Cross plots

The range of the nomographs can be extrapolated or extended for a specific vessel volume by cross-plotting as shown in Figure 7-64, limiting to a constant Pjed for each chart at varying Do not extrapolate below Pjt j of 0.05 bar ga, nor below Pred of 0.1 bar ga. Pr d should not be extrapolated above 2.0 bar ga Pjtai can be extrapolated but it must always be less than P gd by at least 0.05 bar [27]. 

By cross-plotting and interpolation, for 8,000 cfm at 20.7 psia discharge, and 80°F inlet temperature ... 

The first step in the analysis was to assess the effect of mass velocity for the 1000 psia data. By plotting and cross-plotting and finally by computer optimization it was found that 15 of the different geometries conformed very... 

If we let and t2 represent the times corresponding to reaction progress variables and <5J, respectively, the time ratio t2/tl for fixed values of <5 and <5 will depend only on the ratio of rate constants k. One may readily prepare a table or graph of <5 versus k t for fixed k and then cross-plot or cross-tabulate the data to obtain the relation between k and ktt at a fixed value of <5. Table 5.1 is of this type. At specified values of <5 and S one may compute the difference log(fe1t)2 — log f) which is identical with log t2 — log tj. One then enters the table using experimental values of t2 and tx and reads off the value of k = k2/kv One application of this time-ratio method is given in Illustration 5.5. 

In addition to the temporal correlation coefficient, the spatial correlation coefficient was calculated approximately for fixed values of time. Except for one of the mathematical models, all techniques showed a better temporal correlation than spatial correlation. The two correlation coefficients are cross plotted in Figure 5-6. Nappo stressed that correlation coefficients express fidelity in predicting tends, rather than accuracy in absolute concentration predictions. Another measure is used for assessing accuracy in predicting concentrations the ratio of predicted to observed concentration. Nappo averaged this ratio over space and over time and extracted the standard deviation of the data sample for each. The standard deviation expresses consistency of accuracy for each model. For example, a model might have a predicted observed ratio near unity,... 

PLASTOFROST PLASTIC RANGE ( C) Figure 7. Cross plot of dilatometric and Plasto ost plastic ranges. 

The models presented in the previous section are evaluated by comparing their predictions to actual plant data and also to the predictions of correlations proposed by previous researchers. The models give predictions that are close to the actual data. The average relative percent error between the predicted values and the actual values ranges from 0 to 10% for the proposed models. Figure 2.5 represents cross plots of the predicted versus the actual product yields and properties. As can be seen, all data points lie close to the 45° line, indicating a good fit. 

By cross-plotting the data in Figs. 13 and 14, the relation between the specific surface area and the porosity of the carbon rods after reaction at different temperatures can be presented, as in Fig. 15. It is seen that the surface area developed in the rods is not only a function of the porosity developed but also a function of the reaction temperature. The development... 

Using cross plots of the depth-dose curve and 8/So vs. dose (from the nomogram) one can calculate the concentration of contaminants. 

The culprit is the phenomenon of retrograde condensation, which wa6 previously discussed In connection with hydrocarbon dewpoints. This can best be understood by looking at a graph of equilibrium ratios, commonly called K-values, as shown in Figure 2. We have cross-plotted a limited number of curves, to avoid confusion while Illustrating our point. 

Fig. 5. Various parameters of accessibility, twist, and bend plotted vs. sequence number. Part 1 (a) Solvent-accessible area of side chains, (b) Fractional accessibility (referred to full sphere) of backbone carbonyl oxygen and peptide nitrogen. The separate plot for values less than 1% is meant to show that no accessibility was detected for many atoms. The actual nonzero values are not to be taken too literally. Part 2 (c) Backbone angles as normally defined, (d) Angles between sequentially adjacent carbonyl vectors in the backbone plotted between the sequence numbers of the two residues involved. Part 3 (e) Distance in A between the tips, T, of adjacent residues as defined in the text, (f) Distances in A between peptide center, M, and the third sequential peptide center (open circles), and between carbon a and the sixth sequential a-carbon (crosses) plotted opposite the central carbon atom in each case, (g) Angles between lines joining the centers of successive peptide bonds plotted between the residues defining the central bond, (h) Angles between lines joining successive a carbons plotted opposite the central carbon, (Note that the accessibilities were calculated with coordinate set 4 and the other parameters with set 6 see text.)...

The performance ratio or heat economy is a result of the selection of design variables previously discussed, and is not a variable as such. Lines of constant capital cost per daily gallon of capacity are also included in Figure 3. Capital costs have been based on plant capacities in a range of 25,000,000 to 60,000,000 gallons a day and a velocity in the evaporator tubes of 5 feet per second. These lines of constant capital cost per daily gallon are a result of cross plotting the results obtained in the optimization study. 

Results are shown graphically in Figure 4 for a brine temperature of 220°F., condenser tube velocity of 5 feet per second, blowdown temperature of 90°F., and brine concentration of twice sea water. As can be seen, a minimum water cost for these conditions is obtained with a 50-stage plant operating with a terminal temperature difference of about 4°F. Similar calculations were made for a blowdown concentration of 1.5 times sea water and for a once-through system. By cross plotting, it was then possible to determine the optimum blowdown salt concentration for the plant. It was about 1.7 times sea water. However, the curve is almost flat in the range of 1.5 to 2.0 times sea water. 

