Differential methods of data analysis

The use of the differential method of data analysis to determine reaction orders and specific reaction rates is clearly one of the easiest, since it requires only one experiment. However, other effects, such as the presence of a significant reverse reaction, could render the differential method ineffective. In these cases, the method of initial rates could be used to determine the reaction order and the specific rate constant. Here, a series of experiments is carried out at different initial concentrations, C q, and the initial rate of reaction, is determined for each run. The initial rate, can be found by differentiating the data and extrapolating to zero time. For example, in the tfi-tert-butyl peroxide decomposition shown in Example 5-1, the initial rate was found to be... 

Figure 3.1 Schematic representation of the application of the differential method of data analysis.

Use a differential method of data analysis to assess the validity of the proposed rate expression. When appropriate coordinates and scales (linear, log-log, semilog, etc.) are chosen, this method should lead to a linear plot from which k and k can be determined. What are the values of k and k at this temperature ... 

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. 

Differential Method In order to use the differential method of data analysis, it is necessary to differentiate the reactant concentration versus space-time data obtained in a plug-flow PBR. There are three methods of differentiation that are commonly used (i) graphical equal-area differentiation, (ii) numerical differentiation or finite difference formulas, and (iii) polynomial fit to the data followed by analytical differentiation. The aim of differentiation is to obtain point values of the reaction rate ( Ra)p at each reactant concentration Q4 or conversion xa or space time (.W/Fao), as required. All three differentiation methods can introduce some error to the evaluation of -Ra)p- Information on and illustration of the various differentiation techniques are available in the literature [23, 26]. 

The differential method of data analysis is convenient since it requires only one experiment for rate equations containing only one reactant and is readily applied to determine reaction parameters of power law or LHHW rate equations. The rate equations may be linearized to allow the use of linear regression... 

Analysis In this example the differential method of data analysis was used to find the reaction order with respect to triiyl (a = 1.99) and the pseudo rate constant (ft = 0.125 (dmVmol)/min). The reaction order was rounded up to a = 2 and the data was regressed again to obtain ft = 0.122 (dmVmoD/min. again knowing ft and Cbo and the true rate constant is ft = 0.244 (dnvVmol)Vmin. 

Figure 3-17. Sohematios of the differential method for data analysis.

The root time method of data analysis for diffusion coefficient determination was developed by Mohamed and Yong [142] and Mohamed et al. [153]. The procedure used for computing the diffusion coefficient utilizes the analytical solution of the differential equation of solute transport in soil-solids (i.e., the diffusion-dispersion equation) ... 

Factorial design of experiments, combined with statistical methods of data analysis, offers wider and more differentiated information on the system, while conclusions are of greater usability. The results of all the eight runs in the analyzed example serve for determining the factor effects, with seven trials being independent possibilities of testing the effects and one serving for their comparison with the chosen fixed values. Three out of seven independently determined factor effects serve for... 

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 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 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. 

Finally, the realization that spectra of cells do exhibit a natural variance, due to a number of factors such as metabolic activity, stage in the cell cycle, and so on, has necessitated a statistical approach to the interpretation of SCP data. Since spectra of cells may be noisy - due to the small amount of sample in a typical cell ( 10 g) - multivariate methods of data analysis should be employed on the spectral data sets to help differentiate uncorrelated changes (noise) with variance associated with the desired (or suspected) cause of change. The availability of increased computational power to carry out these analyses was among the most important developments to propel SCP toward the commercial realm. 

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

Integrating the rate equation is often diffieult for orders greater than 1 or 2. Therefore, the differential method of analysis is used to seareh the form of the rate equation. If a eomplex equation of the type below fits the data, the rate equation is ... 

The following example illustrates the use of the differential method for the analysis of kinetic data. It also exemplifies some of the problems... 

These concentrations may be used in the various integral and differential methods for the analysis of kinetic data that have been described in previous sections. An example of the use of this approach is given in Illustration 3.5. 

Cycled Feed. The qualitative interpretation of responses to steps and pulses is often possible, but the quantitative exploitation of the data requires the numerical integration of nonlinear differential equations incorporated into a program for the search for the best parameters. A sinusoidal variation of a feed component concentration around a steady state value can be analyzed by the well developed methods of linear analysis if the relative amplitudes of the responses are under about 0.1. The application of these ideas to a modulated molecular beam was developed by Jones et al. ( 7) in 1972. A number of simple sequences of linear steps produces frequency responses shown in Fig. 7 (7). Here e is the ratio of product to reactant amplitude, n is the sticking probability, w is the forcing frequency, and k is the desorption rate constant for the product. For the series process k- is the rate constant of the surface reaction, and for the branched process P is the fraction reacting through path 1 and desorbing with a rate constant k. This method has recently been applied to the decomposition of hydrazine on Ir(lll) by Merrill and Sawin (35). 

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