Equal-area differentiation

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 derivative l-dCj /dt) is determined by calculating and plotting (-AC/ /At) as a function of lime. /, and then using the equal-area differentiation technique (Appendix A.2) to determine l dC /dt as a function of Ca- First, we calculate the ratio (-ACVA/) from the first two columns of Table E7-2.2 the result is written in the third column. 

Next we use Tabic E7-2.2 to plot the third column as a Function of the first column in Figure E7-1.1 [i.e., (-AC /Ar) versus /J. Using equal-area differentiation, the value of [-dCxldt) is read off the figure (represented by the arrows) then it is used to complete the fourth column of Table E7-2.2. 

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 ... 

Electrochemical kinetics of the oxygen differential aeration cell The differential aeration cell shown in Fig. 6.11 consists of identical iron electrodes placed in a two-compartment cell separated by a membrane. Two electrically connected iron electrodes of equal area are... 

This is the well-known equal area condition (Fig. 3). One can also prove that pressure p = p f (p) is constant in both coexisting phases. Indeed, differentiating Eq. (18) yields W p) = and the pressure difference between the two phases is computed as... 

II. This is the equivalent, in the ii, p-plane, of the equal-areas construction in the p, u-plane shown in Fig. 1.8, and is the once differentiated version of the double-tangent construction in Fig. 3.1. 

In applying quantum mechanics to real chemical problems, one is usually faced with a Schrodinger differential equation for which, to date, no one has found an analytical solution. This is equally true for electronic and nuclear-motion problems. It has therefore proven essential to develop and efficiently implement mathematical methods which can provide approximate solutions to such eigenvalue equations. Two methods are widely used in this context- the variational method and perturbation theory. These tools, whose use permeates virtually all areas of theoretical chemistry, are briefly outlined here, and the details of perturbation theory are amplified in Appendix D. 

The term P is a fictitious pressure because the cross-sectional area. A, is not equal to either the surface area of the particles or the actual contact areas. In actual cakes, there is a small area of contact. A, and the solid pressure may be defined as F /A. Taking differentials with respect to x in the interior of the cake, we obtain ... 

This is the fundamental differential equation. The reader who is acquainted with the rules for transforming the variables in a surface integral will observe that it has the geometrical interpretation that corresponding elements of area on the (v, p) and (s, T) diagrams are equal (cf. 43). 

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