Numeral distribution curve

To establish the validity of the numerical scalar technique for RTD analysis, the normalized exit age distribution curve of both counter-current (Figure 1 (a-b)) and cocurrent (Figure 1 (c-d)) flow modes were compared. Table 1 shows that a good agreement was obtained between CFD simulation and experimental data. 

Plotting the number of compounds as a function of percent inhibition at 10 M results in a distribution curve with a slope break point that corresponds to a discontinuity in the first derivative ( DFD ). A change in the distribution density suggests that there are different populations within the percent inhibition values. The DFD point is numerically calculated from the distribution curve and gives an assessment of the suitable threshold for each assay. Most often it is located in the vicinity of 30% inhibition at 10 M (unpublished work). 

FIGURE 18 Countercurrent distribution of enolase from baker s yeast ( ) experimental data (O) theoretical (—) sum of theoretical curves. (From G. Blomquist and S. Wold, Numerical resolution of CCD [counter current distribution] curves. Acta Chem. Scand. B28, 56-60, 1974.)... 

Figure 12. PDF of C (T t ) for different r - t /T and a = 0.8. Diamonds are numerical simulations. Curves are analytical results without fitting For r = 0, Eq. (32) is used (full line) for r = 0.01, 0.1 and 0.5 Eq (35) is used (dashed) and for r = 0.9 and 0.99, Eq. (38) is used (full lines). If compared with the cases a = 0.3 and 0.5, the distribution function exhibits a weaker nonergodic behavior, namely for r = 0 the distribution function peaks on the ensemble average value of 1/2.

As can be seen from the differential curves of the cell volume fraction distribution (Curves 1 in Fig. 2a, b), the microcells occupy a relatively small volume of the foam 5% for y = 40 kg/m and 11% for y = 500 kg/m. They are, however, the most numerous group of cells (Curves 3,... 

In thermoporometry experiments the pore radius is deduced from the measurement of the solidification temperature and the volume of these pores is calculated from the energy involved during the phase transition. The pore radius distribution and the pore surface are then calculated. The pore texture can be described from numerical values (mean pore radius, total pore volume or surface, etc...) or by curves. For example, curves of figure 1 are the cumulative pore volume vs pore radius while curves of figure 2 are the pore radius distributions. Texture modifications are conveniently depicted by the pore size distribution curves. 

The residence time distribution curve (RTD) can be inscribed by its statistical moments, of which the centroid of distribution T and spread of distribution a are the most important numerical values. Thus, for a C curve, the zeroth moment is... 

Comparison of experimental pass duration distribution curves, with the curves calculated using Equations 2.104 and 2.106, results in obtaining the numerical values Bo and n and therefore, an opportunity for the quantitative estimation of the flow structure deviation in reaction zones with different geometry from an ideal model. 

The result of the particle size analysis (screen or sieve analysis) can be represented in a numerical table and/or as a particle size cumulative distribution curve... 

Assuming, that the numerical..vaLu.es of.K andjk arexoughly the same, the importance of the slope of the equilibrium-curve chords can readily be demonstrated. If m is small (equilibrium-distribution curve, veryJJat), so that at equilibrium only a.small,concentration,of.. A in ihe.gas will provide a-very-large concentration in the liquid (solute. A is very soluble in the-liquid), the term wVK. of Eq. (5.7) becomes minor, the majoniesistance is represented by 1/i, and it is said that the rate of mass. transfer is gas phase-controlled. In the extreme, this becomes... 

With time and effort, the radial distribution curves (Section 9.4) for atom-atom contacts in the liquid could be deconvoluted from experimental diffraction patterns taken on liquid samples. The task is not an easy one, as is evident if one just compares the large amount of information contained in the numerous and sharply defined peaks from a crystal with the scarce and ambiguous information in a pattern from a liquid. 

The most practical representation of the dispersity of molar masses is shown in Figure 3.7, which consists in plotting either the number (Ni) of moles of species corresponding to a degree of polymerization (Xi) or their mass (A jAf, ), versus either the degree of polymerization (Xi) or the corresponding molar mass (Af,). This representation of Ni =f(Xi or Mi) affords a curve that gives information about the numeral distribution of chains. 

Hausenhas solved this partial differential equation for the case of steady-state cyclic operation and provided numerous sets of gas-temperature distribution curves as a function of time. (These curves show that the temperature distribution becomes essentially linear when the distance is greater than x = 0.25L.) By using this gas-temperature distribution, a relation for the packing material temperature can then be developed utilizing Eq. (5.67). 

The f-curve and its associated t-plot were originally devised as a means of allowing for the thickness of the adsorbed layer on the walls of the pores when calculating pore size distribution from the (Type IV) isotherm (Chapter 3). For the purpose of testing for conformity to the standard isotherm, however, a knowledge of the numerical thickness is irrelevant since the object is merely to compare the shape of the isotherm under test with that of the standard isotherm, it is not necessary to involve the number of molecular layers n/fi or even the monolayer capacity itself. 

Theoretical efforts a step beyond simply fitting standard statistical curves to fragment size distribution data have involved applications of geometric statistical concepts, i.e., the random partitioning of lines, areas, or volumes into the most probable distribution of sizes. The one-dimensional problem is reasonably straightforward and has been discussed by numerous authors... 

When an experimental value is obtained numerous times, the individual values will symmetrically cluster around the mean value with a scatter that depends on the number of replications made. If a very large number of replications are made (i.e., >2,000), the distribution of the values will take on the form of a Gaussian curve. It is useful to examine some of the features of this curve since it forms the basis of a large portion of the statistical tools used in this chapter. The Gaussian curve for a particular population of N values (denoted x ) will be centered along the abscissal axis on the mean value where the mean (r ) is given by... 

Notice that those distribution functions that satisfy Eq. (4-179) still constitute a convex set, so that optimization of the E,R curve is still straightforward by numerical methods. It is to be observed that the choice of an F(x) satisfying a constraint such as Eq. (4-179) defines an ensemble of codes the individual codes in the ensemble will not necessarily satisfy the constraint. This is unimportant practically since each digit of each code word is chosen independently over the ensemble thus it is most unlikely that the average power of a code will differ drastically from the average power of the ensemble. It is possible to combine the central limit theorem and the techniques used in the last two paragraphs of Section 4.7 to show that a code exists for which each code word satisfies... 

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