Chord method

Figure A-16 Molar enthalpy of solution of NiS04-6H20(cr) at 25°C. Experimental data from Goldberg et al. [66GOL/RID] o dotted curve calculated according to Equation (1X.44) in [97ALL/BAN] dash dotted curve calculated according to the chord method, see Equations (8 - 2 - 19) and (8 - 2 — 20) of Hamed and Owen [58HAR/OWE] solid ciuwe calculated vdth an adjustable Debye-Hilckel parameter, A, and a term linear in /, ((A.27), 298.15 K) derived from dilution enthalpies [56LAN/M1E] x mean value of Aj /7° ((A.27), 298.15 K), dash line.

An alternative to the polynomial method is the chord method. The chord method is recommended for experiments that reach larger extents of reaction (Casado et al., 1986 Piscitelle, 1990 Waley, 1981). This method extracts... 

Example 4.3. Initial rate of scorodite dissolution using the polynomial and chord method... 

Harvey et al. (2006) measured the rate of scorodite (FeAs04 2H20) dissolution using a batch reactor. One of the experiments produced the concentration of arsenic versus time data given in Table 4.1. These data can be analyzed to find the initial rate using either the polynomial fit method or the chord method. 

The chord method is an improvement over the polynomial fit method but it requires slightly more manipulation of the data. First the slopes of the chords shown in Figure 4.5a are calculated and tabnlated. Then the chord slopes are graphed versus time as shown in in Figure 4.5b and these points are fit to a second-order polynomial. 

Either the polynomial method or the chord method can be used to find an initial rate from plug flow reactor data. The chord method is especially convenient in this case because the difference between the concentration in the effluent and feed streams (Am, molal) divided by the transit time for a slug of solution is a chord. The transit time is the reactor volume (V, L) times the porosity (p, no units) divided by the flow rate Q, L/sec). 

The chord method determines the initial rate, R (molal/sec). To convert this rate into the flux of species to or from the surface of the solid, R must be divided by the A/M ratio for the packed bed as described in Chapter 3. 

The calculation of escape probabilities is often most conveniently carried out by means of the chord method developed by Dirac. For this analysis we refer to Fig. 7.10, which shows an arbitrary convex volume V. As before, n is the inward directed normal at dA and s(Q) is the chord length from dA in the direction O. Let us assume now that the number of chords of length between s and s + ds m the solid angle y... 

Finally, for formulation D the flows in the tree branches can be computed sequentially assuming zero chord flows. This initialization procedure was used by Epp and Fowler (E2) who claimed that it led to fast convergence using the Newton-Raphson Method. 

The isotropic chord length distribution (CLD) is of limited practical value if soft matter with only short-range order is studied. Nevertheless, the related notions have been fruitful for the development of new methods for topology visualization from SAXS data. 

The second method is more elegant, because it only involves the numerical computation of moments (cf. Sect. 1.3) of the smeared CLDg2 (rn) followed by moment arithmetics [200], The first step is the computation of the Mellin transform102 of the analytical function gc (rn) which we have selected to describe the needle diameter shape. This is readily accomplished by Mathematica [205], Because the Mellin transform is just a generalized moment expansion, we retrieve for the moments of the normalized chord distribution of the unit-disc103... 

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. 

In the preceding example, the chords have been taken for equal intervals, because the curve changes slope only gradually and the data are given at integral temperatures at equal intervals. Under these circumstances, the method of numerical differentiation is actually preferable. In many cases, however, the intervals will not be equal nor will they occur at whole numbers. For the latter cases, the chord-area method of differentiation may be necessary, although considerable care is required to avoid numerical errors in calculations. 

Computational methods combined with a novel approach in the application of scattering physics were recently employed by Barbi et al. in a synchrotron SAXS study of the nanostructure of Nafion as a function of mechanical load. A new method of multidimensional chord-distribution function (CDF) analysis was used to visualize the multiphase nano-... 

RP-HPLC, using a citric acid-AcONa buffer-MeOH mobile phase and amperometric ELD. The method was used to demonstrate the increase of the H2O2 level in the extracellular liquid after impact injury on the spinal chord. The measured H2O2 levels were about 1... 

In-Plane Shear Properties. The basic lamina in-plane shear stiffness and strength is characterized using a unidirectional hoop-wound (90°) 0.1 -m nominal internal diameter tube that is loaded in torsion. The test method has been standardized under the ASTM D5448 test method for in-plane shear properties of unidirectional fiber-resin composite cylinders. D5448 provides the specimen and hardware geometry necessary to conduct the test. The lamina in-plane shear curve is typically very nonlinear [51]. The test yields the lamina s in-plane shear strength, t12, in-plane shear strain at failure, y12, and in-plane chord shear modulus, G12. 

The local gas holdup and bubble behavior were measured by a reflective optic fiber probe developed by Wang and co-workers [21,22]. It can be known whether the probe is im-merging in the gas. The rate of the time that probe immerg-ing in the gas and the total sample time is gas holdup. Gas velocity can be got by the time difference that one bubble touch two probes and the distance between two probes. Chord length can be obtained from one bubble velocity and the time that the probe stays in the bubble. Bubble size distribution is got from the probability density of the chord length based on some numerical method. The local liquid velocity in the riser was measured by a backward scattering LDA system (system 9100-8, model TSI). Details have been given by Lin et al. [23]. 

The conventional design is the default method of the ICPD tray computer programs. All conventional designs have straight, chord-type downcomer areas. (Compare Figs. 3.2 and 3.3.) Note that the nonconventional downcomer areas may have complex weir configurations as compared to the simple, straight weir runs of the conventional downcomers. 

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