we are looking for the appropriate function of concentration correspondmg to a particular rate law that is linear with time. You should be thoroughly familiar with the methods of obtaining these linear plots for reactions of zero, first, and second order. 

Another possible cause for such a behavior may not be due to the non-ideality of the system but may be rooted in the nature of the experimental technique and the method of data analysis. The binary sorption equilibria (total amount adsorbed and the composition of the adsorbed phase) are not measured directly in the chromatographic method but are deduced from integration of the slopes of the isotherms at different gas phase compositions. Such an integration may give rise to a significant cumulative error which, in turn, may be responsible for underpredicting the binary... 

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 ... 

The previous chapters by Bretz on qualitative research (P) and Sanger on quantitative research 8) have examined methods of data analysis germane to those traditions. In a research endeavor that employs a mixed methods design, the important analysis question focuses on how the data is integrated, related, and mixed 3, 12). How is data transformed from one type into another so that the results from sequential studies may be compared Similarly, how is data transformed during a concurrent study to facilitate interpretation ... 

This simple approach was adopted in order to circumvent the complications that are introduced by the fact that the volume of the liquid phase in the reactor varies with time. When the volume of the aqueous growth medium varies during the course of the reaction, an approach based on integration of a proposed rate law is problematic, although numerical integration would be possible. An additional reason for employing the differential approach below is that for rate laws that are other than those of the simple nth-order form (such as a Monod rate expression) a differential method of data analysis is often adequate for preliminary considerations involved in the design of a bioreactor that is intended to operate in a batch mode. 

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. 

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