Upper-bound curve

Figure 12.9 Upper bound curve and spectrum of solutions...

In order to compare the performance for polymeric and carbon membranes, Figure 15.13 shows a CO2/CH4 trade-offline for P84 and Matrimid precursors and their carbon membranes as reported by Tin et al It is clear that carbon membranes possess excellent permeation properties, where both of the permeability and ideal selectivity access the Robeson upper-bound curve. Moreover, some researchers have also investigated the influence of temperature on the gas permeability.They concluded that the gas permeability values increased with the increase of temperature due to the activated process for the CMS membranes. They also found that the apparent activation energies for CO2 calculated from the Arrhenius equation Pe = Peo Qxp(-EJRT)) was much smaller than the other gas species of O2, N2 and CH4, thereby indicating that CO2 has much higher permeability. 

Each blend has properties which fall near Robeson s upper bound (see Figure 3) (25). Variation in the content of DIA allows one to move nearly parallel to the upper bound curve and control the permeation/separation properties. 

Drawing boundaries around all reported landslide locations in historical earthquakes and calculating the areas enclosed yields a plot of area versus earthquake magnitude (Fig. 5) a well-defined upper bound curve represents the maximum area that can be affected for a given magnitude (Keefer 1984). Average area affected by landslides as a function of earthquake magnitude... 

The maximum distance of the three categories of landslides from the earthquake epicenter and from the closest point on the fault-rupture surface relates closely to earthquake magnitude (Fig. 6). Upper bound curves are well defined and are constrained to pass through the minimum threshold magnitudes shown in Table 2 as distance approaches zero. Although the upper bounds shown have been exceeded a few times in subsequent earthquakes, they remain fairly reliable indicators of the maximum possible distances at which the three classes of landslides could be triggered in earthquakes of various magnitudes. 

Figure 3. Modified PPO and the upper bound curves, a) O2 permeability versus O2/N2 permeability ratio and b) CO2 permeability versus CO2/CH4 permeability ratio - PPO,...

Example 10 Logistics Curve We shall derive the logistics curve representing the ciimiilarive-freqiiency distrihiitions of the normal distrihiition curve defined by Eqs. (9-72) and (9-73). In this case, y varies between a cumulative probability of zero and unity as z varies from — to -l-oo. Since the upper bound is unity, c = 1. From Table 9-10 the area under the right-hand side of the curve between = 0 and z = z may be read. Since the frequency curve is symmetrical about the mean, this is also the area between = 0 and z=z. Hence, the area under the frequency curve, which represents the cumulative probability, is 0.50000 at = 0 and the 80 percentile, for which the area is 0.80000, corresponds to the value = 0.842. We substitute these values into Eqs. (9-92) through (9-94) to give... 

The lower bound confidence limit is the probability that a parameter, x, is less than some value x . The upper bound confidence limit is the probability that a parameter, x, is greater than some value x . Figure 2.5-1 shows that confidence may be obtained from the discrete curve by simply adding the probabilities below or above x, for the lower or upper bound confidence respectively. If the curve is continuous it must be integrated above or below x. These results are normalized, by dividing the partial integral or partial sum by the full integral of the curve or complete sum. 

The experimental and theoretical results for E., are shown in Figure 3-41 for a resin content by weight ranging from 10% to 100%. Because E. is not a function of C, only k was varied — two values were chosen k = 1 and k =. 9. Some experimental results in Figure 3-41 lie above the curve for k=1 (i.e., above the upper bound ) some results lie below k =. 9. However, most results lie between k =. 9 and k = 1 with k =. 9 being a conservative estimate of the behavior. The actual specimens were handmade, so the resin content might not be precise, and fiber misalignment is not unexpected. Thus, the results above the upper bound are not unusual nor is the basic fact of variation in E. ... 

The experimental results for 6 2 ai" also shown in Figure 3-44, along with theoretical results from Equation (3.67) for C = 0,. 2,. 4, and 1. As with the previous moduli, the experimental data are bounded by curves for C = 0 and C =. 4. The upper (parallel-connected phases) and lower (series-connected phases) bounds due to Paul (see Section 3.3) are shown to demonstrate the accuracy of the bounds in the present case where E is much greater than E. The lower bound results of Hashin and Rosen [3-8] correspond to C = 0, but their upper bound is below some of the experimental data in Figure 3-44. 

A typical field test involves several steps (a) transporting the mobile unit to the site (b) instrument warmup (c) system check out, consisting of mobile unit measurements of distilled water and a 1-ppm stock phenol solution and (d) in situ measurements of the well water, repeated three times for statistical analysis. Signal levels recorded at a field site may be reported as equivalents of phenol (or other calibrant) using the calibration curves. Therefore, this method allows us to report the upper bounds of pollution levels. 

Estimated Upper-Bound Human Cancer Risk Levels This is the range associated with the upper-bound for lifetime cancer risk of 1 in 10,000 to 1 in 10,000,000. These risk levels are derived from the EPA s Human Health Assessment Group s upper-bound estimates of the slope of the cancer dose response curve at low dose levels (qi ). 

Qi — The upper-bound estimate of the low-dose slope of the dose-response curve as determined by the multistage procedure. The q, can be used to calculate an estimate of carcinogenic potency, the incremental excess cancer risk per unit of exposure (usually pg/L for water, mg/kg/day for food, and pg/m for air). 

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