The Treatment of Experimental Data

Some quantities in which we are interested can be measured directly. More often, a quantity must be calculated from other quantities that can be measured. This calculation process is called data reduction. The simplest form of data reduction is the use of a formula into which measured values are substituted. Other forms of data reduction include analysis of a set of data that can be represented by data points on a graph. Construction of such a graph and analysis of features of the graph, such as slopes and intercepts of lines, can provide values of variables. Statistical analysis done numerically can replace graphical analysis, providing better accuracy with less effort. We discuss both of these approaches. 

We also discuss the analysis of the accuracy of e qrerimental data. In the case that we can directly measure some desired quantity, we need to estimate the accuracy of the measurement. If data reduction must be carried out, we must study the propagation of errors in measurements through the data reduction process. The two principal types of e qrerimental errors, random errors and systematic errors, are discussed separately. Random errors are subject to statistical analysis, and we discuss this analysis. 

Every measured quantity is subject to experimental error. 

When the value of a measured quantity is reported, an estimate of the expected error should be included. 

Random experimental errors can be analyzed statistically if the measurement can be repeated a number of times. 

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Flow Models. Many flow models have been proposed (10,12), which are useful for the treatment of experimental data or for describing flow behavior (Table 1). However, it is likely that no given model fits the rheological behavior of a material over an extended shear rate range. Nevertheless, these models are useful for summarizing rheological data and are frequently encountered in the Hterature. 

Equation 46 is a general expression that may be applied to the treatment of experimental data to evaluate exponent a. This, however, is a cumbersome approach that can be avoided by rewriting the equation in dimensionless form. Equation 42 shows that there are n = 5 dimensional values, and the number of values with independent measures is m = 3 (m, kg, sec.). Hence, the number of dimensionless groups according to the ir-theorem is tc = 5 - 3 = 2. As the particle moves through the fluid, one of the dimensionless complexes is obviously the Reynolds number Re = w Upl/i. Thus, we may write ... 

This is the same case with which in Eqs. (2)-(4) we demonstrated the elimination of the time variable, and it may occur in practice when all the reactions of the system are taking place on the same number of identical active centers. Wei and Prater and their co-workers applied this method with success to the treatment of experimental data on the reversible isomerization reactions of n-butenes and xylenes on alumina or on silica-alumina, proceeding according to a triangular network (28, 31). The problems of more complicated catalytic kinetics were treated by Smith and Prater (32) who demonstrated the difficulties arising in an attempt at a complete solution of the kinetics of the cyclohexane-cyclohexene-benzene interconversion on Pt/Al203 catalyst, including adsorption-desorption steps. 

The treatment of experimental data constitutes an essential step in any chemical kinetics study. Although a large part of the present section is based on the investigations in transient flow degradation, the procedure should be general enough to be applicable to other experimental flow arrangements. 

Many computer libraries contain programs that perform the necessary statistical calculations and relieve the engineer of this burden. For discussions of the use of weighted least squares methods for the analysis of kinetic data, see Margerison s review (8) on the treatment of experimental data and the treatments of Kittrell et al. (9), and Peterson (10). 

Margerison, D., The Treatment of Experimental Data in The Practice of Kinetics, Volume I of Comprehensive Chemical Kinetics, edited by C. H. Bamford and C. F. H. Tipper, Elsevier, New York, 1969. 

Application of this equation to the treatment of experimental data leads to curve C in Figure 5.8c. At smaller H202 concentrations, its consumption... 

The treatment of experimental data is an essential activity to calculate precise kinetic variables in the equations above. The quality of the kinetic analysis, the identification of relevant phenomena and, subsequently, model parameter fitting, are directly dependent on the initial data. 

A disadvantage of the differential reactor is the inaccuracy in the determination of conversion and selectivity due to the small concentration changes. The second difficulty in the treatment of experimental data is caused by possible flow nonuniformities. Since the average residence time is short and the fluid elements moving with different axial velocities do not mix, the simplified Equation 5.30 may not be valid. This is because the reactor operates as a segregated flow reactor rather than a plug flow or ideal mixed reactor, on which Equation 5.30 is based. 

Our concern with the treatment of experimental data does not end when we have obtained a numerical result for the quantity of interest. We must also answer the question How good is the numerical result Without an answer to this question, the numerical result may be next to nseless. The expression of how good the result may be is usually couched in terms of its accuracy, i.e., a statement of the degree of the uncertainty of the resnlt. A related question, often to be asked before the experiment is begun, is How good does the resnlt need to be The answer to this question may influence important decisions as to the experimental design, equipment, and degree of effort required to achieve the desired accuracy. 

The principal equations in Part B of this chapter and key references to the text are cited here as a practical aid in the treatment of experimental data. This sununary is not intended to be a substitute for thoughtful judgment. Use the equations with care if there is any doubt concerning the applicability of an equation, review the associated text material. 

The potential benefit of impedance studies of porous GDEs for fuel applications has been stressed in Refs. 141, 142. A detailed combined experimental and theoretical investigation of the impedance response of PEFC was reported in Ref. 143. Going beyond these earlier approaches, which were based entirely on numerical solutions, analytical solutions in relevant ranges of parameters have been presented in Ref. 144 which are convenient for the treatment of experimental data. It was shown, in particular, how impedance spectroscopy could be used to determine electrode parameters as functions of the structure and composition. The percolation-type approximations used in Ref. 144, were, however, incomplete, having the same caveats as those used in Ref. 17. Incorporation of the refined percolation-type dependencies, discussed in the previous section, reveals effects due to varying electrode composition and, thus, provides diagnostic tools for optimization of the catalyst layer structure. 

In order to bring uniformity to the studies, as well as to gain more useful information transferable from one environment to another, standardized protocols covering all variables that can enter into photochemical reactions must be introduced. To this end, the apparatus must be defined in detail and the procedures to be followed fully explained. The discussion of these issues can be approached under three headings, namely, (1) the light or photon source (2) the measurement of the number of photons absorbed by the sample by physical and chemical methods and (3) the treatment of experimental data. 

The correction factors in several different possible cases for the actual 2 values observed for a mercury electrode in NaF (66) are shown in Table 13.7.2. Clearly these factors can be quite large, especially at low concentrations of supporting electrolyte and at potentials distant from E. A number of other cases and details about the treatment of experimental data are discussed in more extensive reviews (16, 34). 

Flory (1969) has discussed in detail methods for calculating from knowledge of bond polarizabilities and barriers to internal rotation. Patterson and Flory (1972) have discussed the treatment of experimental data. 

With the advance of numerical methods it is possible to use the general form of eq. (4.88) for the treatment of experimental data. At the same time, the analysis presented above is still important for the preliminary evaluation of the correspondence of experimental data with the proposed reaction mechanism. 

Styrene and DVB isomers by means of nonlinear least-squares analysis (such an approach to the treatment of experimental data was demonstrated to give a smaller error [17]). The new values of r and are also given in Tables 1.1 and 1.2. The numerical values of the copolymerization constants changed, but the general conclusion remained the same the distribution of crosslinks is extremely inhomogeneous in styrene—p-DVB copolymers. It is more homogeneous in styrene—m-DVB networks, although m-DVB is stiU predominantly incorporated into the network. With a probability of 95%, the compositions of styrene—p-DVB and styrene—m-DVB copolymers were found to be described by monomer reactivity ratios of = 0.30 and f2 = 1.02 and fj = 0.62 and f2 = 0.54, respectively. 

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