Graphical Differentiation Method

In addition to the graphical technique used to differentiate the data, tw-o other methods are commonly used dift ereniiation formulas and polynomial fitting. 

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In Figure 3.11, we exclude Ihe use of differential methods with a BR, as described in Section 3.4.1.1.1, This is because such methods require differentiation of experimental ct(t) data, either graphically or numerically, and differentiation, as opposed to integration, of data can magnify the errors. 

If AH is plotted against m at constant 2, a graphical differentiation by the chord-area method will yield L ii as a function of composition. Alternatively, the data could be fitted to a polynomial and the derivative of that pol5momial then could be computed. Differentiation of Equation (18.12) with respect to 2 at constant m yields... 

Numerous procedures have been developed for graphical differentiation. A particularly convenient one (9), which we call the chord-area method, is illustrated using the same data (from Table A.2) to which we previously apphed numerical differentiation. It is clear from Figure A.2 that if we choose a sufficiently small temperature interval, then the slope at the center of that interval will be given approximately by A /Af. In this example, with an interval of 5°C, the approximation is good. Then we proceed to tabulate values of A °/At from 0°C, as illustrated in Table A.6 for the first few data. Note that values of are placed between the values of to which they refer, and the temperature intervals (5°C) are indicated between their extremities. Similarly, as L%°jis an average value (for example, —0.000484) within a particular region (such as 0°C to 5°C), values in the fifth column also are placed between the initial and the final temperatures to which they refer. 

The number of different functions is equal to the number of steps in the sequence. To determine the mechanism we thus have to find a (finite) number of functions which by linear superposition gives the reciprocal velocity. For a preliminary orientation it is often a good practice to find dt/dx by numerical (graphical) differentiation for different values of x in different experiments. This method has been used extensively by Bodenstein and his school. 

This indicates that the required D (c is determined by graphical differentiation of a plot for (1/2) [pa(c °) + ( >< 1°°)] °° vs-ci°°- It is n°t a serious disadvantage of this method that both Da and Dd must be determined experimentally, since, in most work, the measurement of an absorption curve is followed by the determination of the corresponding desorption. 

Three methods are commonly used to estimate this quantity (1) slopes from a plot of n versus f, (2) equal-area graphic differentiation, or (3) Taylor series expansion. For details on these, see a mathematics handbook. The derivatives as found by equal-area graphic differentiation and other pertinent data are shown in the following table ... 

The other approach, the differential method, entails graphical differentiation of the data for comparison with the differential form of Eq. (I). If the data are plotted... 

The Integration of the Differential Equation for Liquid Junction Potential. To account for the results of such measurements as have been described in the previous section there have been a number of integrations of the fundamental differential equation (3). Of these the following will deal only with the integrations by Henderson and by Planck. In addition a graphical integration method devised by Maclnnes and Longsworth will be discussed. 

The treatment of data acquired in sedimentation analysis usually involves graphical differentiating of the sediment accumulation curve. This method of obtaining particle size distribution is based on the Svedberg - Oden equation ... 

For differential methods, moderate mathematical efforts are required. However, the accuracy in calculations of the rate is low since numerical or graphical differentiation is used to determine the rate values from concentration vs time diagrams (Figure 10.10). 

Since the replots of slopes or intercepts versus / are nonlinear, it is not possible to determine directly the values of kinetic constants from the data in Fig. 2 instead, it is necessary to apply a differential method to rate equations, in order to obtain a graphical solution (Cleland, 1967, 1979). By using the differential method, we are raising the horizontal axis in Fig. 2 ensuring that curves become hyperbola that start at the origin ... 

Figure 3. Determination of inhibition constants in hyperbolic inhibition by a differential method. Graphical presentation of Eqs. (6.8) and (6.9), assuming that a-2 and fi = 0.5...

For determining the rate constants, we use similar methodology as presented previously the integral or differential methods. The differential method is frequently used and easily visualized in the graphic solution after transforming the rate equation. For example, for a monomolecular, irreversible, and first-order reaction, the rate is expressed in Equation 10.22. It was deduced assuming that the reaction rate is the limiting step and both reactants and products adsorbed. Thus,... 

The connection between integral and differential reactors and that between integral and differential methods of data evaluations are shown in Fig. 4.16 (after Froment, 1975). Data from integral reactors can be evaluated in the same way as data from differential reactors if the data are first numerically differentiated, or differentiated analytically or, more often, graphically. In cases where integral data are to be evaluated differentially, the following steps should be followed ... 

The two major methods used predominantly in the kinetic analysis of isothermal data on solid-catalyzed reactions conducted in plug-flow PBRs are the differential method and the method of initial rates. The integral method is less frequently used either when data are scattered or to avoid numerical or graphical differentiation. Linear and nonlinear regression techniques are widely used in conjunction with these major methods. 

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]. 

In the differential method of estimating the rate parameters, we first plot [A] as a function of time and differentiate it either graphically or by curve-fitting. The slope thus obtained gives the rate directly for a reaction with no volume change, based on which the kinetic parameters n and k can be determined as shown in Figure 7.3. 

Figure 4.11.16 shows the influence of the initial concentration of naphthalene on its conversion for different residence times and constant initial concentrations of hydrogen and steam. Although the reactor operates in integral mode, we can calculate the reaction order of naphthalene by the differential method, if we determine graphically the initial slopes of the X>j — TefF plot (Figure 4.11.16). Rearrangement of Eq. (4.11.33) for 0 (c>j = c>j,m) yields ...

In the differentiation method, not to be confused with a differential reaction system, solutions have traditionally been obtained using graphical or equivalent means. Consider the elementary irreversible reaction... 

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