The sum of all these costs is shown as the total cost curve. The minimum is approximately 4°F. TTD. A cross plot of the data indicated that a concentration of 1.7 times sea water had a slight economic advantage over either 1.5 or 2.0 times sea... 

Figure 10. Cross plot of torque and temperature from rheocord.

Fig. 74 Cross-plot of yield stress for BPA-PC against volume changes obtained by varying temperature or pressure (From [54])...

Fig. 5. Cranked Shell Model frequency for the proton 1m crossing minus that for the neutron an crossing plotted as a function of neutron number for the Pt isotopes.

Preliminary to a more detailed theoretical treatment of the problem, it may be of interest to express the above observations from another point of view. One characteristic of the surface behavior of a solid is the adsorption isotherm. If it were possible within a few minutes to obtain a sufficient number of points to construct an isotherm, it would be most interesting to follow the changes in the adsorption isotherm of the diamond surface as it decays to its normal surface behavior. Figure 7 indicates that the decay is almost independent of the presence of argon. One possible procedure is to introduce different amounts of argon to the evacuated sample and by making a cross plot of the amounts adsorbed at constant time of cooling construct the desired isotherms. 

An article by Karam (1) gives typical data to illustrate the difference between shear rate and shear stress. Table II is extracted from cross plots of their data, showing the shear rate required with different continuous phase viscosities and one dispersed phase viscosity to break up a second fluid of the same size droplet. This shows that the shear stress in grams per centimeter squared is the basic parameter and the viscosity and shear rate are inversely proportional to give the required shear stress. 

A fit to the experimental results may require a one-plus rate equation with three or more terms in the denominator (e.g., see eqns 7.5). If so, the coefficients can be determined by linear regression or, long-hand, by cross-plotting. Say, the one-plus equation is... 

In the F-O cross-plot, different experimental data sets satisfied the relationship F>(3- e.g.. Fig. 7, Fig. 8. The cementation factor (/w) and the structural parameters (a) were interpreted following two methods i) The Archie method assumes a = 1, and m is determined for each sample from a logF-log curve using Eq. (1), e.g.. Table 1, Fig. 8 ii) The best-fit method is applied for each textural class in order to obtain a and m e.g.. Fig. 7. The values of m obtained by using the first method, range from 1.62 to 2.48. 

Using Figure 48 make a cross plot of the equilibrium constant data for n-hexane. Plot log K versus log P at 50° F, 150° F, and 250° F. Plot log K versus log P at the same temperature assuming ideal-solution behavior. 

On a psychrometric chart (or humidity chart) several properties of a gas-vapor mixture are cross-plotted, providing a concise compilation of a large quantity of physical property data. The most common of these charts—that for the air-water system at 1 atm—is used extensively in the analysis of humidification, drying, and air-conditioning processes. 

By repeated plotting and cross-plotting of the values from Ref. [4] as functions of pressure, temperature, reduced pressure and reduced temperature, the author has been able to prepare a table of Z as a function of Pr (0 < < 0-15) and... 

Figure 5 A cross-plot of Os/ Os versus Sr/ Sr illustrating the isotopic ratios associated with various oceanic inputs. Solid black arrows schematically illustrate the temporal evolution of seawater during the Cenozoic. Data sources (see Table 1 also) are (-) Cenozoic seawater (o) Loess deposits, Peucker-Ehrenbrink and Jahn (2001) (-I-) Indus paleosols, Chesley et al. (2000) and (x) Ganges paleosols, Chesley et al. (2000).

The curves obtained at various concentrations are almost parallel. This observation means that the rate of change of voltage with respect to temperature is independent of the solids concentration. By cross-plotting the results shown in Figure 24, a set of calibration curves can be prepared with temperature as a parameter. When such curves were prepared, they indicated that the value of e at C = 0, obtained by extrapolation, was lower than the corresponding value obtained for tap water at the same temperature. A review of the procedure of this experiment indicated that the only possible reason for this difference was the fact that a small amount of a wetting agent (an anionic surfactant) was added with the solids to increase the wettability of the polystyrene particles. 

Direct observations of sub-surface pressure allow a calibration to be made between the SGR and seal capacity. Ideally, an in situ measurement of the pore-pressure in the reservoir and that inside the fault zone would allow the capillary entry pressure of the fault to be calculated. However, fault-zone pressures are rarely available. Instead, the pressure difference between the two walls of the fault is a more general parameter that can be derived from pressure measurements in pairs of wells across the fault. Fig. 7a shows one such calibration, based on the Nun River dataset of Bouvier et al. (1989). From their strike projections of Fault K , values of SGR have been calculated on a dense grid across the fault surface. On the same grid, minimum across-fault pressure differences have also been derived, using the proven distribution of hydrocarbons in the footwall sands to calculate buoyancy pressures. Fig. 7a shows a cross-plot of these two parameters for the areas of sand-sand contact at the fault surface. The dashed line indicates the inferred relationship between SGR and seal capacity. At SGR < 20%, no fault-sealed hydrocarbons are observed the shale content of the slipped interval... 

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