Equal-Area Graphical Differentiation

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 ... 

A.l Useful Integrals in Reactor Design 921 A.2 Equal-Area Graphical Differentiation 922 A.3 Solutions to Differential Equations 924 A.4 Numerical Evaluation of Integrals 924 A.5 Software Packages 926 ... 

Graphical Method. With this method disparities in the data are easily seen. As explained in Appendix A.2, the graphical method involves plotting AC, /At as a fimction of t and then using equal-area differentiation to obtain dCJdt. An illustrative example is also given in Appendix A.2. 

Graphical technique-equal area differentiation. (Use O s to represent these points on any graphs you make.)... 

There are many ways of differentiating numerical and graphical data. We shall confine our discussions to the technique of equal-area differentiation. In the procedure delineated below we want to find the derivative of y with respect to x. 

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 11-equation system (in the case of a single electrolyte) is solved numerically [26,23,17] to fit experimental curves oq = f(pH) and (, = f(pH) from experimental data on pH, electrolyte concentration, total density of surface sites, total area A and temperature. The other necessary parameters are the differential capacitances, which are considered to be constant in the various zones of the interface, and the equilibria constants (7,4), (7.5), (7.46) and (7.47). The capacitance C2 is considered equal to 0.2 Fm7 and the adjustable parameter is Ci. The equilibria constants are determined graphically or numerically (see below) from the experimentally obtained A s and <7o = f(pH) curves obtained for several electrolyte concentrations. 

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