Ratio , integral

Unlike. 208pb/204pb or 2 pb/204pb, which depend on Th/Pb and U/Pb, respectively, 208p 5 /206p]3 reflects the Th/U ratio integrated over the history of the Earth. The existence of global correlations between neodymium, strontium, and hafnium isotope ratios and, and the absence... 

It is worth while to note that the estimates based on the lunar cratering record, highly siderophile elements and water deuterium to protium (D/H) ratios integrate the late bombardment of Earth over time scales that are not necessarily... 

Ti and A1 partial pressures were determined in the temperature range of the measurements (cf. Fig. 1) over the different samples. Thermodynamic activities of Ti and A1 resulted according to the relation a, = p, / p (i = Ti, AI) by comparing the pressures over the alloy samples, p, with those of pure Ti(s) and Al(l), p- .The results obtained at the mean temperature of 1473 K are given in Fig. 2. The thermodynamic activity ofTi was, additionally, evaluated from the measurement results by the ion intensity ratio integration method. The results obtained by this method are also shown in Fig. 2. The two different evaluation methods agree in general very well. 

On the basis of the information given thus far, one would predict two resonance peaks in the NMR spectrum of 1,1,2-tiichloroethane, with an area ratio (integral ratio) of 2 1. In reality, the high-resolution NMR spectrum of this compound has five peaks a group of three peaks (called a triplet) at 5.77 ppm and a group of two peaks (called a doublet) at 3.95 ppm Figure 3.25 shows this spectrum. The methine (CH) resonance (5.77 ppm) is said to be split into a triplet, and the methylene resonance (3.95 ppm) is split into a doublet. The area under the three triplet peaks is 1, relative to an area of 2 under the two doublet peaks. 

Water-rock interaction can vary between two extremes. When the water/rock ratio is small and the 5 0 in the rock dominates the system it is the fluid composition which is changed, as happens in geothermal systems. On the other hand, when the water/rock ratio is large and the 5 0 of die water dominates, the 5 0 value of the rock is modified. Taylor (1974, 1977) derived mass balance equations from which the water/rock ratio may be calculated from 5 0 values. For a closed system, from which none of the water is lost, the water/rock (W/R) ratio, integrated over the lifetime of the hydrothermal system, is ... 

The theory of light scattering in polymolecular polymer solutions enables one to determine the derivative (d(A/io)/dw).ir using the Rayleigh ratio Integrating this... 

The base-case ratio integrates the best available information from the operating plant with design relationships to predict process changes. It is an inportant and powerful technique with wide application. The base-case ratio, X, is defined as the ratio of a new-case system characteristic, X2, to the base-case system characteristic, x. ... 

Comparing the principal extension ratio integrals, eq. 20a, with the stress integals, eq. 27, for the three cases i = j = x, y, or z, one obtains by inspection the three principal stresses... 

It must be emphasized that it is not worth expending any effort optimizing pressure, feed condition, or reflux ratio until the overall heat-integration picture has been established. These parameters very often change later in the design. 

No attempt should be made to optimize pressure, reflux ratio, or feed condition of distillation in the early stages of design. The optimal values almost certainly will change later once heat integration with the overall process is considered. 

Consider again the simple process shown in Fig. 4.4d in which FEED is reacted to PRODUCT. If the process usbs a distillation column as separator, there is a tradeofi" between refiux ratio and the number of plates if the feed and products to the distillation column are fixed, as discussed in Chap. 3 (Fig. 3.7). This, of course, assumes that the reboiler and/or condenser are not heat integrated. If the reboiler and/or condenser are heat integrated, the, tradeoff is quite different from that shown in Fig. 3.7, but we shall return to this point later in Chap. 14. The important thing to note for now is that if the reboiler and condenser are using external utilities, then the tradeoff between reflux ratio and the number of plates does not affect other operations in the flowsheet. It is a local tradeoff. 

Distillation capital costs. The classic optimization in distillation is to tradeoff capital cost of the column against energy cost for the distillation, as shown in Fig. 3.7. This wpuld be carried out with distillation columns operating on utilities and not integrated with the rest of the process. Typically, the optimal ratio of actual to minimum reflux ratio lies in the range 1.05 to 1.1. Practical considerations often prevent a ratio of less than 1.1 being used, as discussed in Chap. 3. 

If, however, the column is appropriately integrated, then the reflux ratio often can be increased without changing the overall energy... 

Thus the optimal reflux ratio for an appropriately integrated distillation column will be problem-specific and is likely to be quite different from that for a stand-alone column. 

A related measure of the intensity often used for electronic spectroscopy is the oscillator strengdi,/ This is a dimensionless ratio of the transition intensity to tliat expected for an electron bound by Hooke s law forces so as to be an isotropic hanuonic oscillator. It can be related either to the experimental integrated intensity or to the theoretical transition moment integral ... 

The integral cross section is therefore the effective area presented by each field particle B for scattering of the test particles A into all directions. The probability that the test particles are scattered into a given direction v ( /, ([)) is the ratio... 

Fig. 8.2 Simple Monte Carlo integration, (a) The shaded area under the irregular curve equals the ratio of the number of random points under the curve to the total number of points, multiplied by the area of the bounding area, (b) An estimate of tt can be obtained by generating random numbers within the square, v then equals the number of points within the circle divided by the total number of points within the square, multiplied by 4.

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