Key ratio plot

The feed point should give the most equal slopes on both sides of the feed stage in a key ratio plot (Fig. 2,20). Too high a feed causes excessive retrograde distillation below the feed, and too low a feed causes excessive retrograde distillation above it. In order to reach the optimum feed point, the feed stage should be moved from the sharp maximum (or minimum) toward the flat one (Fig. 2.20). 

Figure 2,20 Effect of feed stage on key ratio plot. Depropanizer example, feed composition same as Example 2.4. 20 theoretical stages, R/Rmin - 1.40.

Key ratio plots are primarily for identifying mislocated feed stages in multicomponent distillation. For this purpose, they are superior to x-y diagrams. Key ratio plots are easy to construct all it takes is calculating the key ratio in the liquid for a few stages in the feed region... 

Identify mislocated feed. For binary distillation the feed point should be where the q-line intersects the equilibrium curve. For multicomponent distillation this may or may not be the case. So for multicomponent distillation feed location, key ratio plots and d/b plots are preferred and discussed next. 

Pinching (either due to a mislocated feed, proximity to minimum reflux, or a tangent pinch) is commonly implicated by the above insensitivity. A McCabe-Thiele diagram and a key ratio plot can help identify the cause application of these techniques for this purpose is described elsewhere (193). Any scale-up of such efficiency data must be conservatively performed. 

An alert operating engineer must identify boiler tube leaks before it is too late. The capacity ratio plot (Fig. 5-3) is the key. A gradual increase in pressure drop is an early warning sign. When this happens, check for low steam production rates from the high-pressure boiler. Another tip-off is a low gas outlet temperature from this boiler. If both steam production and outlet temperature are low and pressure drop is high, shut down the plant. There is a tube leak. 

Figure 28 shows the key features of the humidity chart. The chart consists of the following four parameters plotted as ordinates against temperature on the abscissas (1) Humidity H, as pounds of water per pound of dry air, for air of various relative humidities (2) Specific volume, as cubic feet of dry air per pound of dry air (3) Saturated volume in units of cubic feet of saturated mixture per pound of dry air and (4) latent heat of vaporization (r) in units of Btu per pound of water vaporized. The chart also shows plotted hiunid heat (s) as abscissa versus the humidity (H) as ordinates, and adiabatic humidification curves (i.e., humidity versus temperature). Figure 28 represents mixtures of dry air and water vapor, whereby the total pressure of the mixture is taken as normal barometric. Defining the actual pressure of the water vapor in the mixture as p (in units of mm of mercury), the pressure of the dry air is simply 760 - p. The molal ratio of water vapor to air is p/(760-p), and hence the mass ratio is ... 

Figure 5. The ratio S (x)/S (x) plotted as junction of x for the phantom network at different cross-link functionalities, and for a fixed junction network. A is 2. Key Otfis3 n.f s 4 A, f is 6 V, f is 10 O is junctions fixed.

Figure 3. Modulus contributions from chemical cross-links (Cx, filled triangles) and from chain entangling (Gx, unfilled symbols) plotted against the extension ratio during cross-linking, A0, for 1,2-polybutadiene. Key O, GN, equibiaxial extension , G.v, pure shear A, Gx, simple extension Gx°, pseudo-equilibrium rubber plateau modulus for a polybutadiene with a similar microstructure. See Ref. 10.

The range of normal Mg analysis (Figure 6) is shown by the solid line. Allende anor-thites have normal Mg/ Mg ratios but greatly increased 2eMg/2iMg ratios, and plot far to the right of normal Mg samples, indicating a nuclear isotopic effect in Mg. Key , Bl inclusions and , B2 inclusions. 

Metal contents of the sulfide samples collected from the Key Anacon deposits have been classified using the Cu and Zn ratios (Fig. 5). The majority of the massive sulfide samples plot in the Zn-Pb-Cu type (of. Large 1992 Fig. 5) however, the semi-massive to disseminated sulfides in the footwall are enriched in Cu, Co, and Bi and lower in Zn and Pb, typical of stringer zone mineralization. These plots in the Cu type field and one sample contains enough zinc to be placed in the Zn-Cu group (Fig. 5). 

In the REMARK system, estimated effects (e.g., hazard ratios) with confidence intervals for the marker were recommended for univariant and key multivariable analyses, and a Kaplan-Meier plot was recommended to represent the effect of a tumor marker in a time-to-event outcome. The discussion section should interpret the results in the context of the pre-specified hypothesis and describe limitations of the study as well as implications for future research and clinical value. These guidelines were advocated for reporting of tumor marker studies in breast cancer research and treatment (63). 

Figure 3.4 Effect of polysaccharide on protein-stabilized emulsions. The diameter ratio, j43nuxtlire / J43protem is plotted against the molar ratio R (moles polysaccharide / moles protein). Here J43nuxtlire is average droplet diameter in fresh emulsion prepared with protein + polysaccharide, and d43pTOtQm is average diameter in emulsion stabilized by protein alone. Key , , legumin + dextmn (48 kDa) or legumin + dextran (500 kDa), respectively (0.5 w/v % protein, 10 vol% oil, pH = 8.0, /= 0.1 M) (Dickinson and Semenova, 1992) O, , asi-casein + pectinate and p-casein + pectinate at pH = 7.0, / = 0.01 M (2.0 w/v % protein, 40 vol% oil), respectively , p-casein + pectinate at pH = 5.5, / = 0.01 M (2.0 w/v % protein, 40 vol% oil) (Semenova et al, 1999). Reproduced from Semenova (2007) with permission.

Figure 7.42. Comparison between (A) an idealized plot of variation in 8180 and 813C for carbonates subjected to vadose and phreatic meteoric diagenesis (after Lohmann, 1988) with (B) the meteoric alteration trend observed for the Key Largo Limestone, Florida, U.S.A. (after Martin et al., 1986). The critical trend in isotopic composition is termed the meteoric calcite line. This trend may be modified at the water recharge surface where evaporation is an important process, caliche is formed and the diagenetic phases are depleted in 13C derived from soil-gas CO2. Another modification can occur distally to the recharge area where precipitating carbonate cements may have isotopic ratios nearly equivalent to dissolving phases.

Stupin and Lockhart (27) also noted that as reflux is lowered from total to minimum, the separation of nonkeys first worsens (curve 2, Fig, 2,21), then improves (curve 3, Fig. 2,21), The intermediate keys follow the converse pattern, At a reflux ratio of about 1,2 to 1,5 times the minimum, component distribution resembles that of the total reflux component distribution. Detailed discussion is elsewhere (7,27). Figure 2,16 demonstrates that light nonkey6 are fractionated out in the stripping section and heavy nonkeys in the rectifying section. The d/b plot depicts this behavior (Sec. 2,4,2). 

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